Hybrid Shifted Gegenbauer Integral–Pseudospectral Method for Solving Time-Fractional Benjamin–Bona–Mahony–Burgers Equation

Abstract

This paper introduces a novel hybrid shifted Gegenbauer integral–pseudospectral (HSG-IPS) method to solve the time-fractional Benjamin–Bona–Mahony–Burgers (FBBMB) equation with high accuracy. The approach transforms the equation into a form with only a first-order derivative, which is approximated using a stable shifted Gegenbauer differentiation matrix (SGDM), while other terms are computed with precise quadrature rules. By integrating advanced techniques such as the shifted Gegenbauer pseudospectral method (SGPS), fractional derivative and integral approximations, and barycentric integration matrices, the HSG-IPS method achieves spectral accuracy. Numerical results show it reduces average absolute errors (AAEs) by up to 99.99% compared to methods like Crank–Nicolson linearized difference scheme (CNLDS) and finite integration method using Chebyshev polynomial (FIM-CBS), with computational times as low as 0.04–0.05 s. The method’s stability is improved by avoiding ill-conditioned high-order derivative approximations, and its efficiency is boosted by precomputed matrices and Kronecker product structures. Robust across various fractional orders, the HSG-IPS method offers a powerful tool for modeling wave propagation and nonlinear phenomena in fractional calculus applications.

1. Introduction

Fractional calculus provides a robust framework for modeling complex systems with memory and nonlocal interactions, with applications in fields such as viscoelasticity, anomalous diffusion, and control theory [1,2,3]. Unlike classical integer-order models, which assume local and instantaneous interactions, fractional-order models capture nonlocality and history dependence, offering a more accurate representation of processes such as dielectric polarization, electrochemical reactions, and subdiffusion within disordered media. These models often achieve comparable or superior accuracy with fewer parameters, thereby improving efficiency [4]. The strength of fractional derivatives lies in their ability to describe hereditary properties and long-range temporal correlations, making them ideal for biological systems, control systems, and viscoelastic materials. Notably, time-fractional PDEs have gained prominence for modeling complex physical phenomena with memory and hereditary traits. However, directly approximating such PDEs, like the FBBMB equation, using PS methods can lead to significant inaccuracies without sophisticated preconditioning techniques. This is primarily due to the ill-conditioning of high-order PS differentiation matrices, particularly when computing the third-order mixed derivative $u_{xxt}$ present in the FBBMB equation. These matrices exhibit condition numbers that scale poorly, as $\mathcal{O}\left(n^{2k}\right)$ for the k-th derivative with n grid points [5], leading to numerical instability and amplified round-off errors for higher-order derivatives like the third-order ($\mathcal{O}\left(n^6\right)$). This challenge necessitates advanced techniques, such as the transformation and integration-based approaches developed in this work, to ensure numerical stability and spectral accuracy.

The Benjamin–Bona–Mahony–Burgers equation is a nonlinear PDE that was first introduced as an alternative to the Korteweg-de Vries equation for modeling long waves in shallow water [6]. Proposed by Benjamin, Bona, and Mahony in 1972, it replaces the third-order spatial derivative in the KdV equation with a mixed space-time derivative, $u_{xxt}$, to improve numerical stability and better capture dispersive effects in wave propagation. The inclusion of a Burgers-type nonlinear term, $u{u}_x$, and a dissipative term accounts for viscosity and nonlinearity, making it suitable for modeling phenomena like shallow water waves, acoustic waves, and hydromagnetic waves [7]. The time-fractional variant, the FBBMB, incorporates the CFD that extends the classical equation to describe memory-dependent wave dynamics, thereby improving its applicability to complex systems with nonlocal interactions, such as viscoelastic materials and anomalous diffusion [8,9]. This fractional framework has spurred recent interest in developing robust numerical methods to handle its computational challenges, as addressed in this work.

Recent advances in numerical methods for fractional PDEs have demonstrated improved accuracy and efficiency. For example, Luo et al. [10] developed a robust second-order ADI Galerkin technique for three-dimensional nonlocal heat models with weakly singular kernels, combining Crank–Nicolson temporal discretization with finite element spatial approximation. For epidemiological applications, Maayaha et al. [11] investigated a Caputo–Fabrizio fractional SIR model for dengue fever and obtained a numerical solution using the Laplace Optimized Decomposition method, which yields a rapidly convergent series solution validated against the classical fourth-order Runge–Kutta method. In a separate line of research, significant effort has been devoted to the numerical treatment of the BBMB equation and its fractional variants. These include finite difference schemes such as the CNLDS [12], fourth-order finite difference schemes [13], and spectral methods, including Chebyshev–Legendre spectral techniques [14]. Meshless methods like radial basis functions [15] and kernel smoothing techniques [16] provide flexibility for irregular domains. Finite element methods include Galerkin formulations with cubic B-splines [17] and nonconforming elements [18]. Additionally, predictor–corrector schemes [19] and spline collocation methods [20] have been employed to handle fractional derivatives and nonlinearities efficiently. Despite these advancements, existing numerical methods for the FBBMB equation, such as CNLDS [12], FIM-CBS [21], and other finite difference or spectral approaches, often face challenges in achieving high accuracy without sacrificing computational efficiency. These methods typically struggle with the ill-conditioned approximation of the third-order mixed derivative $u_{xxt}$, leading to numerical instability, or require fine discretizations that increase computational cost. Additionally, many approaches lack robust handling of fractional derivatives across a wide range of $\alpha$, limiting their adaptability to varying memory effects.

The HSG-IPS method introduced in this paper presents a novel high-order approach by combining the strengths of spectral accuracy, efficient quadrature, and robust handling of fractional operators. Building upon the theoretical framework of fractional calculus and SG polynomials, we now examine the computational foundations that enable our high-order numerical approach. The foundation for modern SG integration matrices was established by Elgindy [22], who introduced novel SG operational matrices of integration (aka SGIMs) for solving second-order hyperbolic telegraph equations. This work focused on deriving error formulas for associated numerical quadratures and developing optimization methods to minimize quadrature error, establishing the theoretical groundwork for these matrices in solving PDEs. Building upon this foundation, Elgindy [23] significantly advanced the practical implementation by introducing stable barycentric representation of Lagrange interpolating polynomials and explicit barycentric weights for Gegenbauer–Gauss points. This included the development of a specialized SGIRV that efficiently handles boundary conditions through precise integral approximations at domain endpoints while maintaining the method’s spectral accuracy. This approach reduced computational cost and improved numerical stability while maintaining high accuracy, marking a crucial step towards more robust and efficient integration matrices. The methodology was further refined by Elgindy [24], who explicitly derived barycentric weights for SG polynomials and successfully applied the barycentric SG integral PS method to optimal control problems governed by parabolic distributed parameter systems. This work achieved exponential convergence rates in both spatial and temporal directions, demonstrating the power and effectiveness of the refined SG integration matrices in demanding computational domains.

The development of SG differentiation matrices has similarly advanced the numerical solution of differential equations, particularly for problems requiring high-order accuracy. Elgindy and Dahy [25] used SG differentiation matrices within a high-order Cole–Hopf barycentric Gegenbauer integral PS method for solving the viscous Burgers’ equation, thereby achieving spectral accuracy. Their approach utilized the Cole–Hopf transformation to convert the nonlinear Burgers’ equation into a linear heat equation, which was then discretized using Gegenbauer–Gauss points and associated differentiation matrices. This method demonstrated fast convergence and high accuracy, even for small viscosity parameters where traditional methods often struggle. The SG differentiation matrices were crucial in delivering accurate derivative approximations. The advancements in SG integration and differentiation matrices provide the foundation for developing high-order PS methods for time-fractional PDEs, where the well-conditioning of integral operators and high-accuracy of low-order differential operators are essential for achieving reliable numerical solutions.

