Bi-Finite Difference Method to Solve Second-Order Nonlinear Hyperbolic Telegraph Equation in Two Dimensions

Raslan, K. R.; Ali, Khalid K.; Al-Jeaid, Hind K.; Shaalan, M.A.

doi:10.1155/2022/1782229

Cited by 4 publications

(2 citation statements)

References 25 publications

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“…Al-Maskari and Karaa [13] considered the numerical approximation of a generalized fractional Oldroyd-B fluid problem with a semidiscrete scheme based on the piecewise linear Galerkin finite element method. In other engineering fields, Raslan et al [14] present a computational scheme using the bi-finite difference method (Bi-FDM) to solve the hyperbolic telegraph equation in two dimensions. The fluid flow model governed by partial differential equations is reduced to a system of nonlinear ordinary differential equations for solving the calculation in reference [15].…”

Section: Introductionmentioning

confidence: 99%

“…The fractionalderivative-type viscoelasticity principal structure relationship established by the theory on fractional derivatives was a newly developed principal structure model nearly two decades ago. Compared to the Riemann-Liouville [16] fractional derivative, Caputo fractional derivatives [14] are more suitable and convenient for numerical analysis. Khan and Rasheed [17] have adopted the Galerkin finite element method to determine the approximate solution that is uniform with the finite difference approximation of the Caputo fractional time derivative.…”

Section: Introductionmentioning

confidence: 99%

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A Finite Difference Method for Solving Unsteady Fractional Oldroyd-B Viscoelastic Flow Based on Caputo Derivative

Wang

2023

Advances in Mathematical Physics

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In this paper, the effect of a fractional constitutive model on the rheological properties of fluids and its application in numerical simulation are investigated, which is important to characterize the rheological properties of fluids and physical characteristics of materials more accurately. Based on this consideration, numerical simulation and analytical study of unsteady fractional Oldroyd-B viscoelastic flow are carried out. In order to improve the degree of accuracy, the mixed partial derivative including the fractional derivative in the differential equation is converted effectively by integrating by parts instead of by direct discretization. Then, the stability, convergence, and unique solvability of the difference scheme are verified. The validity of the finite difference method is tested by making comparisons with analytical solutions for two kinds of fractional Oldroyd-B viscoelastic flow. Numerical results obtained using the finite difference method are in good agreement with analytical solutions obtained via the variable separation method. Viscoelastic characteristics of the unsteady Poiseuille flow are similar to the second-order fluid or integer-order Oldroyd-B fluid when the parameter is close to 0 or to 1. Oscillation characteristics of fractional viscoelastic oscillatory flow are similar to those of the classical viscoelastic fluid under the same condition. Compared with the previous research, the present work studies the rheological properties of fluids with the finite difference method, and the application of fractional constitutive models in describing the rheological properties of fluids is developed. Meanwhile, more cases reflecting the fractional-order characteristics are given. The present method can accurately capture the flow characteristics of unsteady fractional Oldroyd-B viscoelastic fluid and is applicable for the generalized fractional fluid.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Finite Difference Method for Solving Unsteady Fractional Oldroyd-B Viscoelastic Flow Based on Caputo Derivative

Wang

2023

Advances in Mathematical Physics

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Investigation of the analytical and numerical solutions with bifurcation analysis for the (2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili equation

Ali

Mehanna²,

Shaalan

2023

Opt Quant Electron

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In this work, we investigate the solutions of the (2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili (BKP) equation by three powerful analytical methods: the $$\exp _{a}$$ exp a function method, the $$(\frac{G'}{G})$$ ( G ′ G ) -expansion method, and the Sine-Gordon expansion method. This equation describes the nonlinear wave propagation in many applications like waves of evolutionary shallow water, electrical networks, and engineering devices. Moreover, we study the solutions numerically via the finite difference method. We analyze the bifurcation of dynamical system resulting from the BKP equation. Finally, the majority of our solutions are displayed graphically to present the strength of imposed methods.

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Analytical and numerical solutions with bifurcation analysis for the nonlinear evolution equation in (2+1)-dimensions

Ali

Mehanna²,

Shaalan³

et al. 2023

Results in Physics

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Bi-Finite Difference Method to Solve Second-Order Nonlinear Hyperbolic Telegraph Equation in Two Dimensions

Cited by 4 publications

References 25 publications

A Finite Difference Method for Solving Unsteady Fractional Oldroyd-B Viscoelastic Flow Based on Caputo Derivative

A Finite Difference Method for Solving Unsteady Fractional Oldroyd-B Viscoelastic Flow Based on Caputo Derivative

Investigation of the analytical and numerical solutions with bifurcation analysis for the (2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili equation

Analytical and numerical solutions with bifurcation analysis for the nonlinear evolution equation in (2+1)-dimensions

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