scholarly journals An Efficient Numerical Approach for Solving Nonlinear Coupled Hyperbolic Partial Differential Equations with Nonlocal Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi ◽  
Eman S. Alaidarous
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Collocation Method ◽  
Jacobi Polynomials ◽  
Nonlocal Conditions ◽  
Initial Boundary ◽  
Variable Coefficients ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Hyperbolic Pdes

One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (PDEs) as well as PDEs with variable coefficients. A numerical solution based on a Jacobi collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary nonlocal conservation conditions. This approach, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equation which is far easier to solve. In fact, we deal with initial-boundary coupled hyperbolic PDEs with variable coefficients as well as initial-nonlocal conditions. Using triangular, soliton, and exponential-triangular solutions as exact solutions, the obtained results show that the proposed numerical algorithm is efficient and very accurate.

A NONSTANDARD FINITE DIFFERENCE SCHEME FOR TWO-SIDED SPACE-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250079 ◽  
Author(s):  
SHAHER MOMANI ◽  
ABDULLAH ABU RQAYIQ ◽  
DUMITRU BALEANU
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Boundary ◽  
Variable Coefficients ◽  
Finite Domain ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Left Handed ◽  
Nonstandard Finite Difference Scheme ◽  
Initial Boundary Value

In this paper, we apply the Mickens nonstandard discretization method to solve a class of initial-boundary value fractional partial differential equations with variable coefficients on a finite domain, and thereby increase the accuracy of the solutions. We examine the case when a left-handed and a right-handed fractional spatial derivative may be present in the partial differential equation. Two numerical examples using this method are presented and compared successfully with the exact analytical solutions.

scholarly journals On a Computational Method for Non-integer Order Partial Differential Equations in Two Dimensions

2019 ◽  
Vol 12 (1) ◽  
pp. 39-57
Author(s):  
Muhammad Ikhlaq Chohan ◽  
Kamal Shah
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Jacobi Polynomials ◽  
Two Dimensions ◽  
Computational Method ◽  
Hyperbolic Partial Differential Equations ◽  
Unknown Coefficient ◽  
Operational Matrices ◽  
Partial Differential

This manuscript is concerning to investigate numerical solutions for different classesincluding parabolic, elliptic and hyperbolic partial differential equations of arbitrary order(PDEs). The proposed technique depends on some operational matrices of fractional order differentiation and integration. To compute the mentioned operational matrices, we apply shifted Jacobi polynomials in two dimension. Thank to these matrices, we convert the (PDE) under consideration to an algebraic equation which is can be easily solved for unknown coefficient matrix required for the numerical solution. The proposed method is very efficient and need no discretization of the data for the proposed (PDE). The approximate solution obtain via this method is highly accurate and the computation is easy. The proposed method is supported by solving various examples from well known articles.

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