Spectral Collocation Technique for Solving Two-Dimensional Multi-Term Time Fractional Viscoelastic Non-Newtonian Fluid Model

Abstract

Applications of non-Newtonian fluids have been widespread across industries, accompanied by theoretical developments in engineering and mathematics. This paper studies a two-dimensional multi-term time fractional viscoelastic non-Newtonian fluid model by using two autonomous consecutive spectral collocation strategies. A modification of the spectral approach is implemented, leading to an algebraic system of equations able to obtain an approximate symmetric solution for the model. Numerical examples illustrate the effectiveness of the technique in terms of accuracy and convergence.

1. Introduction

Fractional calculus has been proven more effective than the classical integer calculus for describing a variety of natural scenarios [1], and emerged as a crucial tool within the scope of mathematical modeling [2,3], with a plethora of applications [4,5]. Indeed, several anomalous phenomena and complex systems in multiple areas have been studied using fractional differential equations [6], since they have advantages over the integer-order ones for describing real processes with memory [7].

A Newtonian fluid is a type of fluid that obeys Newton’s law of viscosity, which states that the shear stress of a fluid is directly proportional to its rate of shear strain. In other words, the viscosity of a Newtonian fluid remains constant regardless of the applied shear stress or strain rate. This makes Newtonian fluids relatively simple to model and analyze mathematically. Examples of Newtonian fluids include water, air, and most oils. However, many real-world fluids exhibit non-Newtonian behavior, where the viscosity changes with the applied stress or strain rate. These fluids require more complex models to describe their behavior.

Non-Newtonian fluid models are used to describe fluids that do not follow the traditional Newtonian model of viscosity. These fluids often exhibit complex behavior, such as shear-thinning or shear-thickening, where the viscosity of the fluid changes depending on the applied force. There are many different models used to describe these types of fluids, including the Power-law model, the Herschel-Bulkley model, and the Casson model. These models are often used in engineering and scientific applications, such as in the design of non-Newtonian fluids for use in food products, cosmetics, and even in hydraulic fracturing fluids for oil and gas exploration. By understanding the behavior of non-Newtonian fluids, we can better design and optimize processes that rely on these materials.

Over the last few decades, non-Newtonian fluids have been widely used in engineering applications. Non-Newtonian fluids’ constitutive equations are significantly more complex than those describing Newtonian ones. Fractional differential equations have been adopted to describe non-Newtonian fluids, since they are well-suited for dealing with viscoelastic properties [8], yielding results in accordance to experimental data [9,10].

The flow of an incompressible Oldroyd-B fluid [11] bounded by two rigid plates is described by the following model, which is one of the most significant:

$$ϵ\frac{{\partial }^{\nu +1}\mathbb{Y}}{\partial {\eta }^{\nu +1}}+\frac{\partial \mathbb{Y}}{\partial \eta }=\lambda \Delta \mathbb{Y}+\lambda \beta \frac{{\partial }^{{\nu }_3}\Delta \mathbb{Y}}{\partial {\eta }^{{\nu }_3}},$$

(1)

where $\mathbb{Y}$ is the velocity field (meter/second), $ϵ$ represents the relaxation time (second), $\beta$ is the retardation time (second), $\lambda =\frac{\mu }{\rho }$ and $\mu$ stand for the dynamic viscosity (Pascal·second), and $\rho$ is the density of the fluid.

Sutton [12] summarized its fundamental electromagnetic properties. When exposed to a magnetic field, under the assumption of low magnetic Reynolds number, the flow of a Oldroyd-B fluid between two infinite parallel rigid plates is challenging to model. Khan et al. [13] investigated the generalized Oldroyd-B fluid in a porous medium with the influence of Hall current. Zheng et al. [14] explored the interaction of two plates with slip boundaries and achieved the exact solution for the flow magnetohydrodynamics by some transform techniques.

The exact solution of the generalized Oldroyd-B fluid was obtained using generalized G- or H-functions [14,15]. Fetecau et al. [16] derived an exact solution for a two-dimensional fluid model. However, because multi-term fractional fluid models are difficult to solve analytically, many researchers studied alternative approaches [17,18] based on numerical methods. Finite difference [19,20], finite element [21,22], finite volume [23,24], and spectral methods [25,26] have been the most commonly used for solving fractional equations.