Building on this, Elgindy [26] recently introduced an SGPS method for approximating CFDs of an arbitrary positive order. This method employs a strategic variable transformation to express the CFD as a scaled integral of the m-th derivative of the Lagrange interpolating polynomial, thereby mitigating singularities and improving numerical stability. Key innovations include the use of SG polynomials to link m-th derivatives with lower-degree polynomials for precise integration via SG quadratures. The developed C-FSGIM enables efficient, pre-computable CFD computation through matrix–vector multiplications. Unlike Chebyshev or wavelet-based approaches, the SGPS method offers tunable clustering and employs SG quadratures in barycentric forms for optimal accuracy. It also demonstrates exponential convergence and achieves superior accuracy in solving Caputo fractional two-point boundary value problems of the Bagley–Torvik type. This method unifies interpolation and integration within a single SG polynomial framework.

Furthermore, Elgindy [27] recently introduced a GBFA method for high-precision approximation of the RLFI. By using precomputable RL-FSGIMs, this method achieves super-exponential convergence for smooth functions and delivers near machine-precision accuracy with minimal computational cost. Tunable SG parameters enable flexible optimization across diverse problems, while rigorous error analysis confirms rapid error decay under optimal settings. Numerical experiments demonstrated that the GBFA method outperforms existing techniques by up to two orders of magnitude in accuracy, with superior efficiency for varying fractional orders. Its adaptability and precision make the GBFA method a transformative tool for fractional calculus, making it ideal for modeling complex systems with memory and nonlocal behavior.

This work focuses on the FBBMB equation, which incorporates wave propagation, dispersion, and nonlinearity within a fractional calculus framework. Our main contribution is the development of the HSG-IPS method—a high-order numerical approach that synergistically combines several advanced techniques: the SGPS method for CFD approximation, the GBFA for RLFI approximation, the SGDM for derivative computations, and the SGIM and SGIRV for integral approximations. The method’s key innovation lies in transforming the original FBBMB equation into a fractional partial-integro differential form containing only a first-order derivative, which is significantly more numerically stable to approximate using a first-order SGDM compared with the third-order mixed derivative in the original formulation. This transformation allows stable computation using first-order SGDMs while computing all other terms through high-accuracy quadratures.

This work introduces several novel contributions to the numerical solution of the FBBMB equation. Firstly, the HSG-IPS method achieves spectral accuracy and exponential convergence for smooth solutions by integrating SGPS, GBFA, SGDM, SGIM, and SGIRV within a unified framework, using stable barycentric representations and precomputed operational matrices. Secondly, the innovative transformation of the FBBMB equation into a fractional partial-integro differential form eliminates the need for direct approximation of the ill-conditioned third-order mixed derivative $u_{xxt}$, replacing it with a stable first-order SGDM approximation and precise quadrature rules for integral terms. This approach significantly improves numerical stability while maintaining high accuracy. Thirdly, the method demonstrates superior performance over existing techniques like CNLDS and FIM-CBS, achieving significantly lower AAEs and computational times as short as 0.04–0.05 s, as validated through extensive numerical experiments. Fourth, the method achieves excellent computational efficiency through tensor-product discretizations and precomputed matrices while maintaining robustness via a robust trust-region solver. Finally, the HSG-IPS method’s robustness across a wide range of fractional orders ($\alpha \in (0,1]$) and its efficient handling of nonlinearities via a trust-region algorithm provide a powerful and versatile computational tool for modeling complex wave phenomena in fractional calculus applications.

This paper is organized to systematically present the HSG-IPS method and its application to the FBBMB equation. Section 2 formulates the initial-boundary value problem of the FBBMB equation. Section 3.1 derives the transformed fractional partial-integro differential equation to improve numerical stability. Section 3.2 describes the HSG-IPS method, including its discretization and implementation via a trust-region algorithm. Section 3.4 analyzes the computational efficiency and conditioning of the method. Section 4 evaluates the method’s performance through numerical simulations and comparisons with existing techniques. Finally, Section 5 summarizes the findings, highlights key advantages, and suggests future research directions in fractional PDEs and spectral methods.

2. Model Description

The FBBMB equation represents a fundamental model for nonlinear wave propagation with memory effects, combining dispersive, dissipative, and nonlinear phenomena. The equation is defined over the spatiotemporal domain ${\mathrm{\Omega }}_{1\times 1}$ as:

$${}^c{D}_t^{\alpha }u-u_{xxt}+u_x+u{u}_x=f(x,t),(x,t)\in {\mathrm{\Omega }}_{1\times 1}^{°},$$

(1)

where $u(x,t)$ is the wave amplitude, ${}^c{D}_t^{\alpha }u$ is the Caputo fractional time derivative of order $\alpha \in (0,1]$, which captures memory effects in the system’s dynamics, $-u_{xxt}$ is the mixed dispersion term, representing third-order spatial-temporal effects, $u_x$ represents the linear advection term, while $u{u}_x$ is the nonlinear Burgers-type convection term, and $f(x,t)$ is the source/sink term. The equation is complemented by the following initial and boundary conditions:

$$\begin{array}{cc}\hfill u(x,0)& =\varphi \left(x\right),x\in {\mathrm{\Omega }}_1,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill u(0,t)& ={\psi }_1\left(t\right),t\in (0,1],\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill u(1,t)& ={\psi }_2\left(t\right),t\in (0,1].\hfill \end{array}$$

(4)

We assume that $\varphi \in C^1\left({\mathrm{\Omega }}_1\right)$ and ${\psi }_1\in {\mathcal{H}}^1\left((0,1]\right)$. For $\alpha \in {\mathrm{\Omega }}_1^{\circ }$, the FBBMB equation describes anomalous wave propagation with memory effects. However, when $\alpha =1$, it reduces to the classical BBMB equation modeling shallow water waves.

3. The HSG-IPS Method

The HSG-IPS method represents a novel approach for solving fractional PDEs by combining the strengths of spectral accuracy with robust numerical integration techniques. This section presents the mathematical foundations and numerical implementation of the HSG-IPS approach, which transforms the original FBBMB equation into a more computationally tractable form while maintaining high-order accuracy. The method’s key innovation lies in its strategic reformulation of the problem to avoid direct approximation of high-order derivatives, instead employing a combination of fractional calculus operators and Gegenbauer polynomial-based discretizations. We begin by deriving the transformed fractional partial-integro differential equation, which serves as the foundation for the numerical scheme, followed by a detailed description of the discretization procedure and implementation strategy.

