scholarly journals A matrix approach to solving hyperbolic partial differential equations using Bernoulli polynomials

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 993-1000 ◽  
Author(s):  
Bicer Erdem ◽  
Salih Yalcinbas
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Error Analysis ◽  
Matrix Equation ◽  
Bernoulli Polynomials ◽  
Hyperbolic Partial Differential Equations ◽  
Matrix Approach ◽  
Residual Function ◽  
Partial Differential ◽  
The Matrix

The present study considers the solutions of hyperbolic partial differential equations. For this, an approximate method based on Bernoulli polynomials is developed. This method transforms the equation into the matrix equation and the unknown of this equation is a Bernoulli coefficients matrix. To demostrate the validity and applicability of the method, an error analysis developed based on residual function. Also examples are presented to illustrate the accuracy of the method.

Matrix Approach to Discretization of Ordinary and Partial Differential Equations of Arbitrary Real Order: The Matlab Toolbox

2009 ◽  
Author(s):  
Igor Podlubny ◽  
Tomas Skovranek ◽  
Blas M. Vinagre Jara
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Matrix Approach ◽  
Matlab Toolbox ◽  
Partial Differential ◽  
The Matrix ◽  
Fractional Differential ◽  
Derivatives Of ◽  
Partial Fractional Differential Equations

The method developed recently by Podlubny et al. (I. Podlubny, Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, pp. 359–386; I. Podlubny et al., Journal of Computational Physics, vol. 228, no. 8, 1 May 2009, pp. 3137–3153) makes it possible to immediately obtain the discretization of ordinary and partial differential equations by replacing the derivatives with their discrete analogs in the form of triangular strip matrices. This article presents a Matlab toolbox that implements the matrix approach and allows easy and convenient discretization of ordinary and partial differential equations of arbitrary real order. The basic use of the functions implementing the matrix approach to discretization of derivatives of arbitrary real order (so-called fractional derivatives, or fractional-order derivatives), and to solution of ordinary and partial fractional differential equations, is illustrated by examples with explanations.

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