In the last four decades, spectral methods [27,28,29] have been widely used in several areas. Firstly, techniques based on Fourier expansion were applied, namely for dealing with simple geometric area and periodic boundary conditions. Afterwards, sophisticated techniques were developed, and spectral methods emerged as powerful tools to solve different kinds of problems. Indeed, due to their thoroughness and exponential rates of convergence, spectral methods reveal superiority when compared with their counterparts. They include the collocation [30,31], tau [32], Galerkin [33], and Petrov-Galerkin [34] variants. Regardless of the technique adopted, spectral methods express the problem as a finite series of several functions. Then, the coefficients are chosen such that the absolute error is minimized as well as possible. In the spectral collocation technique [35,36], the numerical solution is implemented in order to almost satisfying the problem. On the other hand, the residuals are permitted to be zero at the collocation points.

Spectral schemes can be applied to a wide range of problems, including integral and integro-differential equations [37], fractional differential equations [38], optimal control, and variational problems. However, global spectral schemes [39] have received little attention, when compared to other classical methods, as finite difference and finite element, for solving fractional order differential equations. Although the approach has disadvantages like the inability to represent physical processes in spectral space and parallelizing difficulties on distributed memory computers, it is highly accurate, converges quickly, and is straightforward.

In this paper we propose a spectral method for approximating a fractional non-Newtonian fluid model (FNNFM). The solution of the model is expressed as a limited expansion of shifted Legendre polynomials for the independent variables, and the residuals are estimated at the shifted Legendre quadrature points. The proposed collocation scheme is investigated for both temporal and spatial discretizations. The shifted Legendre collocation method is proposed, with a suitable modification for treating the intial-boundary, for spatial and temporal discretization. This treatment improves greatly the accuracy of the scheme. The solutions of the problem are approximated as a finite expansion of shifted Legendre polynomials for the discretization of the spatial and temporal variables. Consequently, the spatial and temporal derivatives of this finite expansion are evaluated explicitly at some quadrature nodes. The resulting equations, when combined with the initial conditions, generate a system of algebraic equations that can be solved by any suitable method. In addition, we investigate the effectiveness of the proposed technique in terms of accuracy and convergence.

The paper is structured into 5 sections. Section 2 presents some preliminary tools as well as information about shifted Legendre polynomials. Section 3 introduces the spectral collocation method to solve the time FNNFM. Section 4 presents some numerical examples to illustrate the effectiveness and accuracy of the technique. Section 5 summarizes some conclusions.

2. Adopted Notation and Preliminary Concepts

2.1. Caputo Fractional Derivative

Definition 1. 

Given a function $\mathbb{Y}\left(\zeta \right)$, its Caputo fractional derivative [40] is:

$${}_0^c{\mathbb{D}}_{\xi }^{\theta }\mathbb{Y}\left(\xi \right)=\frac{1}{\Gamma (\vartheta -\theta )}{\int }_0^{\xi }{(\xi -\zeta )}^{\vartheta -\xi -1}\frac{d^{\vartheta }\mathbb{Y}\left(\zeta \right)}{d{\zeta }^{\vartheta }}d\zeta ,\vartheta -1<\theta \le \vartheta ,\xi >0,$$

(2)

where$\vartheta =\lceil \theta \rceil$and

$$\Gamma \left(n\right)={\int }_0^{\infty }{e}^{-t}{t}^{n-1}dt,n>0$$

is the Gamma function.

2.2. Brief Introduction to Shifted Legendre Polynomials

A class of mathematical functions known as Legendre polynomials is crucial in many branches of physics and engineering. They are defined as solutions to the Legendre’s equation, a particular differential equation that frequently appears in spherical symmetry-related issues. The polynomials are orthogonal in the range [−1, 1], so unless they are identical, their inner product over this range is zero. Numerous fields, including quantum mechanics, electromagnetism, and signal processing, as well as approximation theories and numerical methods for solving differential equations, use Legendre polynomials. The Legendre polynomials ${\mathbb{P}}_m\left(t\right)$ fulfil the Rodrigues formula, meaning that [41,42]

$$\begin{array}{c}\hfill {\mathbb{P}}_m\left(t\right)=\frac{{(-1)}^m}{2^{m}m!}{D}^m\left({(1-t^2)}^m\right),\end{array}$$

(3)

with pth derivative given by

$$\begin{array}{c}\hfill {\mathbb{P}}_m^{\left(p\right)}\left(\tau \right)=\sum _{r=0(m+r=even)}^{m-p}{C}_p(m,r){\mathbb{P}}_r\left(t\right),\end{array}$$

