B-Spline Wavelet Scheme for Multi-Term Time–Space Variable-Order Fractional Nonlinear Diffusion-Wave Equation

Abstract

This paper presents a novel B-spline wavelet-based scheme for solving multi-term time–space variable-order fractional nonlinear diffusion-wave equations. By combining semi-orthogonal B-spline wavelets with a collocation approach and a quasilinearization technique, we transform the original problem into a system of algebraic equations. To enhance the computational efficiency, we derive the operational matrix formulation of the proposed scheme. We provide a rigorous convergence analysis of the method and demonstrate its accuracy and effectiveness through numerical experiments. The results confirm the robustness and computational advantages of our approach for solving this class of fractional differential equations.

1. Introduction

Over the past two decades, variable-order fractional partial differential equations (V-FPDEs) have attracted considerable interest due to their superior capability, compared to classical fractional counterparts, for modeling complex dynamical systems. These equations have proven particularly effective in describing phenomena across diverse domains, including viscoelastic mechanics [1,2], solute transport [3], polymer materials [4], and electrode aging [5]. While several definitions of variable-order fractional derivatives exist [6,7,8,9], the Caputo-type derivative has gained prominence owing to its physical interpretability and suitability for modeling real-world processes.

As a significant subclass of V-FPDEs, the multi-term time–space variable-order fractional nonlinear diffusion-wave equation (M-TS-VFNDWE) provides a generalized framework for characterizing anomalous diffusion. This comprehensive formulation encompasses fundamental equations such as the diffusion-wave equation, the Klein–Gordon equation, and the sine-Gordon equation, making it indispensable for modeling diverse anomalous diffusion phenomena. However, deriving analytical solutions for the M-TS-VFNDWE proves exceptionally challenging due to its inherent complexity, particularly the functional freedom and uncertainty intrinsic to variable-order fractional derivatives. Consequently, developing high-precision numerical methods for solving the M-TS-VFNDWE has become a critical research priority.

Reflecting this priority, significant effort has focused on developing numerical methods, given the general intractability of analytical solutions for the M-TS-VFNDWE. Finite difference schemes [10,11,12] have been applied to solving variable-order fractional PDEs, but their accuracy is typically inferior to that of spectral methods, which benefit from global approximation properties. Spectral collocation methods based on Jacobi polynomials [13,14] and shifted Jacobi spectral algorithms [15,16] have been employed for one- and two-dimensional linear/nonlinear V-FPDEs. Additionally, wavelet-based methods, such as Legendre and Chebyshev wavelets [17,18,19], have shown promise in solving various V-FPDEs.

Despite these advances, dedicated numerical methods for the M-TS-VFNDWE remain limited. For instance, weighted average nonstandard finite difference methods [20] and spectral collocation methods using shifted Jacobi polynomials [21] have been explored for specific cases, such as variable-order fractional Klein–Gordon and sine–Gordon equations. More recently, wavelet techniques [18,19,22] and local discontinuous Galerkin methods [23] have been proposed for multi-term variable-order time-fractional diffusion-wave and sine–Gordon equations.

Although wavelets are powerful tools for fractional PDEs, their application to V-FPDEs remains limited. Notably, only a few studies [17,18,19] have employed orthogonal wavelets, which lack simultaneous finite support and symmetry, the key properties for efficient numerical implementation. To address this gap, we propose the use of semi-orthogonal B-spline wavelets [24], which combine compact support, symmetry, and explicit expressions. Our approach involves

• Linearizing the M-TS-VFNDWE via quasilinearization;
• Discretizing the linearized equation using a collocation method to derive an algebraic system;
• Establishing a rigorous convergence analysis for the proposed scheme.

In this paper, we consider a class of M-TS-VFNDWE with Caputo fractional derivatives:

$$\sum _{i=1}^r{a}_i{D}_t^{{\alpha }_i(x,t)}u(x,t)-a_{r+1}{D}_x^{\beta }u(x,t)=f(x,t,u),(x,t)\in Q_{b,T}:=(0,b)\times (0,T),$$

(1)

subject to the following initial conditions and Dirichlet boundary conditions:

$$\begin{array}{cc}\hfill & u(x,0)=g_1\left(x\right),{\left.{\displaystyle \frac{\partial u(x,t)}{\partial t}}\right|}_{t=0}=g_2\left(x\right),x\in [0,b],\hfill \\ \hfill & u(0,t)={\nu }_1\left(t\right),u(b,t)={\nu }_2\left(t\right),t\in [0,T],\hfill \end{array}$$

where $r\in {\mathbb{N}}_{+}$, $1<\beta \le 2$, $a_i(i=1,2,\cdots ,r+1)$, b and T are positive constants, $f(x,t,u)$ is a known nonlinear source function, and $1<{\alpha }_1(x,t)<{\alpha }_2(x,t)<\cdots <{\alpha }_r(x,t)\le 2$, $g_1\left(x\right)$, $g_2\left(x\right)$, ${\nu }_1\left(t\right)$ and ${\nu }_2\left(t\right)$ are given initial and boundary functions. $D_t^{{\alpha }_i(x,t)}u(x,t)$ and $D_x^{\beta }u(x,t)$ denote the variable-order Caputo fractional derivatives of the order ${\alpha }_i(x,t)$ with respect to time and the constant-order Caputo fractional derivative of the order $\beta$ with respect to space, respectively.

This paper is organized as follows: Section 2 reviews the fundamental definitions and properties of fractional derivatives and semi-orthogonal B-spline wavelets on bounded intervals. Section 3 presents the B-spline wavelet scheme (BSWS) for solving the M-TS-VFNDWE and derives its operational matrix form. Section 4 provides a detailed convergence analysis. Section 5 validates the proposed method through numerical experiments. Section 6 concludes this paper with key findings and future directions.

2. Preliminaries

This section presents a concise review of key mathematical foundations, including the classical Caputo fractional derivative [25], the variable-order Caputo fractional derivative [9] in Section 2.1, and semi-orthogonal B-spline wavelets on bounded intervals with their essential properties [26,27,28] in Section 2.2.

2.1. Fractional Definitions

Definition 1. 

The Caputo fractional derivative of order β for $u\left(x\right)$ is defined as

$$D^{\beta }u\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\mathrm{\Gamma }(n-\beta )}}{\int }_0^x{\displaystyle \frac{u^{\left(n\right)}\left(\sigma \right)}{{(x-\sigma )}^{\beta -n+1}}}\mathrm{d}\sigma ,\hfill & n-1<\beta <n,\hfill \\ u^{\left(n\right)}\left(x\right),\hfill & \beta =n,\hfill \end{array}\right.$$

(2)

where $n\in {\mathbb{N}}_{+}$, $u^{\left(n\right)}\left(x\right)$ is the conventional integer derivative of order n for $u\left(x\right)$.

Definition 2. 

The Caputo fractional derivative of $u(x,t)$ with respect to space is defined as

$$D_x^{\beta }u(x,t)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\mathrm{\Gamma }(n-\beta )}}{\int }_0^x{(x-\sigma )}^{n-1-\beta }{\displaystyle \frac{{\partial }^{n}u(\sigma ,t)}{\partial {\sigma }^n}}\mathrm{d}\sigma ,\hfill & n-1<\beta <n,\hfill \\ {\displaystyle \frac{{\partial }^{n}u(x,t)}{\partial x^n}},\hfill & \beta =n,\hfill \end{array}\right.$$

(3)

where $n\in {\mathbb{N}}_{+}$.

Definition 3. 

The variable-order Caputo fractional derivative of $u(x,t)$ with respect to time is defined as

$$D_t^{\alpha (x,t)}u(x,t)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\mathrm{\Gamma }(n-\alpha (x,t\left)\right)}}{\int }_0^t{(t-\sigma )}^{n-1-\alpha (x,t)}{\displaystyle \frac{{\partial }^{n}u(x,\sigma )}{\partial {\sigma }^n}}\mathrm{d}\sigma ,\hfill & n-1<\alpha (x,t)<n,\hfill \\ {\displaystyle \frac{{\partial }^{n}u(x,t)}{\partial t^n}},\hfill & \alpha (x,t)=n,\hfill \end{array}\right.$$

(4)

where $n\in {\mathbb{N}}_{+}$.