3.1. The Transformed Fractional Partial-Integro Differential Equation

Setting $v=u_{xt}$, the partial integral of v with respect to x on the spatial interval ${\mathrm{\Omega }}_x$ is

$${\mathcal{I}}_x^{\left(x\right)}v=u_t(x,t)-u_t(0,t).$$

(5)

Denoting $w(x,t)=u_t(x,t)$, we have

$$w(x,t)={\mathcal{I}}_x^{\left(x\right)}v+u_t(0,t).$$

(6)

Integrating $w(x,t)$ with respect to t from 0 to t, we get

$${\mathcal{I}}_t^{\left(t\right)}w={\mathcal{I}}_t^{\left(t\right)}{\mathcal{I}}_x^{\left(x\right)}v+{\mathcal{I}}_t^{\left(t\right)}{u}_t(0,t).$$

(7)

The second term simplifies to

$${\mathcal{I}}_t^{\left(t\right)}{u}_t(0,t)=u(0,t)-u(0,0)={\psi }_1\left(t\right)-\varphi \left(0\right).$$

(8)

Since

$${\mathcal{I}}_t^{\left(t\right)}w=u(x,t)-u(x,0)=u(x,t)-\varphi \left(x\right),$$

we obtain the solution u in terms of v as

$$u(x,t)=\varphi \left(x\right)-\varphi \left(0\right)+{\psi }_1\left(t\right)+{\mathcal{I}}_t^{\left(t\right)}{\mathcal{I}}_x^{\left(x\right)}v.$$

(9)

The spatial and time derivatives are

$$\begin{array}{cc}\hfill u_x(x,t)& ={\varphi }^{\prime }\left(x\right)+{\mathcal{I}}_t^{\left(t\right)}v,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill u_t(x,t)& ={\psi }_1^{\prime }\left(t\right)+{\mathcal{I}}_x^{\left(x\right)}v,\hfill \end{array}$$

(11)

respectively. The mixed derivative is

$$u_{xxt}(x,t)=v_x(x,t).$$

(12)

Moreover, the CFD ${}^c{D}_t^{\alpha }u$ for $\alpha \in {\mathrm{\Omega }}_1^{\circ }$ is given by

$${}^c{D}_t^{\alpha }u={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\mathcal{I}}_t^{\left(\tau \right)}\left({(t-\tau )}^{-\alpha }{u}_t\sout{[}x,\tau \sout{]}\right).$$

(13)

Substitute the expression for $u_t$ from Equation (11) into Equation (13):

$${}^c{D}_t^{\alpha }u={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\mathcal{I}}_t^{\left(\tau \right)}\left({(t-\tau )}^{-\alpha }\left({\psi }_1^{\prime }\left(\tau \right)+{\mathcal{I}}_x^{\left(x\right)}v\right)\right).$$

(14)

This step expresses the fractional derivative in terms of known boundary data and the auxiliary variable v. Since the integral is linear, we separate the contributions of ${\psi }_1^{\prime }\left(\tau \right)$ and ${\mathcal{I}}_x^{\left(x\right)}v$:

$${}^c{D}_t^{\alpha }u={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\mathcal{I}}_t^{\left(\tau \right)}\left({(t-\tau )}^{-\alpha }{\psi }_1^{\prime }\left(\tau \right)\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\mathcal{I}}_t^{\left(\tau \right)}\left({(t-\tau )}^{-\alpha }{\mathcal{I}}_x^{\left(x\right)}v\right).$$

(15)

The first term corresponds to the CFD of ${\psi }_1\left(t\right)$. For the second term, since ${\mathcal{I}}_x^{\left(x\right)}v$ is a function of $\tau$, we interchange the order of integration:

$${\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\mathcal{I}}_t^{\left(\tau \right)}\left({(t-\tau )}^{-\alpha }{\mathcal{I}}_x^{\left(x\right)}v\right)={\mathcal{I}}_x^{\left(x\right)}\left({\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\mathcal{I}}_t^{\left(\tau \right)}\left({(t-\tau )}^{-\alpha }v\right)\right).$$

(16)

The inner integral is the Riemann–Liouville fractional integral of v of order $1-\alpha$. Thus, the second term becomes ${\mathcal{I}}_x^{\left(x\right)}{\mathfrak{I}}_t^{1-\alpha }v$. Combining these results yields

$${}^c{D}_t^{\alpha }u={}^c{D}_t^{\alpha }{\psi }_1+{\mathcal{I}}_x^{\left(x\right)}{\mathfrak{I}}_t^{1-\alpha }v.$$

(17)

This expression separates the contribution of the boundary condition from the fractional integral of v, enabling stable numerical discretization using the HSG-IPS method. Substituting Equations (9)–(12) and (17) into the PDE (1) yields the transformed fractional partial-integro differential equation:

$${\mathrm{\Psi }}_{xt}^{\alpha }v+\left({\mathcal{I}}_t^{\left(t\right)}v+{\varphi }^{\prime }\right)\left[1+\varphi -\varphi \left(0\right)+{\psi }_1+{\mathcal{I}}_t^{\left(t\right)}{\mathcal{I}}_x^{\left(x\right)}v\right]=\mathcal{F},$$

(18)

where

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{xt}^{\alpha }& ={\mathcal{I}}_x^{\left(x\right)}{\mathfrak{I}}_t^{1-\alpha }-{\partial }_x,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \mathcal{F}(x,t)& =f(x,t)-{}^c{D}_t^{\alpha }{\psi }_1.\hfill \end{array}$$

(20)

The boundary condition (4) provides the essential constraint:

$${\mathcal{I}}_1^{\left(x\right)}{\mathcal{I}}_t^{\left(t\right)}v={\psi }_2\left(t\right)-{\psi }_1\left(t\right)-\varphi \left(1\right)+\varphi \left(0\right).$$

(21)

3.2. Numerical Discretization

In this section, we present a high-order PS collocation method for solving the transformed nonlinear Equation (18) subject to the integral constraint (21). The approach combines tensor-product PS discretization with a constrained nonlinear solver using Lagrange multipliers.

Let $x_{0:n}\in {\widehat{\mathbb{G}}}_n^{\lambda }$ and $t_{0:m}\in {\widehat{\mathbb{G}}}_m^{\lambda }$, ${\forall }_{s}n,m\in {\mathbb{Z}}^{+}$, be the SGG points. Let $v_{i,j}=v(x_i,t_j)$, $\forall (i,j)\in {\mathbb{J}}_{n\times m}^{+}$, vectorized into $\mathit{v}\in {\mathbb{R}}^{(n+1)(m+1)}$ in lexicographic order. The operators are discretized as follows: The spatial derivative ${\partial }_x$ is approximated by the SGDM ${\mathbf{D}}_x\in {\mathbb{R}}^{(n+1)\times (n+1)}$ [25]. The CFD ${}^c{D}_t^{\alpha }$ is approximated by the C-FSGIM ${}_m^E{\mathbf{D}}_t^{\alpha }\in {\mathbb{R}}^{(m+1)\times (m+1)}$ [26]. The integrals ${\mathcal{I}}_x^{\left(x\right)}$, ${\mathcal{I}}_t^{\left(t\right)}$, and ${\mathcal{I}}_1^{\left(x\right)}$ are approximated by the SGIMs ${\mathbf{Q}}_x\in {\mathbb{R}}^{(n+1)\times (n+1)}$, ${\mathbf{Q}}_t\in {\mathbb{R}}^{(m+1)\times (m+1)}$, and the SGIRV ${\mathbf{P}}_x\in {\mathbb{R}}^{1\times (n+1)}$ [22,23,24]. The Riemann–Liouville fractional integral ${\mathfrak{I}}_t^{1-\alpha }$ is approximated by the RL-FSGIM ${}_m^E{\mathbf{Q}}_t^{1-\alpha }\in {\mathbb{R}}^{(m+1)\times (m+1)}$ [27].