(4)

where

$$C_p(m,r)=\frac{2^{p-1}(2r+1)\Gamma \left(\frac{p+m+r+1}{2}\right)\Gamma \left(\frac{p+m-r}{2}\right)}{\Gamma \left(\frac{3-p+m+r}{2}\right)\Gamma \left(\frac{2-p+m-r}{2}\right)\Gamma \left(p\right)}.$$

The property of orthogonality [43] is written as

$${({\mathbb{P}}_m\left(t\right),{\mathbb{P}}_l\left(t\right))}_w=\underset{-1}{\overset{1}{\int }}{\mathbb{P}}_m\left(t\right){\mathbb{P}}_l\left(t\right)w\left(t\right)=h_m{\delta }_{lm},\omega \left(t\right)=1,h_m=\frac{2}{2m+1}.$$

(5)

Using the Legendre-Gauss-Lobatto quadrature, ref. [43] we obtain

$$\begin{array}{c}\hfill \underset{-1}{\overset{1}{\int }}\psi \left(t\right)dt=\sum _{j=0}^{\mathbb{N}}{\varpi }_{\mathbb{N},j}\psi \left(t_{\mathbb{N},j}\right),\end{array}$$

(6)

for $\psi \in S_{2\mathbb{N}-1}$$[-1,1]$.

Let us express the discrete inner product by means of the expression

$${(\psi ,\phi )}_w=\sum _{j=0}^{\mathbb{N}}\psi \left(t_{\mathbb{N},j}\right)\phi \left(t_{\mathbb{N},j}\right){\varpi }_{\mathbb{N},j}.$$

(7)

Definition 2. 

The shifted Legendre polynomial is given by [44]

$$\begin{array}{c}\hfill {\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left(x\right)={\mathbb{P}}_{\varsigma }\left(2{\left(\frac{x}{{\xi }_{end}}\right)}^{\epsilon }-1\right),\varsigma =0,1,\cdots ,0\le \xi \le {\xi }_{end}.\end{array}$$

(8)

Theorem 1. 

For ${\varpi }_{{\xi }_{end},f}\left(x\right)=1$, a complete $L_{{\varpi }_{{\xi }_{end},f}}^2[0,{\xi }_{end}]$-orthogonal system is obtained [45]

$$\underset{0}{\overset{{\xi }_{end}}{\int }}{\mathbb{P}}_i^{{\xi }_{end}}\left(x\right){\mathbb{P}}_j^{{\xi }_{end}}\left(x\right){\varpi }_{{\xi }_{end},f}\left(x\right)dx={\delta }_{ij}{h}_{{\xi }_{end},k},$$

(9)

where$h_{{\xi }_{end},k}=\frac{{\xi }_{end}}{2(2k+1)}$.

Corollary 1. 

Let ${\mathbb{P}}_{\mathcal{M}}=span\{{\mathbb{P}}_{{\xi }_{end},r}:0\le r\le \mathbb{K}\}$ be the finite space of fractional-polynomials. Using (9), the function ${\rm Y}\left(\xi \right)\in L_{{\mathcal{W}}_f}^2[0,{\xi }_{end}]$ can be obtained as

$${\rm Y}\left(\xi \right)=\sum _{r=0}^{\infty }{\varrho }_r{\mathbb{P}}_{{\xi }_{end},r}\left(\xi \right),{\varrho }_r=\frac{1}{h_{{\xi }_{end},r}}\underset{0}{\overset{{\xi }_{end}}{\int }}{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left(\xi \right){\rm Y}\left(\xi \right){\mathcal{W}}_{{\xi }_{end},f}\left(\xi \right)d\xi .$$

3. Solving the Time FNNFM

The time FNNFM [46,47] is dealt with the Gauss–Lobatto and the Gauss–Radau shifted Legendre collocation techniques [43], yielding:

$$\begin{array}{c}\hfill {\varrho }_1\frac{{\partial }^{{\nu }_1}\mathbb{Y}}{\partial {\eta }^{{\nu }_1}}+{\varrho }_2\frac{\partial \mathbb{Y}}{\partial \eta }+{\varrho }_3\frac{{\partial }^{{\nu }_2}\mathbb{Y}}{\partial {\eta }^{{\nu }_2}}+{\varrho }_4\mathbb{Y}={\varrho }_5\Delta \mathbb{Y}+{\varrho }_6\frac{{\partial }^{{\nu }_3}\Delta \mathbb{Y}}{\partial {\eta }^{{\nu }_3}}+{\rm Y}(\xi ,\eta ),(\xi ,\eta )\in {\Omega }^{•}\times {\Omega }^{\diamond },\end{array}$$