2.2. Semi-Orthogonal B-Spline Wavelets on a Bounded Interval

Definition 4. 

Suppose $j\in \mathbb{Z}$ denote the knot sequence as ${\xi }^{\left(j\right)}:={\left\{{\xi }_k^{\left(j\right)}\right\}}_{k=-m+1}^{2^j-m+1}$ in $[0,T]$, with

$$\begin{array}{c}{\xi }_{-m+1}^{\left(j\right)}={\xi }_{-m+2}^{\left(j\right)}=\cdots ={\xi }_0^{\left(j\right)}=0,\\ {\xi }_k^{\left(j\right)}={\displaystyle \frac{kT}{2^j}},k=1,\cdots ,2^j-1,\\ {\xi }_{2^j}^{\left(j\right)}={\xi }_{2^j+1}^{\left(j\right)}=\cdots ={\xi }_{2^j+m-1}^{\left(j\right)}=T.\end{array}$$

The m-th-order B-spline function is defined by

$$B_{i,m,j}\left(t\right)={(-1)}^m({\xi }_{i+m}^{\left(j\right)}-{\xi }_i^{\left(j\right)}){\left[{\xi }_i^{\left(j\right)},{\xi }_{i+1}^{\left(j\right)},\cdots ,{\xi }_{i+m}^{\left(j\right)}\right]}_{\xi }{(t-\xi )}_{+}^{m-1},$$

(5)

where ${\left[·,\cdots ,·\right]}_{\xi }$ is the m-th divided difference in ${(t-\xi )}_{+}^{m-1}$ as regards variable ξ, and ${(t-\xi )}_{+}^{m-1}$ is defined as

$${(t-\xi )}_{+}^{m-1}=\left\{\begin{array}{cc}{(t-\xi )}^{m-1},\hfill & t>\xi ,\hfill \\ 0,\hfill & t\le \xi .\hfill \end{array}\right.$$

Assuming that $j_0$ satisfies

$$2^{j_0}\ge 2m-1$$

(6)

for an inner wavelet function on $[0,T]$, which ensures the method is more widely applicable. 

Definition 5. 

For $j\ge j_0$, the m-th-order scaling function of space $V_j^{[0,T]}$ can be defined as

$${\varphi }_{i,m,j}\left(t\right)=\left\{\begin{array}{cc}{B}_{i,m,j_0}\left(2^{j-j_0}t\right),\hfill & i=-m+1,-m+2,\cdots ,-1,\hfill \\ B_{2^j-m-i,m,j_0}(b/2-2^{j-j_0}t),\hfill & i=2^j-m+1,2^j-m+2,\cdots ,2^j-1,\hfill \\ B_{0,m,j_0}(2^{j-j_0}t-2^{-j_0}i),\hfill & i=0,1,\cdots ,2^j-1.\hfill \end{array}\right.$$

(7)

Definition 6. 

For $j\ge j_0$, the wavelet subspace $W_j^{[0,T]}$ is spanned by the inner wavelet

$${\psi }_{j,i}\left(t\right)=1/2^{m-1+{(j+1)}^m}\sum _{k=0}^{2m-2}{(-1)}^k{N}_{2m}(k+1)B_{2i+k,2m,j+1}^{\left(m\right)}\left(t\right),$$

(8)

$i=0,1,\cdots ,2^j-2m+1$, the boundary wavelet for 0,

$$\begin{array}{c}\hfill {\psi }_{j,i}\left(t\right)=1/2^{m-1+{(j+1)}^m}\sum _{k=-m+1}^{-1}{\left({\tilde{B}}^{-1}{r}_i\right)}_k{B}_{k,2m,j+1}^{\left(m\right)}\left(t\right)+1/2^{m-1+{(j+1)}^m}\sum _{k=0}^{2m-2+2i}{(-1)}^k{N}_{2m}(k+1-2i)B_{k,2m,j+1}^{\left(m\right)}\left(t\right),\end{array}$$

(9)

$i=-m+1,-m+2,\cdots ,1$, and the boundary wavelet for T,

$${\psi }_{j,2^j-2m+1-i}\left(t\right)={\psi }_{j,i}\left(t\right),$$

(10)

$i=-m+1,-m+2,\cdots ,1$, where $N_{2m}(2^{j+1}x-i)=B_{i,2m,j+1}\left(x\right)$ and the matrix $\tilde{B}$ and r are defined in [26].

Property 1. 

The m-th-order semi-orthogonal B-spline wavelet ψ has a vanishing moment of order m as

$${\int }_{-\infty }^{\infty }{t}^p\psi \left(t\right)\mathrm{d}t=0,p=0,1,\cdots ,m-1.$$

(11)

Property 2. 

The semi-orthogonal B-spline wavelet is semi-orthogonal, that is,

$$<{\psi }_{s,k},{\psi }_{l,m}>:={\int }_{-\infty }^{\infty }{\psi }_{s,k}\left(t\right){\psi }_{l,m}\left(t\right)\mathrm{d}t=0,s\ne l,\forall s,k,l,m\in \mathbb{Z}.$$

(12)

3. B-Spline Wavelet Scheme

This section addresses the M-TS-VFNDWE via a B-spline wavelet scheme. The problem is first linearized through quasilinearization iteration, followed by the application of the semi-orthogonal B-spline wavelet collocation method (SOBWCM) at each iteration step. The operational matrix formulation is then derived for efficient computation.

To enhance the computational efficiency before SOBWCM implementation, we linearize the M-TS-VFNDWE (1) via quasilinearization, converting it into a multi-term time-space variable-order fractional linear diffusion-wave equation (M-TS-VFLDWE) as

$$\sum _{i=1}^r{a}_i{D}_t^{{\alpha }_i(x,t)}{u}_{n+1}(x,t)-a_{r+1}{D}_x^{\beta }{u}_{n+1}(x,t)=(u_{n+1}(x,t)-u_n(x,t)){\displaystyle \frac{\partial f(x,t,u_n)}{\partial u_n}},(x,t)\in Q_{b,T},$$

(13)

subject to

$$\begin{array}{cc}\hfill & u_{n+1}(x,0)=g_1\left(x\right),{\left.{\displaystyle \frac{\partial u_{n+1}(x,t)}{\partial t}}\right|}_{t=0}=g_2\left(x\right),x\in [0,b],\hfill \\ \hfill & u_{n+1}(0,t)={\nu }_1\left(t\right),u_{n+1}(b,t)={\nu }_2\left(t\right),t\in [0,T].\hfill \end{array}$$

The linearization process transforms the solution of the M-TS-VFNDWE (1) into that of the M-TS-VFLDWE (13). Subsequently, the SOBWCM is employed to solve (13) for $u_{n+1}(x,t)$.