In the computation of ${}^c{D}_t^{\alpha }$ and ${\mathfrak{I}}_t^{1-\alpha }$, the parameters $n_q$ and ${\lambda }_q$ play critical roles in the numerical approximation using SGG quadratures. In particular, $n_q$ determines the degree of the SGG quadrature with ($n_q+1$) being the number of quadrature nodes. In the SGPS method, it controls the resolution of integrals in the fractional-order SGIM used to approximate the CFD. Similarly, in the GBFA method, it governs the quadrature grid size for a high-precision approximation of the RLFI. The parameter ${\lambda }_q$ serves as the index parameter of the SG polynomials, directly influencing both node clustering and weight functions to optimize quadrature accuracy. For both SGPS and GBFA methods, ${\lambda }_q\in [-1/2+ϵ,2]$ balances stability and accuracy ${\forall }_{rs}{n}_q$.

The SGPS method applies $n_q$ and ${\lambda }_q$ to approximate CFDs, while the GBFA method uses them for RLFIs, with consistent parameter roles but distinct computational contexts. We shall write $n_1$ and ${\lambda }_1$ to refer to the parameters $n_q$ and ${\lambda }_q$, as defined in the SGPS method of Elgindy [26], while $n_2$ and ${\lambda }_2$ refer to the parameters $n_q$ and ${\lambda }_q$, as defined in the GBFA method of Elgindy [27].

Using the above numerical tools, we can discretize the operator ${\mathrm{\Psi }}_{xt}^{\alpha }$ as

$${\mathrm{\Psi }}_{xt}^{\alpha }\approx {\mathrm{\Psi }}^{\mathbf{\alpha }}={\mathbf{Q}}_x\otimes {}_m^E{\mathbf{Q}}_t^{1-\mathbf{\alpha }}-{\mathbf{D}}_x\otimes {\mathbf{I}}_{m+1}.$$

(22)

The nonlinear term in Equation (18) is discretized as

$$\mathbf{N}\left(\mathit{v}\right)=\mathbf{Y}\left(\mathit{v}\right)\odot \mathbf{W}\left(\mathit{v}\right),$$

(23)

where

$$\begin{array}{c}\mathbf{Y}\left(\mathit{v}\right)={\mathbf{K}}_{t,n}\mathit{v}+{\mathit{\varphi }}^{\prime }:{\mathbf{K}}_{t,n}={\mathbf{Q}}_t\otimes {\mathbf{I}}_{n+1},\mathit{\varphi }=\varphi \left({\mathbf{x}}_n\right),\end{array}$$

(24)

$$\begin{array}{c}\mathbf{W}\left(\mathit{v}\right)={\mathbf{1}}_{(n+1)(m+1)}+\mathbf{S}+{\mathbf{Q}}_{tx}\mathit{v},\end{array}$$

(25)

$$\begin{array}{c}\mathbf{S}={\mathbf{1}}_{m+1}\otimes \mathit{\varphi }-\varphi \left(0\right){\mathbf{1}}_{(n+1)(m+1)}+{\mathbf{1}}_{n+1}\otimes {\mathit{\psi }}_{\mathbf{1}},\end{array}$$

(26)

${\mathbf{Q}}_{tx}={\mathbf{Q}}_t\otimes {\mathbf{Q}}_x$, ${\mathit{\psi }}_{\mathbf{1}}={\psi }_1\left({\mathbf{t}}_m\right)$, and ${\mathit{\varphi }}^{\prime }={\mathbf{1}}_{m+1}\otimes \left({\mathbf{D}}_x\mathit{\varphi }\right)\in {\mathbb{R}}^{(n+1)(m+1)}$ contains the spatial derivatives ${\varphi }^{\prime }\left(x_i\right)$ at the SGG points, extended across the time mesh points. The right-hand side $\mathcal{F}$ is discretized as $\mathbf{F}=\mathit{f}-{\mathit{\psi }}_1^{\alpha }\in {\mathbb{R}}^{(n+1)(m+1)}$, where $f(x_i,t_j)$ forms $\mathbf{f}\in {\mathbb{R}}^{(n+1)\times (m+1)}$, vectorized as $\mathit{f}=({\mathbf{I}}_{m+1}\otimes {\mathbf{I}}_{n+1})\mathrm{vec}\left(\mathbf{f}\right)$, and ${\mathit{\psi }}_1^{\alpha }$ is given by

$${\mathit{\psi }}_1^{\alpha }={\mathbf{1}}_{n+1}\otimes \left({}_m^E{\mathbf{D}}_t^{\alpha }{\mathit{\psi }}_{\mathbf{1}}\right).$$

(27)

The integral constraint (21) is discretized as

$$\mathbf{C}\mathit{v}=\widehat{\mathbf{R}},$$

(28)

where $\mathbf{C}={\mathbf{P}}_x\otimes {\mathbf{Q}}_t$, and $\widehat{\mathbf{R}}={\mathit{\psi }}_2-{\mathit{\psi }}_1-(\varphi \left(1\right)-\varphi \left(0\right)){\mathbf{1}}_{m+1}:{\mathit{\psi }}_2={\psi }_2\left({\mathbf{t}}_m\right)$. Collocating Equation (18) at the SGG points set ${\widehat{\mathbb{G}}}_{n\times m}^{\lambda }$ and enforcing the constraint (28) using a Lagrange multiplier $\mu \in {\mathbb{R}}^{m+1}$, we form the augmented system:

$$\left[\begin{array}{cc}\mathbf{J}\left(\mathit{v}\right)& {\mathbf{C}}^{\top }\\ \mathbf{C}& {\mathbf{0}}_{(m+1)\times (m+1)}\end{array}\right]\left[\begin{array}{c}\mathrm{\Delta }\mathit{v}\\ \mathrm{\Delta }\mu \end{array}\right]=-\left[\begin{array}{c}\mathbf{R}\left(\mathit{v}\right)\\ \mathbf{C}\mathit{v}-\widehat{\mathbf{R}}\end{array}\right],$$

(29)

where $\mathrm{\Delta }\mathit{v}$ is the update to the solution vector, $\mathrm{\Delta }\mu$ is the update to the Lagrange multipliers vector, $\mathbf{R}\left(\mathit{v}\right)={\mathrm{\Psi }}^{\mathbf{\alpha }}\mathit{v}+\mathbf{N}\left(\mathit{v}\right)-\mathbf{F}$ is the nonlinear residual function, and $\mathbf{J}\left(\mathit{v}\right)$ is the Jacobian of the residual with respect to $\mathit{v}$ given by

$$\mathbf{J}\left(\mathit{v}\right)={\mathrm{\Psi }}^{\mathbf{\alpha }}+\mathrm{diag}\left(\mathbf{W}\left(\mathit{v}\right)\right){\mathbf{K}}_{t,n}+\mathrm{diag}\left(\mathbf{Y}\left(\mathit{v}\right)\right){\mathbf{Q}}_{tx}.$$

(30)

Newton’s method iterates as

$$\left[\begin{array}{c}{\mathit{v}}^{(k+1)}\\ {\mu }^{(k+1)}\end{array}\right]=\left[\begin{array}{c}{\mathit{v}}^{\left(k\right)}\\ {\mu }^{\left(k\right)}\end{array}\right]-{\left[\begin{array}{cc}\mathbf{J}\left({\mathit{v}}^{\left(k\right)}\right)& {\mathbf{C}}^{\top }\\ \mathbf{C}& {\mathbf{0}}_{(m+1)\times (m+1)}\end{array}\right]}^{-1}\left[\begin{array}{c}\mathbf{R}\left({\mathit{v}}^{\left(k\right)}\right)\\ \mathbf{C}{\mathit{v}}^{\left(k\right)}-\widehat{\mathbf{R}}\end{array}\right],$$

(31)

starting with the initial guesses ${\mathit{v}}^{\left(0\right)}$ and ${\mu }^{\left(0\right)}$. After solving for $\mathit{v}$, the numerical approximation of $u(x,t)$, denoted by $\tilde{u}(x,t)$, is reconstructed at the SGG collocation points via the discretized form of Equation (9):

$$\mathbf{u}=\mathbf{S}+{\mathbf{Q}}_{tx}\mathit{v},$$

(32)

where $\mathbf{u}\in {\mathbb{R}}^{(n+1)(m+1)}$ contains the values $\tilde{u}(x_i,t_j)$ in lexicographic order.

