The Rayleigh–Stokes Problem for a Heated Generalized Second-Grade Fluid with Fractional Derivative: An Implicit Scheme via Riemann–Liouville Integral

Ganie, Abdul Hamid; Saeed, Abdulkafi Mohammed; Saeed, Sadia; Ali, Umair

doi:https://doi.org/10.1155/2022/6948461

Mathematical Problems in Engineering

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Volume 2022 | Article ID 6948461 | https://doi.org/10.1155/2022/6948461

The Rayleigh–Stokes Problem for a Heated Generalized Second-Grade Fluid with Fractional Derivative: An Implicit Scheme via Riemann–Liouville Integral

Abdul Hamid Ganie,¹Abdulkafi Mohammed Saeed,²Sadia Saeed,³and Umair Ali³

Academic Editor: Muhammad Irfan

Received06 Sept 2021

Revised29 Jan 2022

Accepted01 Mar 2022

Published06 May 2022

Abstract

The goal of this study is to use the fast algorithm to solve the Rayleigh–Stokes problem for heated generalized second-grade fluid (RSP-HGSGF) with Riemann–Liouville time fractional derivative using the fast algorithm. The modified implicit scheme, which is formulated by the Riemann–Liouville integral formula and applied to the fractional RSP-HGSGF, is proposed. Numerical experiments will be carried out to demonstrate that the scheme is simple to implement, and the results will reveal the best way to implement the suggested technique. The proposed scheme’s stability and convergence will be examined using the Fourier series. The method is stable, and the approximation solution approaches the exact solution. A numerical demonstration will be provided to demonstrate the applicability and viability of the suggested strategy.

1. Introduction

The study and application of arbitrary-order derivatives and integrals are associated with fractional calculus. The use of fractional-order calculus in a variety of fields of science and engineering, including geometric phenomena, has sparked a lot of interest in this area [1]. The first discussion of fractional calculus took place between Leibniz and L’Hospital at the end of the seventeenth century [2]. The great mathematicians Erdelyi, Abel, Riemann, Laplace, Heaviside, Levy, Liouville, Riesz, Gunwald, Letnikov, and Fourier worked on it and had contributed [3]. Fractional-order integrals and derivatives play an important role in solving some chemical problems, and this field has been paid much attention since 1968. The most well-known book in the field of fractional calculus, originally written by Ross and Miller and Ross [4], Spanier and Oldham [5], Podlubny [6], and Samko et al. [7], explains the underlying theory of fractional calculus as well as its applications and solutions.

Many researchers have solved fractional-order problems using various methods. For example, Shivanian and Jafferabadi [8] used fractional derivatives to find the numerical solution for the RSP-HGSGF using spectral meshless radial point interpolation. The time-fractional derivative has been defined in the Riemann–Liouville sense. The Shape functions are created by using a point interpolation method and radial basis functions as basic functions. An efficient numerical approach for approximating RSP-HGSGF in a bounded domain is described by Liu et al. [9]. They investigated the proposed scheme’s stability and convergence. To solve SFP-HGSGF, Wu [10] used a numerical approach. The stability, convergence, and consistency of the INAS for the SFP HGSGF have been investigated. RSPHGSGF was studied in a flow on a heated flat plate and within a heated edge by Shen et al. [11]. A viscoelastic fluid was described using the fractional calculus technique in the constitutive relationship model. For the exact solution of the velocity and temperature fields, the Fourier transform on fractional-order Laplace operator is used. Yu et al. [12] used the Adomian decomposition method to solve the RSP-HGSGF. In general, without discretizing the problem, such series solutions converge quickly and the Adomian decomposition approach yields very precise numerical solutions. In this study, Chen et al. [13] presented two numerical methods for solving a two-dimensional variable-order subdiffusion anomalous problem. Their stability, convergence, and solvability were investigated using Fourier analysis. The numerical approximation for the Riemann–Liouville fractional-order derivative for the fractional SFP-HGSGF was studied by Yu et al. [14]. They used the implicit scheme with Riemann–Liouville fractional derivative to solve the direct and inverse problems. Lin and Jiang [15] devised a straightforward method for calculating the fractional derivative of an RSP-HGSG. They created the series of the exact solution to the problem using kernel theory and established the approximate solution of its fractional derivative using truncating series, which are uniformly convergent. Meanwhile, their method includes error estimation and stability analysis. Chen et al. [16] proposed the implicit and explicit techniques for solving the RSP-HGSGF of fractional order. The convergence, stability, and solvability of the problem have all been determined. In recent years, Chen et al. [17] discussed Stokes’ initial challenge attention. The variable-order nonlinear RSP-HGSGF is investigated, and the fourth-order numerical technique is discussed. The Fourier approach is used to investigate the numerical scheme's theoretical analysis. Dehghan and Abbas Zadeh [18] developed a numerical solution for 2D fractional-order RSP-HGSGF on rectangular domains such as circular, L-shaped, and a unit square with circular holes. The RL principle is used to calculate the fractional derivatives. They used the Galerkin FEM to obtain a fully discrete scheme for the space direction by integrating the equation for the time variable. Finally, we compare the results of Galerkin FEM to those of other numerical techniques. The Rayleigh–Stokes problem for an edge in a generalized Oldroyd-B fluid was solved by Nikan and Avazzadeh et al. [19] using the radial basis function and fractional derivatives. The temporal derivative terms are discretized using the finite difference technique, while the spatial derivative terms are discretized using the local RBF-FD.