Note that the orthogonal decomposition of $L^2[0,T]$ is derived from a multiresolution analysis as

$$\begin{array}{c}\hfill L^2[0,T]=V_0^{[0,T]}\oplus W_0^{[0,T]}\oplus W_1^{[0,T]}\oplus \cdots =V_0^{[0,T]}\underset{j\in {\mathbb{Z}}_{+}}{\oplus }{W}_j^{[0,T]}.\end{array}$$

(14)

Thus, arbitrary function $u\left(t\right)\in L^2[0,T]$ can be approximated as

$$u\left(t\right)=\sum _{k=-m+1}^{2^{j_0}-1}{c}_{j_0,k}{\varphi }_{j_0,k}\left(t\right)+\sum _{j=j_0}^{\infty }\sum _{k=-m+1}^{2^j-m}{d}_{j,k}{\psi }_{j,k}\left(t\right).$$

(15)

Retaining the lower-frequency components up to $W_M^{[0,T]}$, which is equivalent to truncating Equation (15) at M,

$$u\left(t\right)\approx \sum _{k=-m+1}^{2^{j_0}-1}{c}_{j_0,k}{\varphi }_{j_0,k}\left(t\right)+\sum _{j=j_0}^M\sum _{k=-m+1}^{2^j-m}{d}_{j,k}{\psi }_{j,k}\left(t\right)=C^{\mathrm{T}}\mathrm{\Psi }\left(t\right),$$

(16)

where

$$C={\left[c_{j_0,-m+1},\cdots ,c_{j_0,2^{j_0}-1},d_{j_0,-m+1},\cdots ,d_{M,2^j-m}\right]}^{\mathrm{T}},$$

$$\mathrm{\Psi }={\left[{\varphi }_{j_0,-m+1},\cdots ,{\varphi }_{j_0,2^{j_0}-1},{\psi }_{j_0,-m+1},\cdots ,{\psi }_{M,2^j-m}\right]}^{\mathrm{T}}.$$

The elements of matrix C are determined as

$$\begin{array}{cc}\hfill c_{j_0,k}& ={\int }_0^{T}u\left(t\right){\tilde{\varphi }}_{j_0,k}\left(t\right)\mathrm{d}t,k=-m+1,-m+2,\cdots ,2^{j_0}-m,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill d_{j,k}& ={\int }_0^{T}u\left(t\right){\tilde{\psi }}_{j,k}\left(t\right)\mathrm{d}t,j=j_0,j_0+1,\cdots ,M,k=-m+1,-m+2,\cdots ,2^j-m.\hfill \end{array}$$

(18)

where ${\tilde{\varphi }}_{j_0,k}\left(t\right)$ and ${\tilde{\psi }}_{j,k}\left(t\right)$ are the dual functions of ${\varphi }_{j_0,k}\left(t\right)$ and ${\psi }_{j,k}\left(t\right)$ in $V_{j_0}$ and $W_j$, respectively.

Similarly, the arbitrary function $u_{n+1}(x,t)$ defined on ${\overline{Q}}_{b,T}$ can be expanded using a semi-orthogonal B-spline wavelet of order m and the related scaling function as

$$\begin{array}{c}\hfill u_{n+1}(x,t)=\sum _{k=-m+1}^{2^{j_0}-1}\sum _{j=j_0}^{\infty }\sum _{l=-m+1}^{2^j-m}{u}_{j,l}^k{\varphi }_{j_0,k}\left(x\right){\psi }_{j,l}\left(t\right)+\sum _{k=-m+1}^{2^{j_0}-1}\sum _{j=j_0}^{\infty }\sum _{l=-m+1}^{2^j-m}{u}_k^{j,l}{\varphi }_{j_0,k}\left(t\right){\psi }_{j,l}\left(x\right)+\sum _{i=j_0}^{\infty }\sum _{k=-m+1}^{2^i-m}\sum _{j=j_0}^{\infty }\sum _{l=-m+1}^{2^j-m}{u}_{j,l}^{i,k}{\psi }_{i,k}\left(x\right){\psi }_{j,l}\left(t\right).\end{array}$$

(19)

Denoting ${\tilde{\varphi }}_{j_0,k}\left(x\right)$ and ${\tilde{\psi }}_{j,k}\left(x\right)$ as the dual functions of ${\varphi }_{j_0,k}\left(x\right)$ and ${\psi }_{j,k}\left(x\right)$, then the coefficients can be shown as

$$\begin{array}{cc}\hfill u_{j,l}^k& ={\int \int }_{Q_{b,T}}{u}_{n+1}(x,t){\tilde{\varphi }}_{j_0,k}\left(x\right){\tilde{\psi }}_{j,l}\left(t\right)\mathrm{d}x\mathrm{d}t,\hfill \\ \hfill u_k^{j,l}& ={\int \int }_{Q_{b,T}}{u}_{n+1}(x,t){\tilde{\varphi }}_{j_0,k}\left(t\right){\tilde{\psi }}_{j,l}\left(x\right)\mathrm{d}x\mathrm{d}t,\hfill \\ \hfill u_{j,l}^{i,k}& ={\int \int }_{Q_{b,T}}{u}_{n+1}(x,t){\tilde{\psi }}_{i,k}\left(x\right){\tilde{\psi }}_{j,l}\left(t\right)\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

Equation (19) truncated at M would be

$$u_{n+1}^M(x,t)={\mathrm{\Psi }}^{\mathrm{T}}\left(x\right)U^{n+1}\mathrm{\Psi }\left(t\right),$$

(20)

where U is a $(2^{M+1}+m-1)\times (2^{M+1}+m-1)$ matrix. Using the notation

$$\tilde{\mathrm{\Psi }}={\left[{\tilde{\varphi }}_{j_0,-m+1},\cdots ,{\tilde{\varphi }}_{j_0,2^{j_0}-1},{\tilde{\psi }}_{j_0,-m+1},\cdots ,{\tilde{\psi }}_{M,2^j-m}\right]}^{\mathrm{T}},$$

we have

$$U^{n+1}={\int \int }_{Q_{b,T}}{u}_{n+1}^M(x,t){\tilde{\mathrm{\Psi }}}^{\mathrm{T}}\left(x\right){\tilde{\mathrm{\Psi }}}^{\mathrm{T}}\left(t\right)\mathrm{d}x\mathrm{d}t.$$

(21)

Substituting Equation (20) into the M-TS-VFLDWE (13), we have the following equation:

$$\sum _{i=1}^r{a}_i{\mathrm{\Psi }}^{\mathrm{T}}\left(x\right)U^{n+1}{D}^{{\alpha }_i(x,t)}\mathrm{\Psi }\left(t\right)-a_{r+1}{D}^{\beta }{\mathrm{\Psi }}^{\mathrm{T}}\left(x\right)U^{n+1}\mathrm{\Psi }\left(t\right)=({\mathrm{\Psi }}^{\mathrm{T}}\left(x\right)U^{n+1}\mathrm{\Psi }\left(t\right)-u_n(x,t)){\displaystyle \frac{\partial f(x,t,u_n)}{\partial u_n}},(x,t)\in (0,b)\times (0,T),$$

(22)

subject to

$$\begin{array}{cc}\hfill & {\mathrm{\Psi }}^{\mathrm{T}}\left(x\right)U^{n+1}\mathrm{\Psi }\left(0\right)=g_1\left(x\right),{\mathrm{\Psi }}^{\mathrm{T}}\left(x\right)U^{n+1}D\mathrm{\Psi }\left(0\right)=g_2\left(x\right),x\in [0,b],\hfill \\ \hfill & {\mathrm{\Psi }}^{\mathrm{T}}\left(0\right)U^{n+1}\mathrm{\Psi }\left(t\right)={\nu }_1\left(t\right),{\mathrm{\Psi }}^{\mathrm{T}}\left(b\right)U^{n+1}\mathrm{\Psi }\left(t\right)={\nu }_2\left(t\right),t\in [0,T].\hfill \end{array}$$