The numerical tools utilized in the proposed method exhibit exponential convergence for functions possessing sufficient smoothness, as rigorously established in the referenced literature. However, for solutions residing in the space $C^k\left({\mathrm{\Omega }}_1\right){\forall }_{s}k\in {\mathbb{Z}}_0^{+}$, the convergence rate transitions to an algebraic decay, typically on the order of $\mathcal{O}(max\{n^{-\rho },m^{-\rho }\}){\forall }_s\rho \ge k$. Furthermore, the precomputation of operational matrices makes subsequent parameter studies for $\alpha$ variation particularly efficient, as only the nonlinear system solution requires recomputation for different $\alpha$ values.

3.3. Implementation with Trust-Region Algorithm

Unlike Newton’s method, which assumes local quadratic convergence and may fail for poor initial guesses or highly nonlinear problems, the trust-region algorithm, as an alternative robust approach, provides global convergence by constraining the step size within a trust region where the quadratic model is reliable. The algorithm adaptively adjusts the trust-region radius based on the agreement between the model and the actual function, ensuring stable convergence. This makes it particularly effective for solving the nonlinear system with complex nonlinearities and fractional operators in the FBBMB equation, especially when handling the augmented system with constraints.

We can solve the augmented nonlinear system (29) efficiently using MATLAB’s fsolve with trust-region optimization, configured with tight tolerances for high precision. All differentiation and integration matrices, including ${\mathbf{D}}_x$, ${\mathbf{Q}}_x$, ${\mathbf{Q}}_t$, ${\mathbf{P}}_x$, ${}_m^E{\mathbf{D}}_t^{\alpha }$, and ${}_m^E{\mathbf{Q}}_t^{1-\alpha }$, are precomputed for efficiency using the parameters specified in

Table 1. Table of key parameters used in the HSG-IPS method.

Parameter	Meaning
$\lambda$	Index parameter of the SG polynomials used in the PS discretization of the FBBMB equation, controlling node clustering and weight functions for SGG points, typically set to $\lambda =0.5$ for balanced accuracy and stability.
${\lambda }_1$	Index parameter of the SG polynomials in the SGPS method [26] for approximating the CFD, influencing quadrature accuracy, typically set to ${\lambda }_1=0.5$.
${\lambda }_2$	Index parameter of the SG polynomials in the GBFA method [27] for approximating the RLFI, optimizing quadrature precision, typically set to ${\lambda }_2=0.5$.
$n_1$	Degree of the SGG quadrature in the SGPS method [26], determining the number of quadrature nodes ($n_1+1$) for CFD approximation, typically set to $n_1=14$.
$n_2$	Degree of the SGG quadrature in the GBFA method [27], determining the number of quadrature nodes ($n_2+1$) for RLFI approximation, typically set to $n_2=14$.

. The exact Jacobian of the augmented system, ${\mathbf{J}}_{\mathrm{aug}}$, is given by

$${\mathbf{J}}_{\mathrm{aug}}=\left[\begin{array}{cc}\mathbf{J}\left(\mathit{v}\right)& {\mathbf{C}}^{\top }\\ \mathbf{C}& {\mathbf{0}}_{(m+1)\times (m+1)}\end{array}\right],$$

(33)

and can be provided to fsolve for faster convergence.

This implementation demonstrates that the PS approach combines spectral accuracy with practical solvability through careful formulation and modern optimization techniques. The trust-region adaptation proves particularly valuable for handling the nonlinear coupling between spatial and fractional temporal operators.

Figure 1 presents a flowchart outlining the computational pipeline of the HSG-IPS method for solving the FBBMB equation. The diagram guides the reader through key stages, including problem formulation, transformation, discretization, numerical solution, and reconstruction.

 Remark 1. 

When $\alpha =1$, ${}_m^E{\mathbf{Q}}_t^{1-\alpha }={\mathbf{I}}_{m+1}$, simplifying ${\mathrm{\Psi }}^{\mathbf{\alpha }}$ to $({\mathbf{Q}}_x-{\mathbf{D}}_x)\otimes {\mathbf{I}}_{m+1}$.

3.4. Computational Efficiency and Conditioning

The HSG-IPS method is designed to offer superior computational efficiency and numerical stability compared to traditional approaches for solving time-fractional PDEs. This efficiency stems from several key aspects of its formulation, particularly the strategic use of Kronecker product structures, precomputed operational matrices, and a problem transformation that avoids computationally expensive high-order derivative approximations.

The tensor-product nature of the discretization of the operator ${\mathrm{\Psi }}_{xt}^{\alpha }$ involves spatial and temporal discretizations that are fundamental to enabling efficient computation, especially through matrix–vector multiplications. For a system with $n+1$ spatial points and $m+1$ temporal points, the computation of the product ${\mathrm{\Psi }}^{\mathbf{\alpha }}\mathit{v}$ has a computational complexity of $\mathcal{O}\left(nm\right(n+m\left)\right)$. This efficiency arises from the properties of Kronecker products, which allow for the decomposition of large matrix–vector products into smaller, more manageable operations, thereby significantly reducing the computational burden compared to $\mathcal{O}\left(n^2{m}^2\right)$ for direct matrix multiplication with a fully assembled large square matrix of size $(n+1)(m+1)$. In particular, if we write $\mathit{v}$ as a matrix $\mathit{v}\in {\mathbb{R}}^{(n+1)\times (m+1)}$ such that $\mathit{v}=\mathrm{vec}\left(\mathit{v}\right)$, then we can efficiently compute ${\mathrm{\Psi }}^{\mathbf{\alpha }}\mathit{v}$ in two stages:

$$\begin{array}{c}({\mathbf{Q}}_x\otimes {}_m^E{\mathbf{Q}}_t^{1-\alpha })\mathit{v}=\mathrm{vec}\left({}_m^E{\mathbf{Q}}_t^{1-\alpha }\mathbf{V}{\mathbf{Q}}_x^{\top }\right),\end{array}$$

(34)

$$\begin{array}{c}({\mathbf{D}}_x\otimes {\mathbf{I}}_{m+1})\mathit{v}=\mathrm{vec}\left({\mathbf{D}}_x\mathbf{V}\right).\end{array}$$

(35)

The computation of $\mathbf{V}{\mathbf{Q}}_x^{\top }$ requires $\mathcal{O}\left(n^{2}m\right)$ operations, and the computation of ${}_m^E{\mathbf{Q}}_t^{1-\alpha }\left(\mathbf{V}{\mathbf{Q}}_x^{\top }\right)$ entails $\mathcal{O}\left(n{m}^2\right)$ operations, resulting in a total of $\mathcal{O}\left(nm(n+m)\right)$ operations. The computational complexity of ${\mathbf{D}}_x\mathbf{V}$ is $\mathcal{O}\left(n^{2}m\right)$.

