Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-Diffusion Problem with Source Term

Anley, Eyaya Fekadie; Zheng, Zhoushun

doi:10.3390/math8111878

Cited by 9 publications

(13 citation statements)

References 49 publications

(55 reference statements)

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“…Additionally, we design a MATLAB code for each approach to obtain a numerical solution for three problems to be examined. When compared to earlier analytical [40,41] and numerical [13,42,43] methodologies, the derived numerical results attain excellent efficiency and accuracy. Furthermore, we present some parametric studies to highlight the reliability of our methods with the influence of fractional order derivative, the velocity, and positive diffusion parameters on the results.…”

Section: Introductionmentioning

confidence: 87%

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Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

Mustafa,

Ragb,

Salah

et al. 2023

Fractal Fract

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The aim of this study is to utilize a differential quadrature method with various kernels, such as Lagrange interpolation and discrete singular convolution, to tackle problems related to the Riesz fractional diffusion equation and the Riesz fractional advection–dispersion equation. The governing equation for convection and diffusion depends on both spatial and transient factors. By using the block marching technique, we transform these equations into an algebraic system using differential quadrature methods and the Caputo-type fractional operator. Next, we develop a MATLAB program that generates code capable of solving the fractional convection–diffusion equation in (1+2) dimensions for each shape function. Our goal is to ensure that our methods are reliable, accurate, efficient, and capable of convergence. To achieve this, we conduct two experiments, comparing the numerical and graphical results with both analytical and numerical solutions. Additionally, we evaluate the accuracy of our findings using the L∞ error. Our tests show that the differential quadrature method, which relies mainly on the discrete singular convolution shape function, is a highly effective numerical approach for fractional convective diffusion problems. It offers superior accuracy, faster convergence, and greater reliability than other techniques. Furthermore, we study the impact of fractional order derivatives, velocity, and positive diffusion parameters on the results.

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Section: Introductionmentioning

confidence: 87%

“…The expression "C x ≥ 0" represents the coefficient for the velocity parameter. Subject to the initial and boundary conditions are taken as follows [42]:…”

Section: The One-dimensional Riesz Space Fractional Advection Equationmentioning

confidence: 99%

Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

Mustafa,

Ragb,

Salah

et al. 2023

Fractal Fract

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show abstract

“…Pandey et al 38 introduced a method for solving STFRDE that combines the Laplace transform, HPM and He's polynomials. Anley and Zheng 39 presented a numerical approach based on a combined weighted from the shifted Grünwald–Letnikov difference operator with the Crank–Nicolson scheme for solving two‐dimensional Riesz space fractional convection–diffusion equations. Owolabi and Atangana 40 proposed the Fourier pseudo‐spectral method combined with the exponential time differencing scheme to solve the SFRD systems.…”

Section: Introductionmentioning

confidence: 99%

Numerical method for solving two‐dimensional of the space and space–time fractional coupled reaction‐diffusion equations

Hadhoud

Rageh

Agarwal³

2022

Math Methods in App Sciences

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This paper proposes the shifted Legendre Gauss–Lobatto collocation (SL‐GLC) scheme to solve two‐dimensional space‐fractional coupled reaction–diffusion equations (SFCRDEs). The proposed method is implemented by expressing the function and its spatial fractional derivatives as a finite expansion of shifted Legendre polynomials. Then the expansion coefficients are determined by reducing the SFCRDEs with their initial and boundary conditions to a system of ordinary differential equations for these coefficients. Subsequently, we applied the proposed method to discretize the temporal and spatial variables to convert the two‐dimensional spacetime fractional coupled reaction–diffusion equations (STFCRDEs) to a system of algebraic equations. Some results regarding the error estimation are obtained. Several examples are discussed to validate the capability and efficiency of the proposed scheme.

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“…Several methods have been developed, e.g., the force method, the slope deflection method, and the direct stiffness method, etc. Anley et al [1] considered a numerical difference approximation for solving two-dimensional Riesz space fractional convection-diffusion problem with source term over a finite domain. Kindelan et al [2] presented a method to obtain optimal finite difference formulas which maximize their frequency range of validity.…”

Section: Introductionmentioning

confidence: 99%

Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

Fogang¹

2021

Preprint

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This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations (or partial differential equations). The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional points or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness, etc.). The introduction of additional points permits us to satisfy boundary conditions and continuity conditions. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. Tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, the damping being considered. The efforts and displacements could be determined at any time.

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Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-Diffusion Problem with Source Term

Cited by 9 publications

References 49 publications

Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

Numerical method for solving two‐dimensional of the space and space–time fractional coupled reaction‐diffusion equations

Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

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