(10)

where ${\varrho }_1,{\varrho }_2,{\varrho }_3,{\varrho }_4,{\varrho }_5$ and ${\varrho }_6$ are constants and ${\Omega }^{•}\equiv [0,{\xi }_{end}]$, ${\Omega }^{\diamond }\equiv [0,{\eta }_{end}]$, and

$$\begin{array}{cc}\hfill \frac{\partial \mathbb{Y}}{\partial \eta }{|}_{\eta =0}=& {\omega }_0\left(\xi \right),\mathbb{Y}(\xi ,0)={\omega }_1\left(\xi \right),x\in {\Omega }^{•},\hfill \\ \hfill \mathbb{Y}(0,\eta )=& {\omega }_2\left(\eta \right),\mathbb{Y}({\xi }_{end},\eta )={\omega }_3\left(\eta \right),\eta \in {\Omega }^{\diamond }.\hfill \end{array}$$

(11)

Expression (10) represents various types of fractional diffusion equations. The generalized Oldroyd-B fluid model (1), the time fractional diffusion-wave equation [48] (${\varrho }_2={\varrho }_4=0$), and the generalized Maxwell fluid model [49] (${\varrho }_3=0$) are some particular cases. A detailed derivation of the flow problem is discussed in reference [47].

The truncated solution is

$$\begin{array}{cc}\hfill {\mathbb{Y}}_{\mathbb{N},\mathbb{M}}(\xi ,\eta )& =\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left(\xi \right){\mathbb{P}}_{\iota }^{{\eta }_{end}}\left(\eta \right),\hfill \end{array}$$

(12)

where ${\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left(\xi \right)$ and ${\mathbb{P}}_{\iota }^{{\eta }_{end}}\left(\eta \right)$ are shift Legendre polynomials (see [50,51] for more details).

The time derivatives are then computed as

$$\begin{array}{cc}\hfill \frac{\partial \mathbb{Y}}{\partial \eta }=& \sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left(\xi \right){\mathbb{P}}_{\iota }^{{\eta }_{end},1}\left(\eta \right),\hfill \\ \hfill \frac{{\partial }^2\mathbb{Y}}{\partial {\eta }^2}=& \sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left(\xi \right){\mathbb{P}}_{\iota }^{{\eta }_{end},2}\left(\eta \right),\hfill \end{array}$$

(13)

where ${\mathbb{P}}_{\iota }^{{\eta }_{end},s}\left(\eta \right)={\mathbb{D}}_t^s{\mathbb{P}}_{\iota }^{{\eta }_{end}}\left(\eta \right)$ [52]. Moreover, the space derivatives are

$$\begin{array}{cc}\hfill \frac{\partial \mathbb{Y}}{\partial \xi }=& \sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end},1}\left(\xi \right){\mathbb{P}}_{\iota }^{{\eta }_{end}}\left(\eta \right),\hfill \\ \hfill \frac{{\partial }^2\mathbb{Y}}{\partial {\xi }^2}=& \sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end},2}\left(\xi \right){\mathbb{P}}_{\iota }^{{\eta }_{end}}\left(\eta \right).\hfill \end{array}$$

(14)

The derivative $\frac{{\partial }^{{\nu }_1}\mathbb{Y}}{\partial {\eta }^{{\nu }_1}}$ is calculated as:

$$\begin{array}{cc}\hfill \frac{{\partial }^{{\nu }_1}\mathbb{Y}}{\partial {\eta }^{{\nu }_1}}=& \sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left(\xi \right){}_0^c{\mathbb{D}}_{{\eta }_{end}}^{{\nu }_1}\left({\mathbb{P}}_{\iota }^{{\eta }_{end}}\left(\eta \right)\right)\hfill \\ \hfill =& \sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left(\xi \right){\mathbb{P}}_{\iota }^{{\eta }_{end},{\nu }_1}\left(\eta \right),\hfill \end{array}$$