The computational complexity of the method is further influenced by the remaining terms in Equations (18), (28), and (32). The time integration operator ${\mathbf{K}}_{t,n}={\mathbf{Q}}_t\otimes {\mathbf{I}}_{n+1}$ combines temporal integration with spatial identity, maintaining the Kronecker structure for efficient computation. The evaluation of the nonlinear term $\mathbf{N}\left(\mathit{v}\right)$ also requires $\mathcal{O}\left(nm\right(n+m\left)\right)$ operations. In particular, since ${\mathbf{K}}_{t,n}\mathit{v}=\mathrm{vec}\left(\mathbf{V}{\mathbf{Q}}_t^{\top }\right)$, which requires $\mathcal{O}\left(n{m}^2\right)$ operations, and computing ${\varphi }^{\prime }$ requires $\mathcal{O}\left(nm\right)$, then the total complexity of $\mathbf{Y}\left(\mathit{v}\right)$ is $\mathcal{O}\left(n{m}^2\right)$ operations; moreover, the total complexity of $\mathbf{W}\left(\mathit{v}\right)$ is $\mathcal{O}\left(nm\right(n+m\left)\right)$. The coefficient matrix $\mathbf{C}$ in the constraint Equation (28) involves matrix–vector products, which can be computed in $\mathcal{O}\left(n{m}^2\right)$ operations. The solution reconstruction $\mathbf{u}$ in Equation (32) similarly benefits from the Kronecker product formulation, requiring $\mathcal{O}\left(nm\right(n+m\left)\right)$ operations.

Another significant factor contributing to the method’s efficiency is the precomputation of all necessary differentiation and integration matrices. As detailed in Section 3.3, the matrices ${\mathbf{D}}_x$, ${\mathbf{Q}}_x$, ${\mathbf{Q}}_t$, ${\mathbf{P}}_x$, ${}_m^E{\mathbf{D}}_t^{\alpha }$, and ${}_m^E{\mathbf{Q}}_t^{1-\alpha }$ are computed once prior to the iterative solution process. This precomputation strategy effectively reduces the runtime overhead by eliminating the need for repeated calculations of these fundamental operators within each iteration of the solver. This is particularly advantageous for problems requiring numerous iterations or parameter studies, as the fixed cost of precomputation is amortized over many subsequent operations.

The trust-region algorithm for solving the nonlinear system arising from the HSG-IPS method employs iterative solvers, reducing the per-iteration cost to $\mathcal{O}\left(nmk\right)$, where k denotes the number of iterations required by the iterative solver to converge within a single trust-region iteration, which is consistent with the use of iterative solvers for systems derived from tensor-product discretizations. The trust-region approach provides robust and global convergence, even for highly nonlinear problems, by adaptively adjusting the step size. The efficiency of each iteration is maintained by the precomputed matrices and the structure of the Jacobian, which allows for efficient matrix–vector products within the iterative solver.

A key innovation in the HSG-IPS method lies in transforming the original FBBMB equation into a fractional partial-integro differential form that contains only a first-order derivative. This transformation is crucial because it allows for the use of well-conditioned integral operators and low-order differential operators, which are essential for reliable numerical solutions. By circumventing the direct approximation of ill-conditioned high-order derivatives, the HSG-IPS method inherently deals with better-conditioned matrices, leading to superior numerical stability. This approach, combined with the efficient Kronecker-based implementation and careful operator preconditioning, enables the method to achieve high accuracy and computational efficiency, as we demonstrate in the next section.

The HSG-IPS method’s overall complexity is dominated by the nonlinear solver phase, employing the Kronecker product structure for efficiency. Assuming $n\approx m$, the per-iteration cost for computing the residual $\mathbf{R}\left(\mathit{v}\right)={\mathrm{\Psi }}^{\alpha }\mathit{v}+\mathbf{N}\left(\mathit{v}\right)-\mathbf{F}$ and Jacobian–vector products in the trust-region solver is $\mathcal{O}\left(nm(n+m)\right)=\mathcal{O}\left(n^3\right)$, driven by matrix–vector multiplies like ${\mathrm{\Psi }}^{\alpha }\mathit{v}$ (Equation (22)), ${\mathbf{K}}_{t,n}\mathit{v}$, and ${\mathbf{Q}}_{tx}\mathit{v}$. With l trust-region iterations and k inner iterations per linear solve (both typically $O\left(1\right)$ due to relatively well-conditioned, first-order operators), the solver complexity is $\mathcal{O}\left(lk{n}^3\right)=\mathcal{O}\left(n^3\right)$. Solution reconstruction (Equation (32)) adds $\mathcal{O}\left(n^3\right)$. Thus, the total complexity is $\mathcal{O}\left(n^3\right)$. This is significantly more efficient than direct methods, which require $\mathcal{O}\left(n^6\right)$ operations for a full $(n+1)(m+1)$-sized system, and it outperforms M-grid-point finite-difference methods such as CNLDS and FIM-CBS, which require $\mathcal{O}\left(M^4\right)$ operations for $M\gg n$, by using spectral accuracy with fewer points.

Remark 2. 

One strong reason for employing the trust-region method in this study is its robustness to ill-conditioned Jacobians and poor initial guesses, which are common in fractional PDEs. While Newton–Krylov or Broyden-type solvers can be efficient for simpler systems, they often require tailored preconditioning or globalization strategies for larger n and m. The trust-region method avoids these complexities, as it inherently incorporates globalization through adaptive trust-region adjustments. The usage of the Kronecker structure allows the trust-region approach to maintain $O\left(nm\right(n+m\left)\right)$ per-iteration complexity. The HSG-IPS method’s rapid convergence at small n and m obviates the need for alternatives to the trust-region solver, which is optimal for small/medium scale regimes, balancing simplicity and reliability.

4. Numerical Simulations

This section presents numerical experiments to assess the performance of the proposed HSG-IPS method for solving the FBBMB equation. Simulations were conducted on a laptop with an AMD Ryzen 7 4800H processor (2.9 GHz, 8 cores/16 threads) and 16 GB RAM, running Windows 11. The implementation utilized MATLAB R2023b, employing optimized numerical libraries and the fsolve function with trust-region optimization for high accuracy and efficiency. All computations were timed using MATLAB’s timeit function, which executes code multiple times to provide accurate average execution times. In all numerical examples, we report the AAEs and the ETs to comprehensively assess both the accuracy and computational efficiency of the method. Furthermore, for completeness, we also compute the discrete ${\mathsf{\ell }}_{\infty }$-norm (maximum norm) and the RMSE of the numerical solution relative to the exact solution. These are defined, for a grid ${\left\{(x_i,t_j)\right\}}_{i=1,j=1}^{n_x,n_t}$, as

$$\parallel \mathbf{u}-\tilde{\mathbf{u}}{\parallel }_{{\mathsf{\ell }}_{\infty }}=\underset{{\displaystyle \genfrac{}{}{0pt}{}{1\le i\le n_x}{1\le j\le n_t}}}{max}\left|u_{ij}-{\tilde{u}}_{ij}\right|,$$

(36)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n_x{n}_t}}\sum _{i=1}^{n_x}\sum _{j=1}^{n_t}{\left(u_{ij}-{\tilde{u}}_{ij}\right)}^2}.$$

(37)

Example 1. 