To enhance the computational tractability, we reformulate the problem in discrete matrix form, adopting the following notation:

$$\begin{array}{cc}\hfill & {\mathrm{\Psi }}_x={[\mathrm{\Psi }\left(x_1\right),\mathrm{\Psi }\left(x_2\right),\cdots ,\mathrm{\Psi }\left(x_{N_x}\right)]}^{\mathrm{T}},{\mathrm{\Psi }}_t=[\mathrm{\Psi }\left(t_1\right),\mathrm{\Psi }\left(t_2\right),\cdots ,\mathrm{\Psi }\left(t_{N_t}\right)],{\mathrm{\Psi }}_{X_x}=\mathrm{diag}{\left([\mathrm{\Psi }\left(x_1\right),\mathrm{\Psi }\left(x_2\right),\cdots ,\mathrm{\Psi }\left(x_{N_x}\right)]\right)}^{\mathrm{T}},\hfill \\ \hfill & U_X^{n+1}=\mathrm{diag}\left([U^{n+1},U^{n+1},\cdots ,U^{n+1}]\right),{\mathrm{\Psi }}_{X_t}^i={[D^{{\alpha }_i(x_1,t)}{\mathrm{\Psi }}_t,D^{{\alpha }_i(x_2,t)}{\mathrm{\Psi }}_t,\cdots ,D^{{\alpha }_i(x_{N_x},t)}{\mathrm{\Psi }}_t]}^{\mathrm{T}},{\left[F_1\right]}_{i,j}={\displaystyle \frac{\partial f(x_i,t_j,u_n)}{\partial u_n}},\hfill \\ \hfill & {\left[F_2\right]}_{i,j}=-u_n(x_i,t_j){\displaystyle \frac{\partial f(x_i,t_j,u_n)}{\partial u_n}},g_k={[g_k\left(x_1\right),\cdots ,g_k\left(x_{N_x}\right)]}^{\mathrm{T}},v_k=[v_k\left(t_1\right),\cdots ,v_k\left(t_{N_t}\right)],(k=1,2)\hfill \end{array}$$

where $x_i(i=1,2,\cdots ,N_x)$, $t_j(j=1,2,\cdots ,N_t)$ are the collocation points, and then we have

$$\sum _{i=1}^r{a}_i·{\mathrm{\Psi }}_{X_x}{U}_X^{n+1}{\mathrm{\Psi }}_{X_t}^i-a_{r+1}·D^{\beta }{\mathrm{\Psi }}_x{U}^{n+1}{\mathrm{\Psi }}_t-{\mathrm{\Psi }}_x{U}^{n+1}{\mathrm{\Psi }}_t\ast F_1=F_2,$$

(23)

where

$${\mathrm{\Psi }}_x{U}^{n+1}\mathrm{\Psi }\left(0\right)=g_1,{\mathrm{\Psi }}_x{U}^{n+1}D\mathrm{\Psi }\left(0\right)=g_2,$$

$${\mathrm{\Psi }}^{\mathrm{T}}\left(0\right)U^{n+1}{\mathrm{\Psi }}_t={\nu }_1,{\mathrm{\Psi }}^{\mathrm{T}}\left(b\right)U^{n+1}{\mathrm{\Psi }}_t={\nu }_2.$$

Here, ‘*’ indicates the Hadamard product [29]. We solve the algebraic equation system (23) using the fixed point method while $∥U^{n+1}-U^n∥<ϵ$ with the given tolerance $ϵ$, and substituting $U^{n+1}$ into (20) yields the approximation of $u(x,t)$.

Here, we give the step-by-step Algorithm 1 of this method.

Algorithm 1 Semi-orthogonal B-spline wavelet collocation method.

0 Preparation: Construct the functions $\mathrm{\Psi }\left(x\right),\mathrm{\Psi }\left(t\right),\tilde{\mathrm{\Psi }}\left(x\right),\tilde{\mathrm{\Psi }}\left(t\right)$ using semi-orthogonal B-spline wavelet basis functions. Give a tolerance $ϵ$ for the fixed point method. 1 Linearize the M-TS-VFNDWE (1) via quasilinearization, converting it into the M-TS-VFLDWE (13). Compute functions $D^{{\alpha }_i(x,t)}\mathrm{\Psi }\left(t\right)$ and $D^{\beta }\mathrm{\Psi }\left(x\right)$. 2 Compute $U_0$ using the given $u_0(x,t)$, and solve $U_1$ by (22). Compute $u_1(x,t)$ using (20). 3 Enter the following loop for  $n\ge 1$ do     if $∥U^n-U^{n-1}∥>ϵ$ then         $n\leftarrow n+1$         Solve $U_n$ using (22) with the previous result $U_{n-1}$     end if end for 4 Suppose $U=U^n$: compute $u(x,t)$ by (20).

4. Convergence Analysis

We now establish the absolute error bound for the SOBWCM in approximating solutions to the M-TS-VFLDWE. Since quasilinearization convergence implies convergence of the B-spline wavelet scheme (BSWS), we prove this result by directly quoting two lemmas from [27].

Lemma 1. 

Denote the dual function ${\tilde{\psi }}_{j,k}\left(t\right)$ expansed by the basis ${\left\{{\psi }_{j,l}\left(t\right)\right\}}_{l=-m+1}^{2^j-m}$ as

$${\tilde{\psi }}_{j,k}\left(t\right)=\sum _{l=-m+1}^{2^j-m}{\tilde{p}}_{l,k}^{\left(j\right)}{\psi }_{j,l}\left(t\right).$$

(24)

Then, we have

$$\left|{\tilde{p}}_{l,k}^{\left(j\right)}\right|\le C_1{2}^j,$$

(25)

where $C_1$ only depends on $j_0$ and ${\int }_0^1{\psi }_{j,k}\left(t\right){\psi }_{j,l}\left(t\right)\mathrm{d}t$.

Lemma 2. 

Denote the dual function ${\tilde{\psi }}_{j,k}\left(x\right)$ expanded by the basis ${\left\{{\psi }_{j,l}\left(x\right)\right\}}_{l=-m+1}^{2^j-m}$ as

$${\tilde{\psi }}_{j,k}\left(x\right)=\sum _{l=-m+1}^{2^j-m}{\tilde{q}}_{l,k}^{\left(j\right)}{\psi }_{j,l}\left(x\right).$$

(26)

Then, we have

$$\left|{\tilde{q}}_{l,k}^{\left(j\right)}\right|\le C_2{2}^j,$$

(27)

where $C_2$ only depends on $j_0$ and ${\int }_0^1{\psi }_{j,k}\left(x\right){\psi }_{j,l}\left(x\right)\mathrm{d}x$.

Now, we can estimate the coefficients $u_{j,l}^k$, $u_k^{j,l}$, $u_{j,l}^{i,k}$ in Equation (19).

Lemma 3. 

Let $u(x,t)$ be a smooth function defined on ${\overline{Q}}_{b,T}$, $u(·,t)\in C^m(0,T)$, and $u(x,·)\in C^{\tilde{m}}(0,b)$, which is represented using a semi-orthogonal B-spline wavelet of order m in time and order $\tilde{m}$ in space as Equation (19); then, we have

$$\left|u_{j,l}^k\right|\le C_3·{\displaystyle \frac{2^{-jm}}{m!}}{∥u∥}_{C^m(0,T)},\left|u_k^{j,l}\right|\le C_4·{\displaystyle \frac{2^{-j\tilde{m}}}{\tilde{m}!}}{∥u∥}_{C^{\tilde{m}}(0,b)},\left|u_{j,l}^{i,k}\right|\le C_5·{\displaystyle \frac{2^{-i\tilde{m}-jm}}{m!\tilde{m}!}}{∥u∥}_{C^{\tilde{m},m}\left(Q_{b,T}\right)},$$

where $C_3$ depends on $j_0$ and ${\int }_0^T{\psi }_{j,k}\left(t\right){\psi }_{j,l}\left(t\right)\mathrm{d}t$, $C_4$ depends on $j_0$ and ${\int }_0^b{\psi }_{j,k}\left(x\right){\psi }_{j,l}\left(x\right)\mathrm{d}x$, $C_5$ depends on $j_0$, ${\int }_0^T{\psi }_{j,k}\left(t\right){\psi }_{j,l}\left(t\right)\mathrm{d}t$ and ${\int }_0^b{\psi }_{i,k}\left(x\right){\psi }_{i,l}\left(x\right)\mathrm{d}x$.

Proof. 