(15)

where [45,51]

$${\mathbb{P}}_{\iota }^{{\eta }_{end},{\nu }_1}\left(\eta \right)=\sum _{k=1}^{\iota }\frac{{(-1)}^{\iota -k}\left(\Gamma (\iota +k+1)\right){}_0^c{\mathbb{D}}_{{\eta }_{end}}^{{\nu }_1}\left({\eta }^k\right)}{{\eta }_{end}^k{(\Gamma (k+1))}^2\Gamma (\iota -k+1)},$$

with ${}_0^c{\mathbb{D}}_{{\eta }_{end}}^{{\nu }_1}\left({\eta }^k\right)=\left\{\begin{array}{cc}0,\hfill & k\in {\mathbb{N}}_0\mathrm{and}k<\lceil {\nu }_1\rceil ;\hfill \\ \frac{\Gamma (k+1)}{\Gamma \left(k-{\nu }_1+1\right)}{\eta }^{k-{\nu }_1},\hfill & k\in {\mathbb{N}}_0\mathrm{and}k\ge \lceil {\nu }_1\rceil .\hfill \end{array}\right.$

Also, we get

$$\begin{array}{cc}\hfill \frac{{\partial }^{{\nu }_2}\mathbb{Y}}{\partial {\eta }^{{\nu }_2}}=& \sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left(\xi \right){}_0^c{\mathbb{D}}_{{\eta }_{end}}^{{\nu }_2}\left({\mathbb{P}}_{\iota }^{{\eta }_{end}}\left(\eta \right)\right)\hfill \\ \hfill =& \sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left(\xi \right){\mathbb{P}}_{\iota }^{{\eta }_{end},{\nu }_2}\left(\eta \right),\hfill \end{array}$$

(16)

$$\begin{array}{c}\hfill \frac{{\partial }^{{\nu }_3}\Delta \mathbb{Y}}{\partial {\eta }^{{\nu }_3}}=\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end},2}\left(\xi \right){\mathbb{P}}_{\iota }^{{\eta }_{end},{\nu }_3}\left(\eta \right).\end{array}$$

(17)

At selected quadrature nodes, we get

$$\begin{array}{cc}\hfill {\mathbb{Y}}_{\mathbb{N},\mathbb{M}}({\xi }_{\mathbb{N},n}^{{\xi }_{end}},{\eta }_{\mathbb{M},m}^{{\eta }_{end}})=& \sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right){\mathbb{P}}_{\iota }^{{\eta }_{end}}\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right),\hfill \\ \hfill {\left(\frac{\partial \mathbb{Y}}{\partial \eta }\right)}_{{\eta }_{\mathbb{M},m}^{{\eta }_{end}}}^{\xi ={\xi }_{\mathbb{N},n}^{{\xi }_{end}}}=& \sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right){\mathbb{P}}_{\iota }^{{\eta }_{end},2}\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right),\hfill \\ \hfill {\left(\frac{{\partial }^2\mathbb{Y}}{\partial {\xi }^2}\right)}_{{\eta }_{\mathbb{M},m}^{{\eta }_{end}}}^{\xi ={\xi }_{\mathbb{N},n}^{{\xi }_{end}}}=& \sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end},2}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right){\mathbb{P}}_{\iota }^{{\eta }_{end}}\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right),\hfill \\ \hfill {\left(\frac{{\partial }^{{\nu }_1}\mathbb{Y}}{\partial {\eta }^{{\nu }_1}}\right)}_{{\eta }_{\mathbb{M},m}^{{\eta }_{end}}}^{\xi ={\xi }_{\mathbb{N},n}^{{\xi }_{end}}}=& \sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right){\mathbb{P}}_{\iota }^{{\eta }_{end},{\nu }_1}\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right),\hfill \\ \hfill {\left(\frac{{\partial }^{{\nu }_2}\mathbb{Y}}{\partial {\eta }^{{\nu }_2}}\right)}_{{\eta }_{\mathbb{M},m}^{{\eta }_{end}}}^{\xi ={\xi }_{\mathbb{N},n}^{{\xi }_{end}}}=& \sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right){\mathbb{P}}_{\iota }^{{\eta }_{end},{\nu }_2}\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right),\hfill \\ \hfill {\left(\frac{{\partial }^{{\nu }_3}\Delta \mathbb{Y}}{\partial {\eta }^{{\nu }_3}}\right)}_{{\eta }_{\mathbb{M},m}^{{\eta }_{end}}}^{\xi ={\xi }_{\mathbb{N},n}^{{\xi }_{end}}}=& \sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end},2}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right){\mathbb{P}}_{\iota }^{{\eta }_{end},{\nu }_3}\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right),\hfill \end{array}$$

(18)

where $m=1,\cdots ,\mathbb{M}-1,n=1,\cdots ,\mathbb{N}-1,$ and ${\xi }_{\mathbb{N},n}^{{\xi }_{end}}{t}_{\mathbb{M},m}^{{\eta }_{end}}$ are Gauss-Lobatto and Gauss-Radau shifted Legendre collocation nodes, respectively.