To maintain a constant number of nodes, they evaluate the distribution of data nodes within the local support area. The stability and convergence of the proposed method are also investigated. The RBF-FD results are compared to those of previous approaches on irregular domains, demonstrating the novel methodology’s viability and efficiency. RSP-HGSGF flow was investigated by Zhai et al. [20] on a heated flat plate and within a heated edge. To describe such a viscoelastic fluid, a fractional calculus methodology was used in the constitutive relationship model. The velocity and temperature fields were solved in closed form using the Fourier transform and the fractional Laplace operator. Another study looked at the same model to describe a viscoelastic fluid [21, 22]. For the finite difference/finite element technique, Guan et al. [23] provided an enhanced version of a nonlinear source term with a fractional RSP. The backward difference formula and second-order Grünwald–Letnikov derivative are used to discretize the first-order time derivative. They use the Galerkin finite element approach to define a fully discrete strategy for the fractional RSP-HGSGF with a nonlinear source term in the space direction. A novel analytical technique is used to calculate the level of accuracy in the L2 norm in great detail. For the 2D modified anomalous fractional subdiffusion equation, Ali et al. [24] used a modified implicit difference approximation. The proposed scheme’s convergence and stability are investigated using the Fourier series approach. It is shown that the scheme is unconditionally stable, and that the approximate solution converges to the exact solution. Bazhlekova et al. [25] investigated the RSP-HGSGF in time using the RL fractional derivative, and the problem was analysed in space using semidiscrete, continuous, and completely discrete formulations. Mohebbi et al. [26] compared the meshless approach to a fourth-order approximation for 2D fractional RSP and generated a completely discrete implicit scheme. Sun et al [27] contributed a review article on important fractional calculus information. They talked about the most important real-world applications as well as powerful mathematical tools. The numerical solution of a nonlinear fractional-order reaction-subdiffusion model was investigated by Nikan et al. [28]. For spatial discretization, they used the radial base function-finite difference method, and for time discretization, they used a weighted discrete scheme. They discussed theoretical analysis and tested two numerical examples for the computational efficiency of the proposed scheme, which yielded accurate results. In a separate study [29], the author proposed a meshless scheme for the fractional-order diffusion model. They eliminated the time derivative by integrating both sides of the proposed model and used local hybridization of cubic and radial basis functions for space derivatives. Nikan et al. [30] investigated the local hybrid kernel meshless approach for fractional-order model approximation. To approximate the time and space directions, they used the central difference approximation and Gaussian kernels, respectively. They verified the validity of the proposed method using numerical examples that are both accurate and efficient. Liu et al. [31] discussed the fractional dynamics modelled from the fractional-order PDEs. Fractional-order systems have importance in the field of electrochemistry, chaotic systems, biology etc. Ahmad et al. [32] formulated a new methodology named as variational iteration method I and successfully applied to a nonlinear model. They explained the compactness of the method and compared their results with the existed literature and found that the proposed method is more productive and reliable than others. Khan et al. [33] considered the numerical approach based on the collocation method for the inverse heat source problem and tasted the method both on regular and irregular domain. Different researchers discussed various numerical approaches for time and space fractional-order models in the research [27, 30, 34–38]. The goal of this research is to propose a new scheme for this model modified implicit scheme for fractional-order RSP-HGSGF. It lowers the computational cost and allows for easy theoretical analysis using any method for the final scheme. In the procedure, the discretized form of the Riemann–Liouville integral operator is used to replace the Riemann–Liouville derivative with the first-order time derivative. The partial derivative with respect to time is then eliminated using backward difference approximation. Additionally, we use the Fourier series method to investigate the established method's stability and convergence criterion. Finally, numerical examples are presented and solved using the proposed method to verify the method’s accuracy and feasibility. Maple 15 is used to code the numerical examples.