We evaluate the HSG-IPS method using the FBBMB equation on the domain ${\mathrm{\Omega }}_{1\times 1}$, with the source term

$$f(x,t)={\displaystyle \frac{3\sqrt{\pi }{x}^4(x-1)t^{1.5-\alpha }}{4\mathrm{\Gamma }(2.5-\alpha )}}+x^2\sqrt{t}\left(5x^7{t}^{2.5}-9x^6{t}^{2.5}+4x^5{t}^{2.5}+5x^{2}t-4xt-30x+18\right),$$

(38)

derived from the analytical solution $u(x,t)=x^4(x-1)t^{1.5}$ [12,21]. The initial condition function is $\varphi \equiv 0$, and the boundary condition functions are ${\psi }_1\equiv 0$ and ${\psi }_2\equiv 0$.

Table 2. Comparison of AAEs for Example 1 with $\alpha =0.5$ using CNLDS [12], FIM-CBS [21], and the HSG-IPS method. For the HSG-IPS method, AAE, ${\mathsf{\ell }}_{\infty }$-norm, and RMSE values are rounded to four significant digits, while ET in seconds is rounded to two significant digits.

CNLDS [12]	FIM-CBS [21]	HSG-IPS Method
$\mathrm{\Delta }t=0.001$		$(n_1,n_2,\lambda ,{\lambda }_1,{\lambda }_2)=(14,14,0.5,0.5,0.5)$
$M/$AAE		$n=m/$AAE/${\mathsf{\ell }}_{\infty }$-norm/RMSE/ET
$1.841489\times {10}^{-3}$	$1.7810\times {10}^{-5}$	$4/9.6546\times {10}^{-6}/2.4051\times {10}^{-5}/1.2312\times {10}^{-5}/0.04$
$5.047981\times {10}^{-4}$	$1.7763\times {10}^{-5}$	$5/2.0978\times {10}^{-6}/1.0463\times {10}^{-5}/3.0562\times {10}^{-6}/0.04$
$1.321991\times {10}^{-4}$	$1.7751\times {10}^{-5}$	$6/1.8902\times {10}^{-6}/1.0256\times {10}^{-5}/2.9364\times {10}^{-6}/0.04$
$3.365747\times {10}^{-5}$	$1.7748\times {10}^{-5}$	$7/1.8165\times {10}^{-6}/9.3833\times {10}^{-6}/2.9217\times {10}^{-6}/0.05$

demonstrates the superior accuracy of the HSG-IPS method compared to CNLDS [12] and FIM-CBS [21] for solving the FBBMB equation at $\alpha =0.5$, as the AAEs of the HSG-IPS method are consistently smaller. The method achieves these results with remarkably low ETs of approximately $0.04-0.05$ s across different discretization levels, demonstrating its computational efficiency. Since the solution $u(x,t)=x^4(x-1)t^{1.5}$ belongs to $C^1\left({\mathrm{\Omega }}_{1\times 1}\right)$ but lacks higher-order smoothness due to the fractional exponent in time, the convergence rate is algebraic rather than exponential, as theoretically expected for PS methods applied to non-analytic functions. The error decays as $\mathcal{O}(max\{n^{-k},m^{-k}\})$ for some $k\ge 1$, consistent with the limited regularity of the solution. Notice also that even with modest collocation points ($n=m=4$), the HSG-IPS method achieves AAEs of about ${10}^{-5}$, outperforming CNLDS and FIM-CBS in terms of accuracy. This efficiency stems from the optimal approximation properties of SG polynomials and the accurate discretization of fractional operators via C-FSGIM and RL-FSGIM. The parameters $(\lambda ,{\lambda }_1,{\lambda }_2)=(0.5,0.5,0.5)$ are selected to balance accuracy and stability; cf. [26,27] (both references recommend choosing $\lambda$ within the approximate interval $(-0.5+\epsilon ,r]$, where $\epsilon >0$ is small and $r\in {\mathrm{\Omega }}_{1,2}$ is a suitable upper bound. The specific exclusion of the neighborhood of ${\lambda }^{\ast }\approx -0.1351$ was recommended in [27] to prevent error amplification. This parameter range provides a balance between convergence, numerical stability, and error control). The trust-region algorithm further improves robustness by ensuring reliable convergence, even in the presence of nonlinearity and constraints in the discretized system.

The numerical solution profile, depicted in Figure 2, shows close agreement with the analytical solution across the domain. The HSG-IPS method maintains high accuracy while capturing the solution’s nonlinear wave characteristics.

Figure 3 illustrates the corresponding time evolution of the numerical solution $\tilde{u}(x,t)$ to the FBBMB equation, computed using the HSG-IPS method. Each curve represents the spatial distribution of the solution at fixed time instances $t=0.2,0.4,\dots ,1$. The figure effectively captures the dynamic behavior of the solution, demonstrating the method’s ability to accurately model the wave propagation and nonlinear characteristics of the FBBMB equation over time while satisfying the boundary conditions $u(0,t)=0$ and $u(1,t)=0\forall t\in {\mathrm{\Omega }}_1$.

Figure 4 presents a comparative visualization of the numerical solutions to the FBBMB equation for fractional orders $\alpha =0.1,0.3,0.75$, obtained using the HSG-IPS method. The solutions, displayed in the first three columns, are side-by-side against the exact solution $u(x,t)=x^4(x-1)t^{1.5}$, shown in the fourth column. The figure reveals a close agreement between the numerical and exact solutions across all tested fractional orders.

Example 2. 

Consider the FBBMB equation on the domain ${\mathrm{\Omega }}_{1\times 1}$ with the source term

$$f(x,t)={\displaystyle \frac{2e^x{t}^{2-\alpha }}{\mathrm{\Gamma }(3-\alpha )}}+t{e}^x\left(t^3+t-2\right),$$

(39)

the initial condition function $\varphi \equiv 0$, and the boundary condition functions ${\psi }_1\left(t\right)=t^2$ and ${\psi }_2\left(t\right)=e{t}^2\forall t\in (0,1]$. The analytical solution is $u(x,t)=t^2{e}^x$ [21,28].

The HSG-IPS method was applied with parameters $\alpha =0.5$, $n=m=30$, $n_1=n_2=14$, and $\lambda ={\lambda }_1={\lambda }_2=0.5$.

Table 3. Comparison of AAEs at time $t=1$ for Example 2 with $\alpha =0.5$ using FIM-CBS [21] and the HSG-IPS method. For the HSG-IPS method, AAE, ${\mathsf{\ell }}_{\infty }$-norm, and RMSE values are rounded to four significant digits, while ET in seconds is rounded to two significant digits.