Besides, the provided conditions can be written as

$$\begin{array}{cc}\hfill \sum _{l=0}^{\mathbb{N}}\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left(\xi \right){\mathbb{P}}_{\iota }^{{\eta }_{end},1}\left(0\right)=& {\omega }_0\left(\xi \right),\hfill \\ \hfill \sum _{l=0}^{\mathbb{N}}\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left(\xi \right){\mathbb{P}}_{\iota }^{{\eta }_{end}}\left(0\right)=& {\omega }_1\left(\xi \right),\hfill \\ \hfill \sum _{l=0}^{\mathbb{N}}\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left(0\right){\mathbb{P}}_{\iota }^{{\eta }_{end}}\left(\eta \right)=& {\omega }_2\left(\eta \right),\hfill \\ \hfill \sum _{l=0}^{\mathbb{N}}\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left({\xi }_{end}\right){\mathbb{P}}_{\iota }^{{\eta }_{end}}\left(\eta \right)=& {\omega }_3\left(\eta \right).\hfill \end{array}$$

(19)

The Equation (10) is closest to zero at the $(\mathbb{N}-1)\times (\mathbb{M}-1)$ nodes.

$$\begin{array}{c}\hfill {\Omega }_{n,m}^{\mathbb{N},\mathbb{M}}={\Xi }_{n,m}^{\mathbb{N},\mathbb{M}},\end{array}$$

(20)

where

$$\begin{array}{cc}\hfill {\Omega }_{n,m}^{\mathbb{N},\mathbb{M}}=& {\varrho }_1\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right){\mathbb{P}}_{\iota }^{{\eta }_{end},{\nu }_1}\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right)+{\varrho }_2\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right){\mathbb{P}}_{\iota }^{{\eta }_{end},2}\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right)+\hfill \\ & {\varrho }_3\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right){\mathbb{P}}_{\iota }^{{\eta }_{end},{\nu }_2}\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right)+{\varrho }_4\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right){\mathbb{P}}_{\iota }^{{\eta }_{end}}\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\Xi }_{n,m}^{\mathbb{N},\mathbb{M}}=& {\varrho }_5\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end},2}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right){\mathbb{P}}_{\iota }^{{\eta }_{end}}\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right)+{\varrho }_6\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end},2}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right){\mathbb{P}}_{\iota }^{{\eta }_{end},{\nu }_3}\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right)+\hfill \\ & {\rm Y}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}},{\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right).\hfill \end{array}$$

Additionally

$$\begin{array}{cc}\hfill \sum _{l=0}^{\mathbb{N}}\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right){\mathbb{P}}_{\iota }^{{\eta }_{end},1}\left(0\right)=& {\omega }_0\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right),k=1,\cdots ,\mathbb{N}-1,\hfill \\ \hfill \sum _{l=0}^{\mathbb{N}}\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right){\mathbb{P}}_{\iota }^{{\eta }_{end}}\left(0\right)=& {\omega }_1\left({\xi }_{\mathbb{N},n}^{{\xi }_{end}}\right),k=1,\cdots ,\mathbb{N}-1,\hfill \\ \hfill \sum _{l=0}^{\mathbb{N}}\sum _{\varsigma =0}^{\mathbb{N}}\sum _{\iota =0}^{\mathbb{M}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left(0\right){\mathbb{P}}_{\iota }^{{\eta }_{end}}\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right)=& {\omega }_2\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right),l=0,\cdots ,\mathbb{M},\hfill \\ \hfill \sum _{\begin{array}{c}l,\varsigma =0,\cdots ,\mathbb{N}\\ \iota =0,\cdots ,\mathbb{M},\end{array}}{\pi }_{\varsigma ,\iota }{\mathbb{P}}_{\varsigma }^{{\xi }_{end}}\left({\xi }_{end}\right){\mathbb{P}}_{\iota }^{{\eta }_{end}}\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right))=& {\omega }_3\left({\eta }_{\mathbb{M},m}^{{\eta }_{end}}\right),l=0,\cdots ,\mathbb{M}.\hfill \end{array}$$

(21)

This results in a system of algebraic equations that is simple to solve.