The following is how the rest of the paper is organized: The methodology of the proposed scheme is discussed in Section 2, followed by stability and convergence analysis in Sections 2.1 and 2.2. The numerical experiments and results are presented in Section 3 and discussed in Section 4. The conclusion is discussed in Section 5 of the report.

The aim of this study is to propose a modified implicit scheme for fractional RSP-HGSGF based on the formulated Riemann–Liouville integral operator. The partial derivative w.r.t. time is eliminated by backward difference approximation. Additionally, we investigate the stability and convergence criterion of the established method by the Fourier series method.

Here, we consider the following two-dimensional RSP-HGSGF with fractional derivative [22].

Initial and boundary conditions are as follows:where represents the fractional-order Riemann–Liouville derivative of order .

Lemma 1. The -order Riemann–Liouville fractional integral of the function on [0,T] can be defined in discretized form as

Lemma 2. The coefficients constant fulfils the following properties [29]:(i)(ii)(iii)There exists a positive constant , such that (iv)

2. Methodology of the Proposed Scheme

The 2D RSP-HGSGF in equations (1)–(3) is solved by the modified implicit scheme. We utilized the Riemann–Liouville approximation for time-fractional and central difference for space derivative and partitioned the bounded domain into subintervals of lengths and . The space steps are , in the x-direction with , in the y-direction with . The time step is where . Let be the numerical approximation to by applying (2) to (1), we obtain

Applying Lemma 1 and backward difference approximation w.r.t. time, we obtainwhere

The simplified form of the proposed scheme for 2D RSP-HGSGF (1)-(3) and the conditions are as follows:where .

2.1. Stability

We find the stability of the proposed scheme by Fourier technique. Let the approximate solution be for (9); we have

Next, the error is defined aswhere satisfies (11) and

The error initial and boundary conditions are given as

Define the following grid functions for :

Then, can be expanded in Fourier series such aswhere

From the definition of norm and Parseval equality, we have

Suppose thatwhere , and substituting (19) in (13), we get

After simplifying, we getwhere and

Proposition 1. If satisfies (20), then .

Proof: . By using mathematical induction, we take in (20).and as Now, assume thatand as 0 , from (20) and Lemma 2, we obtainThis completes the proof.
Based on the above proof, it can be summarized that the solution of (5) satisfies the following inequality:

And, we demonstrated that the proposed scheme is unconditionally stable.

2.2. Convergence

Here, we use a similar method to examine the convergence of the scheme. Let represent the exact solution; then, the truncation error of the scheme is obtained as follows: from (3),

From (1), we have

Since are finite, there is a positive constant , for all , which then have

The error is defined as

From (25), we have

To obtain the error equation, subtract (29) from (5) to obtain

With error boundary conditions,

And, the initial condition

Next, we define the following grid functions for :

Here, the can be expanded in Fourier series such aswhere

From the definition of norm and the Parseval equality, we have

Based on the previous equations, suppose that

Respectively, we have ; substitute (40) and (41) into (30) and we get

Simplifying the previous equation, we obtainwhere

Proposition 2. Let be the solution of (42); then, there is a positive constant so that

Proof: . From and (36), we haveFrom (37) and (39), then there is a positive constant , such thatUsing mathematical induction, for , then from (42) and (44), we obtainSince from (45), we getNow, suppose thatAs 0 .
From (41) and (44) and Lemma 2, we haveThe proof is completed via the induction method.