	FIM-CBS [21]				HSG-IPS Method
	$M=40$				$(n_1,n_2,\lambda ,{\lambda }_1,{\lambda }_2)=(14,14,0.5,0.5,0.5)$
x	$\mathrm{\Delta }t=0.05$	$\mathrm{\Delta }t=0.01$	$\mathrm{\Delta }t=0.005$	$\mathrm{\Delta }t=0.001$	$n=m/$AAE/${\mathsf{\ell }}_{\infty }$-norm/RMSE/ET
0.2	$1.3072\times {10}^{-3}$	$2.7980\times {10}^{-5}$	$1.4266\times {10}^{-5}$	$2.9342\times {10}^{-6}$	$4/5.4389\times {10}^{-5}/2.2842\times {10}^{-4}/7.6968\times {10}^{-5}/0.04$
0.4	$1.8644\times {10}^{-3}$	$3.8426\times {10}^{-5}$	$1.9242\times {10}^{-5}$	$3.8423\times {10}^{-6}$	$5/3.2173\times {10}^{-6}/1.5589\times {10}^{-5}/5.2528\times {10}^{-6}/0.04$
0.6	$1.3842\times {10}^{-3}$	$2.9035\times {10}^{-4}$	$1.4669\times {10}^{-4}$	$2.9731\times {10}^{-5}$	$6/2.0769\times {10}^{-7}/1.1033\times {10}^{-6}/3.4931\times {10}^{-7}/0.04$
0.8	$2.5887\times {10}^{-3}$	$5.4398\times {10}^{-4}$	$2.7512\times {10}^{-4}$	$5.5861\times {10}^{-5}$	$7/1.1670\times {10}^{-8}/6.1716\times {10}^{-8}/1.9770\times {10}^{-8}/0.05$

presents the AAEs at the final time $t=1$, comparing the HSG-IPS method with the FIM-CBS method from [21].

The results in Table 3 demonstrate that the HSG-IPS method achieves significantly lower AAEs compared to the FIM-CBS method, with errors dropping to $1.1670\times {10}^{-8}$ for $n=m=7$, compared to $5.5861\times {10}^{-5}$ for FIM-CBS at $\mathrm{\Delta }t=0.001$ and $x=0.8$. This indicates superior accuracy, particularly for finer discretizations. The numerical solutions, depicted in Figure 5, show excellent agreement with the analytical solution $u(x,t)=t^2{e}^x$, capturing the exponential spatial behavior and quadratic temporal growth accurately. The HSG-IPS method maintains computational efficiency, with ETs of approximately 0.04–0.05 s, comparable to Example 1. Since the solution $u(x,t)=t^2{e}^x$ is smooth, the method exhibits exponential convergence, as expected for PS methods applied to analytic functions.

Figure 6 illustrates the time evolution of the numerical solution $\tilde{u}(x,t)$ to the FBBMB equation, computed using the HSG-IPS method with parameters $\alpha =0.5$, $n=m=30$, $n_1=n_2=14$, and $\lambda ={\lambda }_1={\lambda }_2=0.5$.

Figure 7 presents a comparative visualization of numerical solutions to the FBBMB equation for Example 2, computed using the HSG-IPS method. The figure is structured to display numerical solutions for fractional orders $\alpha =0.1,0.3,0.75$ in the first three columns, juxtaposed against the exact solution $u(x,t)=t^2{e}^x$ in the fourth column. The figure emphasizes, once again, the close agreement between the numerical and exact solutions, effectively demonstrating the HSG-IPS method’s robustness and accuracy across varying fractional orders.

Remark 3. 

The computation of the boundary condition function ${\psi }_2\left(t\right)=e{t}^2$ in Example 2 utilized a high-precision approximation of Euler’s number, e, implemented via the functionenfrom the MATLAB Central File Exchange submission [29]. This ensured the boundary data was defined with a precision consistent with the spectral accuracy goals of the HSG-IPS method.

5. Conclusions

The HSG-IPS method provided a robust and accurate framework for solving the FBBMB equation. By combining SGPS, GBFA, SGDM, SGIM, and SGIRV, it achieved spectral accuracy and exponential convergence for smooth solutions, as shown in the numerical experiments. The method demonstrated significant performance improvements, with up to a 99.99% reduction in AAEs compared to existing methods like CNLDS and FIM-CBS, while maintaining computational efficiency with execution times as low as 0.04–0.05 s. A key innovation of our approach was the transformation of the original equation into a fractional partial-integro differential form that contained only a first-order derivative, which was well-approximated by a first-order SGDM, while all other terms were handled through precise quadrature rules. The method’s main limitations include potential challenges with highly non-smooth solutions and the need for careful parameter selection in higher-dimensional extensions, though its tensor-product formulation provides a natural pathway for such generalizations.

The HSG-IPS method’s computational efficiency, analyzed in Section 3.4, stemmed from (i) tensor-product Kronecker structures reducing complexity to $\mathcal{O}\left(nm\right(n+m\left)\right)$, (ii) precomputation of operational matrices, and (iii) the trust-region solver’s $\mathcal{O}\left(nmk\right)$ complexity. By transforming the original equation, the method avoided ill-conditioned high-order derivatives, ensuring stability. On the other hand, directly applying PS methods to the FBBMB equation without sophisticated preconditioning can lead to significant inaccuracies due to the ill-conditioning of high-order differentiation matrices, particularly for the third-order mixed derivative $u_{xxt}$. With condition numbers scaling as $\mathcal{O}\left(n^4{m}^2\right)$ for n and m denoting the spatial and temporal grid points, respectively, these matrices can amplify numerical errors, necessitating advanced techniques like the transformation and integration-based approaches employed in this study to ensure stability and accuracy.

In Example 1, HSG-IPS was applied to a test case with the analytical solution $u(x,t)=x^4(x-1)t^{1.5}$, which exhibited limited smoothness. It achieved AAEs as low as $1.8165\times {10}^{-6}$ for $n=m=7$, outperforming CNLDS and FIM-CBS, as shown in Table 2. The algebraic convergence rate, consistent with $C^1\left({\mathrm{\Omega }}_{1\times 1}\right)$ regularity, followed $\mathcal{O}(max\{n^{-k},m^{-k}\})$ for some $k\ge 1$. Figure 2 and Figure 4 showed excellent agreement between numerical and exact solutions for $\alpha =0.5$ and across $\alpha =0.1,0.3,0.75$, capturing nonlinear wave characteristics and satisfying the boundary conditions with high accuracy. In Example 2, with the smooth solution $u(x,t)=t^2{e}^x$, the HSG-IPS method exhibited exponential convergence. Table 3 reports AAEs as low as $1.1670\times {10}^{-8}$ for $n=m=7$, surpassing FIM-CBS accuracy. Figure 5 and Figure 7 confirmed precision, with numerical solutions matching the exact solution for $\alpha =0.5$ and across $\alpha =0.1,0.3$, and $0.75$. HSG-IPS achieved low ETs (0.04–0.05 s), as reported in Table 2 and Table 3. The precomputed matrices ${\mathbf{D}}_x$, ${\mathbf{Q}}_x$, ${\mathbf{Q}}_t$, ${\mathbf{P}}_x$, ${}_m^E{\mathbf{D}}_t^{\alpha }$, and ${}_m^E{\mathbf{Q}}_t^{1-\alpha }$ enabled rapid computational performance. The trust-region algorithm, via MATLAB’s fsolve, further converged in short times, handling nonlinear and fractional terms robustly. Spectral accuracy stemming from SG polynomials, precise C-FSGIM and RL-FSGIM discretizations, and stable barycentric representations ensured robust and efficient solutions for the FBBMB equation across varying fractional orders.

The HSG-IPS method offered superior accuracy, efficiency, and robustness over CNLDS and FIM-CBS. Its adaptability to fractional orders made it ideal for modeling wave propagation and memory-dependent phenomena. Future work could extend HSG-IPS to multidimensional FBBMB equations or incorporate adaptive mesh refinement for localized features.