4. Some Numerical Examples

We illustrate the proposed spectral approach by solving two examples.

Example 1. 

We consider the multi-term time fractional viscoelastic non-Newtonian fluid model [53]

$$\begin{array}{c}\hfill \frac{{\partial }^{{\nu }_1}\mathbb{Y}}{\partial {\eta }^{{\nu }_1}}+\frac{\partial \mathbb{Y}}{\partial \eta }+\frac{{\partial }^{{\nu }_2}\mathbb{Y}}{\partial {\eta }^{{\nu }_2}}+\mathbb{Y}=\Delta \mathbb{Y}+\frac{{\partial }^{{\nu }_3}\Delta \mathbb{Y}}{\partial {\eta }^{{\nu }_3}}+{\rm Y}(\xi ,\eta ),(\xi ,\eta )\in [0,1]\times [0,1],\end{array}$$

(22)

The included conditions are specified in a way that $\mathbb{Y}(\xi ,\eta )=\left({\eta }^3+1\right)sin\left(\pi \xi \right)$ where

$${\rm Y}(\xi ,\eta )=sin\left(\pi \xi \right)\left({\pi }^2\left({\eta }^3+1\right)+(\eta +3){\eta }^2+6{\eta }^3\left(\frac{{\eta }^{-{\nu }_1}}{\Gamma (4-{\nu }_1}+\frac{{\eta }^{-{\nu }_2}}{\Gamma (4-{\nu }_2)}+\frac{{\pi }^2{\eta }^{-{\nu }_3}}{\Gamma (4-{\nu }_3)}\right)+1\right).$$

Relying on $L_{\infty }$-errors, the results obtained with the proposed technique are compared with those in [53], and are summarized in

Table 1. The $L_{\infty }$-errors of Example 1 for various values of $({\nu }_1,{\nu }_2,{\nu }_3)$.

Finite difference method [53]
h	$(1.5,0.7,0.6)$	$(1.6,0.7,0.8)$	$(1.6,0.5,0.3)$
$\frac{1}{40}$	$9.9671\times {10}^{-3}$	$1.5408\times {10}^{-2}$	$7.8520\times {10}^{-3}$
$\frac{1}{80}$	$4.5553\times {10}^{-3}$	$7.0946\times {10}^{-3}$	$3.7583\times {10}^{-3}$
$\frac{1}{160}$	$2.1071\times {10}^{-3}$	$3.2759\times {10}^{-3}$	$1.8190\times {10}^{-3}$
$\frac{1}{320}$	$9.8700\times {10}^{-4}$	$1.5191\times {10}^{-3}$	$8.8841\times {10}^{-4}$
$\frac{1}{640}$	$4.6799\times {10}^{-4}$	$7.0814\times {10}^{-4}$	$4.3707\times {10}^{-4}$
New method
$(\mathbb{N},\mathbb{M})$	$(1.5,0.7,0.6)$	$(1.6,0.7,0.8)$	$(1.6,0.5,0.3)$
$(6,6)$	$2.06323\times {10}^{-2}$	$2.08022\times {10}^{-2}$	$2.01163\times {10}^{-2}$
$(10,10)$	$1.56512\times {10}^{-5}$	$1.57477\times {10}^{-5}$	$1.5443\times {10}^{-5}$
$(14,14)$	$3.01187\times {10}^{-9}$	$3.02924\times {10}^{-9}$	$2.99857\times {10}^{-9}$
$(18,18)$	$2.04059\times {10}^{-13}$	$2.01617\times {10}^{-13}$	$2.03171\times {10}^{-13}$
$(22,22)$	$4.38933\times {10}^{-14}$	$5.90639\times {10}^{-14}$	$1.58207\times {10}^{-14}$

. We verify that the new procedure yields better numerical results. The tables show that the numerical results and exact solution are well-aligned, proving the efficacy of our numerical approach and supporting the theoretical analysis. We plot a few graphs to show the dynamic properties of the generalized non-Newtonian fluid and to observe the effects of various physical parameters on the velocity field.