Theorem 1. The modified implicit difference scheme is convergent, and the order of convergence is

Proof: . From (27) and (39), we obtainAs thuswhere

3. Numerical Experiment

Example 1. Consider the following two-dimensional Rayleigh–Stokes problem for heated generalized second-grade fluid with the fractional derivative [22]:with initial and boundary conditionsHere, and the exact solution of (52) is given byThe error between the numerical solution and exact solution is defined as follows:And, the rate of convergence for space variable can be defined asThe developed modified implicit scheme is applied to problems (52) to (54).
Tables 1–4 show the errors for values of space step size and . Here, time step is defined by .
Tables 1–4 indicate that, as we reduce the time and space step size and , the error decreases for a fixed value of .This shows that the method converges to the exact solution.

Example 2. Consider the following Cable equation:with initial and boundary conditionsThe exact solution is

4. Results and Discussion

A modified implicit scheme is developed and applied on RSP-HGSGF. Numerical example is given to support theoretical study. The error between the exact and numerical solution is calculated using different values of and . Also, at different values of , Tables 1–4 are created to show the comparison of the numerical scheme with the exact solution in terms of maximum error. In example 2, we solved the fractional-order Cable equation, and the numerical results are shown in Table 5 for various values of space and time step size. The values of are also changed, and the obtained results are converging with reduced step sizes. Here, the error is calculated using Maple15 software with the increase in the number of space and time steps. Figures 1–3 are plotted for different values of , and fractional order shows good agreement with the exact solution.

5. Conclusion

A modiﬁed implicit diﬀerence scheme is formulated for 2D RSP-HGSGF, and a derivative of fractional order has been described in this paper. The modiﬁed scheme has the improvement of low computational cost and can be easily applied. The Fourier technique has been used for the theoretical analysis stability, and convergence with order is unconditionally stable and convergent. The numerical experiment for the 2D RSP-HGSGF and 1D Cable equation is conducted, which shows that the modified implicit scheme is easy to implement, and the results show good performance of the proposed schemes [39].

Abbreviations

RSP-HGSGF:	Rayleigh–Stokes problem for heated generalized second-grade fluid
SFP:	Stokes first problem
INAS:	Implicit numerical approximation scheme
RBF-FD:	Radial basis function finite difference
FEM:	Finite element method
2D:	Two-dimensional
RL:	Riemann–Liouville.

Data Availability

All the related materials have been cited in the paper.

Conflicts of Interest

The authors declare no conflicts of interest.