Firstly, estimate $u_{j,l}^k$. Using the expansion

$${\tilde{\varphi }}_{j,k}\left(x\right)=\sum _{l=-m+1}^{2^j-m}{\tilde{\rho }}_{l,k}^{\left(j\right)}{\varphi }_{j,l}\left(x\right),$$

(28)

and according to Lemma 1, we have

$$u_{j,l}^k=\sum _{s_1=-\tilde{m}+1}^{2^{j_0}-1}\sum _{s_2=-m+1}^{2^j-m}{\int \int }_{Q_{b,T}}{\tilde{\rho }}_{s_1,k}^{\left(j_0\right)}{\tilde{p}}_{s_2,l}^{\left(j\right)}u(x,t){\varphi }_{j_0,s_1}\left(x\right){\psi }_{j,s_2}\left(t\right)\mathrm{d}x\mathrm{d}t,$$

(29)

and thus

$$\left|u_{j,l}^k\right|\le \left|{\tilde{\rho }}_{s_1,k}^{\left(j_0\right)}\right|\left|{\tilde{p}}_{s_2,l}^{\left(j\right)}\right|\sum _{s_1=-\tilde{m}+1}^{2^{j_0}-1}\sum _{s_2=-m+1}^{2^j-m}\left|{\int \int }_{Q_{b,T}}u(x,t){\varphi }_{j_0,s_1}\left(x\right){\psi }_{j,s_2}\left(t\right)\mathrm{d}x\mathrm{d}t\right|.$$

(30)

According to the Taylor expansion,

$$u(x,t)=u(0,0)+x{\displaystyle \frac{\partial u(0,0)}{\partial x}}+t{\displaystyle \frac{\partial u(0,0)}{\partial t}}+\cdots +{\displaystyle \frac{1}{n!}}{(x{\displaystyle \frac{\partial }{\partial x}}+t{\displaystyle \frac{\partial }{\partial t}})}^{n}u(0,0)+{\displaystyle \frac{1}{(n+1)!}}{(x{\displaystyle \frac{\partial }{\partial x}}+t{\displaystyle \frac{\partial }{\partial t}})}^{n+1}u(\xi ,\eta )$$

(31)

where ${(x{\displaystyle \frac{\partial }{\partial x}}+t{\displaystyle \frac{\partial }{\partial t}})}^n={\displaystyle \sum _{s_3=0}^n}{C}_n^{s_3}{x}^{s_3}{t}^{n-s_3}{\displaystyle \frac{{\partial }^{n}u}{\partial x^{s_3}\partial t^{n-s_3}}}{|}_{(0,0)}$, $C_n^{s_3}={\displaystyle \frac{n!}{s_3!(n-s_3)!}}$, and $(\xi ,\eta )\in {\overline{Q}}_{b,T}$. Then, using Properties 1 and 2, we have

$$\begin{array}{cc}\hfill & \sum _{s_1=-\tilde{m}+1}^{2^{j_0}-1}\sum _{s_2=-m+1}^{2^j-m}\left|{\int \int }_{Q_{b,T}}u(x,t){\varphi }_{j_0,s_1}\left(x\right){\psi }_{j,s_2}\left(t\right)\mathrm{d}x\mathrm{d}t\right|\hfill \\ \hfill =& \sum _{s_1=-\tilde{m}+1}^{2^{j_0}-1}\sum _{s_2=-m+1}^{2^j-m}\left|{\int \int }_{Q_{b,T}}{t}^m{\displaystyle \frac{{\partial }^{m}u({\xi }_1,{\eta }_1)}{m!\partial t^m}}{\psi }_{j,s_2}\left(t\right){\varphi }_{j_0,s_1}\left(x\right)\mathrm{d}x\mathrm{d}t\right|\hfill \\ \hfill \le & {\displaystyle \frac{1}{m!}}\left|{\displaystyle \frac{{\partial }^{m}u({\xi }_1,{\eta }_1)}{\partial t^m}}\right|\sum _{s_1=-m+1}^{2^{j_0}-1}\left|{\int }_0^b{\varphi }_{j_0,s_1}\left(x\right)\mathrm{d}x\right|\sum _{s_2=-m+1}^{2^j-m}\left|{\int }_0^T{t}^m{\psi }_{j,s_2}\left(t\right)\mathrm{d}t\right|,\hfill \end{array}$$

(32)

where $({\xi }_1,{\eta }_1)\in {\overline{Q}}_{b,T}$. Using the result of Theorem 1 in [27], we have

$$\sum _{s_2=-m+1}^{2^j-m}\left|{\int }_0^T{t}^m{\psi }_{j,s_2}\left(t\right)\mathrm{d}t\right|\le C_6·{\displaystyle \frac{2^{-j(m+1)}}{m!}}{∥u∥}_{C^m(0,T)},$$

(33)

where $C_6$ depends on $j_0$ and ${\int }_0^T{\psi }_{j,k}\left(t\right){\psi }_{j,l}\left(t\right)\mathrm{d}t$. According to Equations (30), (32), and (33) and Lemma 1, we have

$$\left|u_{j,l}^k\right|\le C_3·{\displaystyle \frac{2^{-jm}}{m!}}{∥u∥}_{C^m(0,T)},$$

(34)

where $C_3$ depends on $j_0$ and ${\int }_0^T{\psi }_{j,k}\left(t\right){\psi }_{j,l}\left(t\right)\mathrm{d}t$.

Secondly, similarly, we estimate $u_k^{j,l}$ as follows:

$$\left|u_k^{j,l}\right|\le C_4·{\displaystyle \frac{2^{-j\tilde{m}}}{\tilde{m}!}}{∥u∥}_{C^{\tilde{m}}(0,b)},$$

(35)

where $C_4$ depends on $j_0$ and ${\int }_0^b{\psi }_{j,k}\left(x\right){\psi }_{j,l}\left(x\right)\mathrm{d}x$.

Thirdly, we estimate $u_{j,l}^{i,k}$. Using Lemmas 1 and 2, we have

$$\left|u_{j,l}^{i,k}\right|\le \left|{\tilde{q}}_{s_1,k}^{\left(i\right)}\right|\left|{\tilde{p}}_{s_2,l}^{\left(j\right)}\right|\sum _{s_1=-\tilde{m}+1}^{2^i-\tilde{m}}\sum _{s_2=-m+1}^{2^j-m}\left|\underset{Q_{b,T}}{\int \int }u(x,t){\psi }_{i,s_1}\left(x\right){\psi }_{j,s_2}\left(t\right)\mathrm{d}x\mathrm{d}t\right|.$$

(36)

Using the notation

$$I=\sum _{s_1=-\tilde{m}+1}^{2^i-\tilde{m}}\sum _{s_2=-m+1}^{2^j-m}\left|\underset{Q_{b,T}}{\int \int }u(x,t){\psi }_{i,s_1}\left(x\right){\psi }_{j,s_2}\left(t\right)\mathrm{d}x\mathrm{d}t\right|,$$

and according to Taylor expansion (31) and Property 1, we have

$$\begin{array}{cc}\hfill I& =\sum _{s_1=-\tilde{m}+1}^{2^i-\tilde{m}}\sum _{s_2=-m+1}^{2^j-m}\left|{\int \int }_{Q_{b,T}}{t}^m{x}^{\tilde{m}}{\displaystyle \frac{{\partial }^{\tilde{m}+m}u({\xi }_2,{\eta }_2)}{m!\tilde{m}!\partial t^m\partial x^{\tilde{m}}}}{\psi }_{i,s_1}\left(x\right){\psi }_{j,s_2}\left(t\right)\mathrm{d}x\mathrm{d}t\right|\hfill \\ \hfill & \le \left|{\displaystyle \frac{{\partial }^{m+\tilde{m}}u({\xi }_2,{\eta }_2)}{m!\tilde{m}!\partial t^m\partial x^{\tilde{m}}}}\right|\sum _{s_1=-\tilde{m}+1}^{2^i-\tilde{m}}\left|{\int }_0^b{x}^{\tilde{m}}{\psi }_{i,s_1}\left(x\right)\mathrm{d}x\right|·\sum _{s_2=-m+1}^{2^j-m}\left|{\int }_0^T{t}^m{\psi }_{j,s_2}\left(t\right)\mathrm{d}t\right|\hfill \\ \hfill & \le C_7·{\displaystyle \frac{2^{-i(\tilde{m}+1)-j(m+1)}}{m!\tilde{m}!}}{∥u∥}_{C^{\tilde{m},m}\left(Q_{b,T}\right)}.\hfill \end{array}$$

(37)

According to Lemmas 1 and 2, we have

$$\left|u_{j,l}^{i,k}\right|\le C_5·{\displaystyle \frac{2^{-i\tilde{m}-jm}}{m!\tilde{m}!}}{∥u∥}_{C^{\tilde{m},m}\left(Q_{b,T}\right)}.$$

(38)

where $C_5$ depends on $j_0$, ${\int }_0^T{\psi }_{j,k}\left(t\right){\psi }_{j,l}\left(t\right)\mathrm{d}t$, and ${\int }_0^b{\psi }_{i,k}\left(x\right){\psi }_{i,l}\left(x\right)\mathrm{d}x$. □

Based on Lemma 3, we can prove the main theorem as follows.

