A Jacobi collocation approximation for nonlinear coupled viscous Burgers’ equation

Doha, E. H.; Bhrawy, A. H.; Abdelkawy, Mohamed; Hafez, Ramy M.

doi:10.2478/s11534-014-0429-z

Cited by 11 publications

(10 citation statements)

References 38 publications

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“…Since the Jacobi spectral collocation method approximates the initial-boundary values problems in physical space and it is a global method, it is very useful to implement and adapt it to various problems, including variable-orders and nonlinear problems (see, for instance [53][54][55][56][57][58]). In this section, two new algorithms for solving one-and two-dimensional variable-order fractional cable equations are proposed based on Jacobi spectral collocation approximation together with the Jacobi operational matrix for variable-order fractional derivative.…”

Section: Jacobi Spectral Collocation Methodsmentioning

confidence: 99%

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Bhrawy

Zaky²

2014

Nonlinear Dyn

208

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The cable equation plays a central role in many areas of electrophysiology and in modeling neuronal dynamics. This paper reports an accurate spectral collocation method for solving one-and twodimensional variable-order fractional nonlinear cable equations. The proposed method is based on shifted Jacobi collocation procedure in conjunction with the shifted Jacobi operational matrix for variable-order fractional derivatives, described in the sense of Caputo. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations. In addition, the method reduces the variable-order fractional nonlinear cable equation to a simpler problem that consists of solving a system of algebraic equations. The validity and effectiveness of the method are demonstrated by solving three numerical examples. The convergence of the method is graphically analyzed. The results demonstrate that the proposed method is a powerful algorithm with high accuracy for solving the variable-order nonlinear partial differential equations.Keywords One-dimensional cable equation · Two-dimensional variable-order nonlinear cable equation · Collocation method · Jacobi polynomials · Operational matrix of fractional derivative · Variable-order derivative

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Section: Jacobi Spectral Collocation Methodsmentioning

confidence: 99%

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Bhrawy

Zaky²

2014

Nonlinear Dyn

208

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“…The Bernoulli collocation approach was utilized by Doha et al in [12] for the purpose of solving hyperbolic telegraph equations. In their study [13], Dascioglu and Sezer looked at the use of BPM in the resolution of high-order modified pantograph models.…”

Section: Introductionmentioning

confidence: 99%

Sixth-Kind Chebyshev and Bernoulli Polynomial Numerical Methods for Solving Nonlinear Mixed Partial Integrodifferential Equations with Continuous Kernels

Al-Bugami,

Abdou,

Mahdy

2023

Journal of Function Spaces

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In the present paper, a new efficient technique is described for solving nonlinear mixed partial integrodifferential equations with continuous kernels. Using the separation of variables, the nonlinear mixed partial integrodifferential equation is converted to a nonlinear Fredholm integral equation. Then, using different numerical methods, the Bernoulli polynomial method and the Chebyshev polynomials of the sixth kind, the nonlinear Fredholm integral equation has been reduced into a system of nonlinear algebraic equations. The Banach fixed-point theory is utilized in order to have a conversation about the nonlinear mixed integral equation’s solution, namely, its existence and uniqueness. In addition, we talk about the convergence and stability of the solution. Finally, a comparison between the two different methods and some other famous methods is presented through various examples. All the numerical results are calculated and obtained using the Maple software.

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“…In recent years there has been a high level of interest of employing spectral methods for numerically solving many types of integral and differential equations, due to their ease of applying them for finite and infinite domains [1][2][3][4][5][6][7][8]. The speed of convergence is one of the great advantages of spectral method.…”

Section: Introductionmentioning

confidence: 99%

A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation

Bhrawy

2014

Abstract and Applied Analysis

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A new spectral shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method is developed and analyzed to solve a class of two-dimensional initial-boundary fractional diffusion equations with variable coefficients. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomialsPL,n(x)PL,m(y), for the function and its space-fractional derivatives occurring in the partial fractional differential equation (PFDE), is assumed; the expansion coefficients are then determined by reducing the PFDE with its boundary and initial conditions to a system of ordinary differential equations (SODEs) for these coefficients. This system may be solved numerically by using the fourth-order implicit Runge-Kutta (IRK) method. This method, in contrast to common finite-difference and finite-element methods, has the exponential rate of convergence for the two spatial discretizations. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier.

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A Jacobi collocation approximation for nonlinear coupled viscous Burgers’ equation

Cited by 11 publications

References 38 publications

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Sixth-Kind Chebyshev and Bernoulli Polynomial Numerical Methods for Solving Nonlinear Mixed Partial Integrodifferential Equations with Continuous Kernels

A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation

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