References

U. Ali, A. Iqbal, M. Sohail, F. A. Abdullah, and Z. Khan, “Compact implicit difference approximation for time-fractional diffusion-wave equation,” Alexandria Engineering Journal, vol. 61, no. 5, pp. 4119–4126, 2022.
View at: Publisher Site | Google Scholar
U. Ali, M. A. Khan, M. M. Khater, A. A. Mousa, and R. A. Attia, “A new numerical approach for solving 1D fractional diffusion-wave equation,” Journal of Function Spaces, vol. 2021, Article ID 6638597, 2021.
View at: Publisher Site | Google Scholar
U. Ali, “Numerical Solutions for Two Dimensional Time-Fractional Differential Sub-diffusion Equation,” University Sains Malaysia, Penang, Malaysia, 2019, Doctoral dissertation, Ph. D. Thesis.
View at: Google Scholar
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Diﬀerential Equations, John Wiley, New York, 1993.
K. B. Oldham and J. Spanier, Fractional Calculus, Academic Press, New York and London, 1974.
I. Podlubny, Fractional Diﬀerential Equations, Academic Press, New York, 1999.
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivaives, Theory and Applications, Gordon and Breach Science Pub- lishers, USA, 1993.
E. Shivanian and A. Jafarabadi, “Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives: a stable scheme based on spectral meshless radial point interpolation,” Engineering with Computers, vol. 34, no. 1, pp. 77–90, 2018.
View at: Publisher Site | Google Scholar
J. Liu, H. Li, and Y. Liu, “A new fully discrete ﬁnite diﬀerence/element ap- proximation for fractional cable equation,” J. Appl. Math. Comput., pp. 1–17, 2015.
View at: Google Scholar
C. Wu, “Numerical solution for Stokes’ first problem for a heated generalized second grade fluid with fractional derivative,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2571–2583, 2009.
View at: Publisher Site | Google Scholar
F. Shen, W. Tan, Y. Zhao, and T. Masuoka, “The Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative model,” Nonlinear Analysis: Real World Applications, vol. 7, no. 5, pp. 1072–1080, 2006.
View at: Publisher Site | Google Scholar
B. Yu, X. Jiang, and H. Qi, “An inverse problem to estimate an unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes’ first problem for a heated generalized second grade fluid,” Acta Mechanica Sinica, vol. 31, no. 2, pp. 153–161, 2015.
View at: Publisher Site | Google Scholar
C.-M. Chen, F. Liu, I. Turner, and V. Anh, “Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation,” Numerical Algorithms, vol. 54, no. 1, pp. 1–21, 2010.
View at: Publisher Site | Google Scholar
F. Liu, Q. Yang, and I. Turner, “Two new implicit numerical methods for the fractional cable equation,” Journal of Computational and Nonlinear Dynamics, vol. 6, no. 1, 7 pages, 2014.
View at: Google Scholar
B. Yu, X. Jiang, and H. Qi, “An inverse problem to estimate an unknown order of a Riemann-Liouville fractional derivative for a fractional Stocks’ ﬁrst problem for a heated generalized second grade ﬂuid,” Acta Mechanica Sinica, vol. 31, no. 2, pp. 153–161, 2015.
View at: Publisher Site | Google Scholar
C. M. Chen, F. Liu, V. Anh, and I. Turner, “Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation,” Mathematics of Computation, vol. 277, no. 81, pp. 345–366, 2011.
View at: Google Scholar
M. C. Chen, F. Liu, I. Turner, and V. Anh, “Numerical methods with fourth-order spatial accuracy for variable-order nonlinear Stokes’ first problem for a heated generalized second grade fluid,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 971–986, 2013.
View at: Google Scholar
M. Dehghan and M. Abbaszadeh, “A finite element method for the numerical solution of Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives,” Engineering with Computers, vol. 33, no. 3, pp. 587–605, 2017.
View at: Publisher Site | Google Scholar
O. Nikan and Z. Avazzadeh, “An improved localized radial basis-pseudospectral method for solving fractional reaction-subdiffusion problem,” Results in Physics, vol. 23, Article ID 104048, 2021.
View at: Publisher Site | Google Scholar
S. Zhai, X. Feng, and Y. He, “An unconditionally stable compact ADI method for three-dimensional time-fractional convection-diffusion equationiﬀusion equation,” Journal of Computational Physics, vol. 269, pp. 138–155, 2014.
View at: Publisher Site | Google Scholar
C. M. Chen, F. Liu, K. Burrage, and Y. Chen, “Numerical method of the variable-order Rayleigh-Stocks’ problem for a heated generalized second grade ﬂuid with fractional derivative,” IMA Journal of Applied Mathematics, pp. 1–21, 2005.
View at: Google Scholar