Theorem 1. 

Let $u(x,t)$ be a smooth function defined on ${\overline{Q}}_{b,T}$, $u(·,t)\in C^m(0,T)$ and $u(x,·)\in C^m(0,b)$, which is approximated using the SOBWCM of order m in both time and space; the truncation error at M is

$${∥E_M(x,t)∥}_{L_{\infty }\left(Q_{b,T}\right)}\le C·{\displaystyle \frac{2^{-(m-1)(M+1)}}{(m-1)!}}\left[1+{\displaystyle \frac{2^{-(m-1)(M+1)}}{(m-1)!}}\right]{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)},$$

(39)

where C depends on $j_0$, ${\int }_0^T{\psi }_{j,k}\left(t\right){\psi }_{j,l}\left(t\right)\mathrm{d}t$ and ${\int }_0^b{\psi }_{i,k}\left(x\right){\psi }_{i,l}\left(x\right)\mathrm{d}x$.

Proof. 

Using Equations (19) and (20), we have

$$E_M(x,t)=u_{r+1}(x,t)-u_{r+1}^M(x,t)=e_1+e_2+e_3+e_4+e_5,$$

(40)

where

$$e_1=\sum _{k=-m+1}^{2^{j_0}-1}\sum _{j=M+1}^{\infty }\sum _{l=-m+1}^{2^j-m}{u}_{j,l}^k{\varphi }_{j_0,k}\left(x\right){\psi }_{j,l}\left(t\right),$$

$$e_2=\sum _{k=-m+1}^{2^{j_0}-1}\sum _{j=M+1}^{\infty }\sum _{l=-m+1}^{2^j-m}{u}_k^{j,l}{\varphi }_{j_0,k}\left(t\right){\psi }_{j,l}\left(x\right),$$

$$e_3=\sum _{i=j_0}^M\sum _{k=-m+1}^{2^i-m}\sum _{j=M+1}^{\infty }\sum _{l=-m+1}^{2^j-m}{u}_{j,l}^{i,k}{\psi }_{i,k}\left(x\right){\psi }_{j,l}\left(t\right),$$

$$e_4=\sum _{i=M+1}^{\infty }\sum _{k=-m+1}^{2^i-m}\sum _{j=j_0}^M\sum _{l=-m+1}^{2^j-m}{u}_{j,l}^{i,k}{\psi }_{i,k}\left(x\right){\psi }_{j,l}\left(t\right),$$

$$e_5=\sum _{i=M+1}^{\infty }\sum _{k=-m+1}^{2^i-m}\sum _{j=M+1}^{\infty }\sum _{l=-m+1}^{2^j-m}{u}_{j,l}^{i,k}{\psi }_{i,k}\left(x\right){\psi }_{j,l}\left(t\right).$$

Using Lemma 3, we have

$$\begin{array}{cc}\hfill \left|e_1\right|& \le \sum _{k=-m+1}^{2^{j_0}-1}\sum _{j=M+1}^{\infty }\sum _{l=-m+1}^{2^j-m}\left|u_{j,l}^k\right|\left|{\varphi }_{j_0,k}\left(x\right)\right|\left|{\psi }_{j,l}\left(t\right)\right|\le \sum _{k=-m+1}^{2^{j_0}-1}\sum _{j=M+1}^{\infty }\sum _{l=-m+1}^{2^j-m}\left|u_{j,l}^k\right|\hfill \\ \hfill & \le \sum _{k=-m+1}^{2^{j_0}-1}\sum _{j=M+1}^{\infty }\sum _{l=-m+1}^{2^j-m}{C}_3·{\displaystyle \frac{2^{-jm}}{m!}}{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)}\le \sum _{k=-m+1}^{2^{j_0}-1}\sum _{j=M+1}^{\infty }{C}_3·{\displaystyle \frac{2^{-j(m-1)}}{m!}}{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)}\hfill \\ \hfill & =(2j_0+m-1)\sum _{j=M+1}^{\infty }{C}_3·{\displaystyle \frac{2^{-j(m-1)}}{m!}}{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)}\le C·{\displaystyle \frac{2^{-(M+1)(m-1)}}{(m-1)!}}{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)}.\hfill \end{array}$$

(41)

Similarly, we have

$$\begin{array}{cc}\hfill & \left|e_2\right|\le \sum _{k=-m+1}^{2^{j_0}-1}\sum _{j=M+1}^{\infty }{C}_4·{\displaystyle \frac{2^{-j(m-1)}}{m!}}{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)}\le C·{\displaystyle \frac{2^{-(M+1)(m-1)}}{(m-1)!}}{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)},\hfill \\ \hfill & \left|e_3\right|\le \sum _{i=j_0}^M\sum _{k=-m+1}^{2^i-m}\sum _{j=M+1}^{\infty }{C}_5·{\displaystyle \frac{2^{-(i+j)(m-1)-i}}{m!}}{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)}\le C·{\displaystyle \frac{2^{-(M+1)(m-1)}}{(m-1)!}}{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)},\hfill \\ \hfill & \left|e_4\right|\le \sum _{i=M+1}^{\infty }\sum _{j=j_0}^M\sum _{l=-m+1}^{2^j-m}{C}_5·{\displaystyle \frac{2^{-(i+j)(m-1)-i}}{m!}}{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)}\le C·{\displaystyle \frac{2^{-(M+1)(m-1)}}{(m-1)!}}{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)},\hfill \\ \hfill & \left|e_5\right|\le \sum _{i=M+1}^{\infty }\sum _{j=M+1}^{\infty }{C}_5·{\displaystyle \frac{2^{-(i+j)(m-1)}}{m!}}{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)}\le C·{\displaystyle \frac{2^{-2(M+1)(m-1)}}{(m-1)!}}{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)}.\hfill \end{array}$$

Finally, we can obtain

$${∥E_M(x,t)∥}_{L_{\infty }\left(Q_{b,T}\right)}\le C·{\displaystyle \frac{2^{-(m-1)(M+1)}}{(m-1)!}}\left[1+{\displaystyle \frac{2^{-(m-1)(M+1)}}{(m-1)!}}\right]{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)},$$

(42)

which ends the proof. □

Theorem 2. 

Let $u(x,t)$ be a smooth function defined on ${\overline{Q}}_{b,T}$, $u(·,t)\in C^m(0,T)$, and $u(x,·)\in C^m(0,b)$, which is approximated using the SOBWCM of order m in both time and space; the truncation error at M is

$${∥E_M(x,t)∥}_{L_2\left(Q_{b,T}\right)}\le C·{\displaystyle \frac{2^{-(m-1)(M+1)}}{(m-1)!}}\left[1+{\displaystyle \frac{2^{-(m-1)(M+1)}}{(m-1)!}}\right]{∥u∥}_{C^{m,m}\left(Q_{b,T}\right)},$$

(43)

where C depends on $j_0$, ${\int }_0^T{\psi }_{j,k}\left(t\right){\psi }_{j,l}\left(t\right)\mathrm{d}t$, and ${\int }_0^b{\psi }_{i,k}\left(x\right){\psi }_{i,l}\left(x\right)\mathrm{d}x$.