Figure 1 and Figure 2 depict the numerical solution and absolute error of Example 1, respectively. The similarity between the numerical and the exact solutions is illustrated in Figure 3. Also, Figure 3 depicts the relationship between time and velocity, and the increase in flow velocity. Figure 4 and Figure 5 reveal the $\xi$- and $\eta$-curves associated with the absolute errors, respectively. In addition, the convergence error decay is shown in Figure 6.

Example 2. 

We consider the multi-term time fractional diffusion equation [47]

$$\begin{array}{c}\hfill \frac{{\partial }^{{\nu }_1}\mathbb{Y}}{\partial {\eta }^{{\nu }_1}}+\frac{\partial \mathbb{Y}}{\partial \eta }=\Delta \mathbb{Y}+\frac{{\partial }^{{\nu }_3}\Delta \mathbb{Y}}{\partial {\eta }^{{\nu }_3}}+{\rm Y}(\xi ,\eta ),(\xi ,\eta )\in [0,1]\times [0,1],\end{array}$$

(23)

The included conditions are specified in a way that $\mathbb{Y}(\xi ,\eta )=e^{\xi }{\eta }^3$ where

$${\rm Y}(\xi ,\eta )={\eta }^2{e}^{\xi }\left(6\eta \left(\frac{{\eta }^{-{\nu }_1}}{\Gamma (4-{\nu }_1)}-\frac{{\eta }^{-{\nu }_2}}{\Gamma (4-{\nu }_2)}\right)-\eta +3\right).$$

The results with the proposed technique are compared with those in [47] and listed in

Table 2. The $L_{\infty }$-error of Example 2 for various values of $({\nu }_1,{\nu }_2)$.

Finite difference method [47]
h	$(1.1,0.6)$	$(1.6,0.3)$	$(1.8,0.3)$
$\frac{1}{20}$	$2.2385\times {10}^{-2}$	$3.2559\times {10}^{-2}$	$3.8225\times {10}^{-2}$
$\frac{1}{40}$	$1.1489\times {10}^{-2}$	$1.6140\times {10}^{-2}$	$1.8773\times {10}^{-2}$
$\frac{1}{80}$	$5.8498\times {10}^{-3}$	$7.9949\times {10}^{-3}$	$9.2139\times {10}^{-3}$
$\frac{1}{160}$	$2.9630\times {10}^{-3}$	$3.9631\times {10}^{-3}$	$4.5272\times {10}^{-3}$
$\frac{1}{320}$	$1.4955\times {10}^{-3}$	$1.9670\times {10}^{-3}$	$2.2278\times {10}^{-3}$
New method
$(\mathbb{N},\mathbb{M})$	$(1.1,0.6)$	$(1.6,0.3)$	$(1.8,0.3)$
(4,4)	$2.1782\times {10}^{-3}$	$2.16526\times {10}^{-3}$	$2.16149\times {10}^{-3}$
(8,8)	$4.24672\times {10}^{-8}$	$4.2443\times {10}^{-8}$	$4.24348\times {10}^{-8}$
(12,12)	$1.47438\times {10}^{-13}$	$1.45661\times {10}^{-13}$	$1.45661\times {10}^{-13}$
(14,14)	$5.32907\times {10}^{-15}$	$4.88498\times {10}^{-14}$	$2.22045\times {10}^{-14}$

in terms of $L_{\infty }$-error. We verify that the accuracy of the solutions with the new technique is superior to that with the alternative method.

Figure 7 and Figure 8 depict the numerical solution and the absolute errors of Example 2, respectively. The similarity between the numerical and exact solutions is illustrated in Figure 9. Also, Figure 9 depicts the relationship between time and velocity, and the increase in flow velocity. Figure 10 and Figure 11 reveal the $\xi$- and $\eta$-curves associated with absolute errors, respectively. In addition, the decay of the convergence error is shown in Figure 12.

5. Conclusions

We proposed an accurate technique for solving two-dimensional multi-term time fractional viscoelastic non-Newtonian fluid model. To exemplify the technique’s implementation and effectiveness, a theoretical analysis was provided, along with a set of numerical tests. Based on the results obtained though the examples, we can deduce that our method is extremely precise and dependable. More fractional order concerns can be incorporated into the existing discussion. The results are completely compatible with the expected values of the technique, evidencing exponential convergence.