W. Chunhong, “Numerical solution for Stocks’ ﬁrst problem for a heated generalized second grade ﬂuid with fractional derivative,” Applied Numerical Mathematics, vol. 59, pp. 2571–2583, 2009.
View at: Google Scholar
Z. Guan, X. Wang, and J. Ouyang, “An improved finite difference/finite element method for the fractional Rayleigh–Stokes problem with a nonlinear source term,” Journal of Applied Mathematics and Computing, vol. 65, no. 1, pp. 451–479, 2021.
View at: Publisher Site | Google Scholar
U. Ali, F. A. Abdullah, and S. T. Mohyud-Din, “Modified implicit fractional difference scheme for 2D modified anomalous fractional sub-diffusion equation,” Advances in Difference Equations, vol. 2017, no. 1, pp. 1–14, 2017.
View at: Publisher Site | Google Scholar
E. Bazhlekova, B. Jin, R. Lazarov, and Z. Zhou, “An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid,” Numerische Mathematik, vol. 131, no. 1, pp. 1–31, 2015.
View at: Publisher Site | Google Scholar
A. Mohebbi, M. Abbaszadeh, and M. Dehghan, “Compact finite difference scheme and RBF meshless approach for solving 2D Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives,” Computer Methods in Applied Mechanics and Engineering, vol. 264, pp. 163–177, 2013.
View at: Publisher Site | Google Scholar
Z.-Z. Sun and X. Wu, “A fully discrete difference scheme for a diffusion-wave system,” Applied Numerical Mathematics, vol. 56, no. 2, pp. 193–209, 2006.
View at: Publisher Site | Google Scholar
O. Nikan, Z. Avazzadeh, and J. A. T. Machado, “Numerical study of the nonlinear anomalous reaction-subdiffusion process arising in the electroanalytical chemistry,” Journal of Computational Science, vol. 53, Article ID 101394, 2021.
View at: Publisher Site | Google Scholar
O. Nikan, Z. Avazzadeh, and J. T. Machado, “A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass transfer,” Journal of Advanced Research, vol. 32, 2021.
View at: Publisher Site | Google Scholar
O. Nikan, Z. Avazzadeh, and J. A. Tenreiro Machado, “Numerical investigation of fractional nonlinear sine-Gordon and Klein-Gordon models arising in relativistic quantum mechanics,” Engineering Analysis with Boundary Elements, vol. 120, pp. 223–237, 2020.
View at: Publisher Site | Google Scholar
F. Liu, P. Zhuang, Q. Liu, M. Zheng, and V. V. Anh, “Fractional-order systems, numerical techniques, and applications: finite element and spectral methods for multiterm time-and time-space fractional dynamic systems and applications,” In Fractional Order Systems, Academic Press, vol. 1, pp. 179–256, 2022.
View at: Publisher Site | Google Scholar
H. Ahmad, A. R. Seadawy, A. H. Ganie, S. Rashid, T. A. Khan, and H. Abu-Zinadah, “Approximate Numerical solutions for the nonlinear dispersive shallow water waves as the Fornberg–Whitham model equations,” Results in Physics, vol. 22, Article ID 103907, 2021.
View at: Publisher Site | Google Scholar
M. N. Khan, I. Hussain, I. Ahmad, and H. Ahmad, “A local meshless method for the numerical solution of space‐dependent inverse heat problems,” Mathematical Methods in the Applied Sciences, vol. 44, no. 4, pp. 3066–3079, 2021.
View at: Publisher Site | Google Scholar
F. M. Salama, N. N. A. Hamid, N. N. A. Hamid, and N. H. M. Ali, “An efficient modified hybrid explicit group iterative method for the time-fractional diffusion equation in two space dimensions,” AIMS Mathematics, vol. 7, no. 2, pp. 2370–2392, 2022.
View at: Publisher Site | Google Scholar
A. Ali and N. H. M. Ali, “On skewed grid point iterative method for solving 2D hyperbolic telegraph fractional differential equation,” Advances in Difference Equations, vol. 2019, no. 1, pp. 1–29, 2019.
View at: Publisher Site | Google Scholar
S. Patnaik and F. Semperlotti, “A generalized fractional-order elastodynamic theory for non-local attenuating media,” Proceedings of the Royal Society A, vol. 476, no. 2238, Article ID 20200200, 2020.
View at: Publisher Site | Google Scholar
S. Patnaik, M. Jokar, and F. Semperlotti, “Variable-order Approach to Nonlocal Elasticity: Theoretical Formulation, Order Identification via Deep Learning, and Applications,” Computational Mechanics, pp. 1–32, 2021.
View at: Google Scholar
W. Sumelka, “Fractional viscoelasticity,” Mechanics Research Communications, vol. 56, pp. 31–36, 2014.
View at: Publisher Site | Google Scholar
O. Nikan and Z. Avazzadeh, “An improved localized radial basis-pseudospectral method for solving fractional reaction-subdiffusion problem,” Results in Physics, vol. 23, Article ID 104048, 2021.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Abdul Hamid Ganie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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