Proof. 

Due to the fact that

$$\begin{array}{c}\hfill {∥E_M(x,t)∥}_{L_2\left(Q_{b,T}\right)}^2={\int \int }_{Q_{b,T}}{\left[E_M(x,t)\right]}^2\mathrm{d}x\mathrm{d}t\le bT{∥E_M(x,t)∥}_{L_{\infty }\left(Q_{b,T}\right)}^2,\end{array}$$

we can obtain the conclusion. □

Remark 1. 

If $M\to \infty$ or $m\to \infty$, the error ${∥E_M(x,t)∥}_{L_{\infty }\left(Q_{b,T}\right)}\to 0$ and ${∥E_M(x,t)∥}_{L_2\left(Q_{b,T}\right)}\to 0$. This fact implies the convergence of approximating the M-TS-VFLDWE using the SOBWCM, that is, $u_{n+1}^M(x,t)$ converges to $u_{n+1}(x,t)$ as $M\to \infty$ or $m\to \infty$. According to the convergence analysis of the quasilinearization method in [30], $u_{n+1}(x,t)$ converges to $u(x,t)$ as $n\to \infty$. Therefore, the B-spline wavelet scheme is convergent.

5. Numerical Results

In this section, several numerical examples are given to compare our proposed scheme with the existing methods in the literature [18,23].

Example 1. 

Consider the following M-TS-VFLDWE [18]:

$$D_t^{\alpha (x,t)}u(x,t)+\sum _{j=1}^4{d}_j{D}_t^{{\alpha }_j(x,t)}u(x,t)+u_t(x,t)={\displaystyle \frac{1}{3}}{u}_{xx}(x,t)+f(x,t),(x,t)\in (0,1)\times (0,1)$$

subject to the following conditions:

$$\begin{array}{cc}\hfill & u(x,0)=0,{\left.{\displaystyle \frac{\partial u(x,t)}{\partial t}}\right|}_{t=0}=0,x\in [0,1],\hfill \\ \hfill & u(0,t)=0,u(1,t)=t^{3}sin\left(\pi \right),t\in [0,1],\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & f(x,t)=\left({\displaystyle \frac{6}{\mathrm{\Gamma }(4-\alpha (x,t\left)\right)}}{t}^{-\alpha (x,t)}+\sum _{j=1}^4{d}_j{\displaystyle \frac{6}{\mathrm{\Gamma }(4-{\alpha }_j(x,t))}}{t}^{-{\alpha }_j(x,t)}\right)t^{3}sin\left(\pi x\right)+\left(3+{\displaystyle \frac{{\pi }^2}{3}}t\right)t^{2}sin\left(\pi x\right),\hfill \\ \hfill & \alpha (x,t)=2-0.3e^{-xt},{\alpha }_1(x,t)=2-0.4e^{-xt},{\alpha }_2(x,t)=2-0.5e^{-xt},{\alpha }_3(x,t)=2-0.6e^{-xt},\hfill \\ \hfill & {\alpha }_4(x,t)=2-0.7e^{-xt},d_j={\displaystyle \frac{1}{j+1}}(j=1,2,3,4).\hfill \end{array}$$

The analytical solution for the above problem is $u(x,t)=t^{3}sin\left(\pi x\right)$.

Here, we solve this problem based on the BSWS. Figure 1 presents the numerical solution and the absolute error for the proposed scheme with $m=6$ and $M=4$.

Table 1. The absolute error of BSWS and CWM [18] for Example 1 while $m=4$.

	BSWS				CWM
$({\mathbf{x}}_{\mathbf{i}},{\mathbf{t}}_{\mathbf{i}})$	$\mathbf{M}=\mathbf{3}$	$\mathbf{M}=\mathbf{4}$	$\mathbf{M}=\mathbf{5}$	$\mathbf{M}=\mathbf{6}$	$\widehat{\mathbf{m}}=\mathbf{4}$	$\widehat{\mathbf{m}}=\mathbf{7}$	$\widehat{\mathbf{m}}=\mathbf{10}$
(0.1,0.1)	4.4070 $\times {10}^{-8}$	1.1338 $\times {10}^{-9}$	4.7947 $\times {10}^{-11}$	4.1355 $\times {10}^{-14}$	3.2005 $\times {10}^{-2}$	1.3326 $\times {10}^{-4}$	9.4321 $\times {10}^{-8}$
(0.2,0.2)	3.0324 $\times {10}^{-8}$	5.8055 $\times {10}^{-9}$	9.5953 $\times {10}^{-11}$	2.7010 $\times {10}^{-11}$	3.1341 $\times {10}^{-2}$	1.9025 $\times {10}^{-4}$	3.9248 $\times {10}^{-8}$
(0.3,0.3)	7.6473 $\times {10}^{-8}$	2.9594 $\times {10}^{-8}$	7.9989 $\times {10}^{-10}$	1.1969 $\times {10}^{-10}$	5.3652 $\times {10}^{-3}$	5.4423 $\times {10}^{-5}$	5.1393 $\times {10}^{-9}$
(0.4,0.4)	9.8438 $\times {10}^{-7}$	2.8004 $\times {10}^{-8}$	5.4350 $\times {10}^{-9}$	1.3484 $\times {10}^{-10}$	3.0927 $\times {10}^{-2}$	3.3837 $\times {10}^{-6}$	3.2847 $\times {10}^{-8}$
(0.5,0.5)	3.6230 $\times {10}^{-6}$	2.5739 $\times {10}^{-7}$	1.6525 $\times {10}^{-8}$	1.0421 $\times {10}^{-9}$	2.6813 $\times {10}^{-2}$	3.7564 $\times {10}^{-5}$	8.2481 $\times {10}^{-9}$
(0.6,0.6)	3.8095 $\times {10}^{-6}$	9.7979 $\times {10}^{-8}$	1.8443 $\times {10}^{-8}$	4.5529 $\times {10}^{-10}$	1.5741 $\times {10}^{-2}$	2.0331 $\times {10}^{-6}$	3.1398 $\times {10}^{-8}$
(0.7,0.7)	1.3502 $\times {10}^{-6}$	4.1117 $\times {10}^{-7}$	1.0687 $\times {10}^{-8}$	1.5270 $\times {10}^{-9}$	4.9913 $\times {10}^{-3}$	5.4594 $\times {10}^{-5}$	1.0362 $\times {10}^{-8}$
(0.8,0.8)	4.1153 $\times {10}^{-6}$	3.7428 $\times {10}^{-7}$	7.8339 $\times {10}^{-9}$	1.7238 $\times {10}^{-9}$	8.3473 $\times {10}^{-3}$	2.5894 $\times {10}^{-5}$	3.1942 $\times {10}^{-8}$
(0.9,0.9)	7.1112 $\times {10}^{-6}$	2.2003 $\times {10}^{-7}$	1.7268 $\times {10}^{-8}$	3.4461 $\times {10}^{-10}$	5.1754 $\times {10}^{-4}$	9.4243 $\times {10}^{-7}$	1.4991 $\times {10}^{-8}$

shows the absolute error when $m=4$ with various values of M and a comparison of the errors obtained by utilizing the BSWS and utilizing the Chebyshev wavelet method (CWM) in [18], where $\widehat{m}$ is the order of the Chebyshev wavelet. From these results, we can conclude that this proposed approach has a higher accuracy compared with the CWM. Moreover, the data in Table 1 imply that the absolute errors are reduced by increasing M.

In addition,

Table 2. The $L_2$- and $L_{\infty }$-error of BSWS with various values of m for Example 1.

Error	$\mathit{m}=4$	$\mathit{m}=5$	$\mathit{m}=6$	$\mathit{m}=7$
$L_2$-error	3.6721 $\times {10}^{-7}$	2.1475 $\times {10}^{-8}$	3.1287 $\times {10}^{-10}$	2.5657 $\times {10}^{-10}$
$L_{\infty }$-error	2.1118 $\times {10}^{-6}$	1.7879 $\times {10}^{-7}$	8.0679 $\times {10}^{-9}$	5.9673 $\times {10}^{-9}$

and Figure 2 presents the $L_2$- and $L_{\infty }$-error approximated using the BSWS with $M=4$ when $m=4,5,6$. In Figure 3, we show the absolute error using this proposed approach, with $M=4$ based on various values of m. From these results, we can find that the $L_2$- and $L_{\infty }$-errors decrease with an increase in m, which verifies the convergence analysis in Section 4.

Example 2. 

Consider the following initial boundary problem [23]:

$$\begin{array}{cc}\hfill & D_t^{\alpha \left(t\right)}u(x,t)-{\displaystyle \frac{\mathrm{\Gamma }(3-\beta )}{\mathrm{\Gamma }\left(3\right)}}{D}_x^{\beta }u(x,t)+\mu sin\left(u(x,t)\right)=f(x,t),(x,t)\in (-1,1)\times (0,1)\hfill \\ \hfill & u(x,0)=0,{\left.{\displaystyle \frac{\partial u(x,t)}{\partial t}}\right|}_{t=0}=0,x\in [-1,1],\hfill \\ \hfill & u(-1,t)=0,u(1,t)=t^{3}sin\left(\pi \right)=0,t\in [0,1],\hfill \end{array}$$

The given analytical solution for the problem is $u(x,t)=t^5{(x+1)}^3(x-1)$. We utilize the BSWS with $m=6$ and $M=4$ to deal with the above problem in two cases, i.e., $\mu =0,1$. The absolute errors in the approximate results of the BSWS for $\alpha \left(t\right)={\displaystyle \frac{t^2+3}{2}}$ and different β when $\mu =0$ and $\mu =1$ are described in Figure 4 and Figure 5, respectively. The results show that the absolute error of the BSWS can achieve 1 × 10⁻¹². To further verify the effectiveness of this proposed method, we apply it to solving the above problem for different choices of $a\left(t\right)$ and β and compare the results with those obtained by using the local discontinuous Galerkin method (LDGM) [23]. Their corresponding $L_2$- and $L_{\infty }$-errors are presented in

Table 3. The $L_2-$ and $L_{\infty }-$error of the BSWS and the LDGM [23] when $\mu =0$ for Example 2.

		BSWS		LDGM
$\mathbf{\alpha }\left(\mathbf{t}\right)$	$\mathbf{\beta }$	${\mathbf{L}}_{\mathbf{2}}$-Error	${\mathbf{L}}_{\infty }$-Error	${\mathbf{L}}_{\mathbf{2}}$-Error	${\mathbf{L}}_{\infty }$-Error
1.5	1.5	2.9439 $\times {10}^{-14}$	5.3268 $\times {10}^{-14}$	5.01 $\times {10}^{-6}$	5.39 $\times {10}^{-5}$
1.2	1.8	5.4306 $\times {10}^{-15}$	1.0356 $\times {10}^{-14}$	5.65 $\times {10}^{-6}$	7.56 $\times {10}^{-6}$
$2-t$	1.1	5.7490 $\times {10}^{-15}$	1.6542 $\times {10}^{-14}$	6.27 $\times {10}^{-6}$	6.83 $\times {10}^{-6}$
$2-t$	1.5	3.6974 $\times {10}^{-15}$	8.7014 $\times {10}^{-15}$	4.76 $\times {10}^{-6}$	6.64 $\times {10}^{-6}$
${\displaystyle \frac{-log(t+1)+3}{2}}$	1.8	7.8701 $\times {10}^{-15}$	1.5864 $\times {10}^{-14}$	5.01 $\times {10}^{-6}$	6.85 $\times {10}^{-5}$
${\displaystyle \frac{sin\left(t\right)+6}{4}}$	1.1	4.7225 $\times {10}^{-13}$	9.0807 $\times {10}^{-13}$	3.78 $\times {10}^{-3}$	3.83 $\times {10}^{-3}$
${\displaystyle \frac{sin\left(t\right)+6}{4}}$	1.7	5.4626 $\times {10}^{-14}$	1.0390 $\times {10}^{-13}$	3.58 $\times {10}^{-3}$	3.75 $\times {10}^{-3}$
${\displaystyle \frac{t^2+3}{2}}$	1.1	4.7200 $\times {10}^{-13}$	2.5482 $\times {10}^{-12}$	4.16 $\times {10}^{-3}$	4.27 $\times {10}^{-3}$
${\displaystyle \frac{t^2+3}{2}}$	1.7	1.2201 $\times {10}^{-13}$	3.7397 $\times {10}^{-13}$	4.00 $\times {10}^{-3}$	4.19 $\times {10}^{-3}$

and

Table 4. The $L_2$- and $L_{\infty }$-error of the BSWS and the LDGM [23] when $\mu =1$ for Example 2.

		BSWS		LDGM
$\mathbf{\alpha }\left(\mathbf{t}\right)$	$\mathbf{\beta }$	${\mathbf{L}}_{\mathbf{2}}$-Error	${\mathbf{L}}_{\infty }$-Error	${\mathbf{L}}_{\mathbf{2}}$-Error	${\mathbf{L}}_{\infty }$-Error
1.8	1.5	8.6691 $\times {10}^{-14}$	2.1058 $\times {10}^{-13}$	3.97 $\times {10}^{-6}$	5.31 $\times {10}^{-6}$
$2-t$	1.3	4.8126 $\times {10}^{-15}$	1.2434 $\times {10}^{-14}$	2.40 $\times {10}^{-3}$	2.36 $\times {10}^{-3}$
$2-t$	1.8	3.0802 $\times {10}^{-15}$	6.5317 $\times {10}^{-15}$	2.74 $\times {10}^{-3}$	2.77 $\times {10}^{-3}$
${\displaystyle \frac{exp\left(t\right)}{2}}$	1.3	7.6324 $\times {10}^{-15}$	2.6229 $\times {10}^{-14}$	2.44 $\times {10}^{-3}$	2.41 $\times {10}^{-3}$
${\displaystyle \frac{exp\left(t\right)}{2}}$	1.8	7.6085 $\times {10}^{-15}$	3.8211 $\times {10}^{-14}$	2.76 $\times {10}^{-3}$	2.80 $\times {10}^{-3}$
${\displaystyle \frac{t^2+3}{2}}$	1.3	2.2781 $\times {10}^{-13}$	6.8752 $\times {10}^{-13}$	3.65 $\times {10}^{-6}$	5.05 $\times {10}^{-6}$
${\displaystyle \frac{t^2+3}{2}}$	1.8	2.2781 $\times {10}^{-13}$	6.8752 $\times {10}^{-13}$	3.95 $\times {10}^{-6}$	5.62 $\times {10}^{-6}$
${\displaystyle \frac{cos\left(t\right)+2}{2}}$	1.5	2.3111 $\times {10}^{-14}$	3.9605 $\times {10}^{-14}$	4.41 $\times {10}^{-6}$	6.22 $\times {10}^{-6}$

in the case of $\mu =0$ and $\mu =1$, respectively. Investigating these data, it is quite obvious that the BSWS is more accurate and valid compared with the LDGM for every choice of $a\left(t\right)$ and β with different μ values.

6. Conclusions

This study presents a B-spline wavelet scheme for solving multi-term time–space variable-order fractional nonlinear diffusion-wave equations. The proposed approach effectively transforms the original nonlinear problem into a system of linear algebraic equations by constructing an efficient operational matrix formulation. Both theoretical analysis and numerical experiments demonstrate the convergence of the method, with the latter further confirming its high computational accuracy and overall effectiveness.

Despite these advantages, the method is subject to certain limitations. Although computationally more efficient than some existing techniques, the cost increases at very high resolution levels M due to the growing size of the operational matrix. Moreover, the current formulation is mainly applicable to rectangular domains; extending it to complex geometries would require additional strategies, such as constructing the wavelet bases on unstructured grids or integrating domain decomposition methods. These challenges offer promising directions for future research.