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.

scholarly journals Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
Yongjin Li ◽  
Kamal Shah
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Coupled Systems ◽  
Test Problems ◽  
Operational Matrices ◽  
Partial Differential ◽  
Established Technique

We develop a numerical method by using operational matrices of fractional order integrations and differentiations to obtain approximate solutions to a class of coupled systems of fractional order partial differential equations (FPDEs). We use shifted Legendre polynomials in two variables. With the help of the aforesaid matrices, we convert the system under consideration to a system of easily solvable algebraic equation of Sylvester type. During this process, we need no discretization of the data. We also provide error analysis and some test problems to demonstrate the established technique.

scholarly journals Analysis of Fractional Systems using Haar Wavelet

2019 ◽  
Vol 8 (9S) ◽  
pp. 455-459
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Haar Wavelet ◽  
Computational Technique ◽  
Operational Matrices ◽  
Partial Differential ◽  
Fractional Systems ◽  
Mathematical Expressions ◽  
Modeling Systems

Wavelets are relatively new tool and have quite been thriving domain in mathematical research. Numerical solutions of differential and integral equations require development of accurate and fast algorithms based on wavelets. This is more pertinent for those problems having localized solutions, both in position and scale. Haar wavelet offers a promising solution bases due to simple mathematical expressions and multi-resolution properties. In this paper, A Haar wavelet based method to solve partial differential equations (PDE) modeling fractional systems is presented. Operational approach is based on representing various integro-differential mathematical operations in terms of matrices. In this article, firstly introduction of Haar wavelet and different operational matrices used for the analysis of fractional systems are presented. A modified computational technique is explained to solve variety of partial differential equations modeling systems of fractional order. This method achieves the solutions by solving Sylvester equation using MATLAB. Demonstrations are provided with the help of two illustrative examples by suitable comparisons with exact solutions.

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.

Steady Gravity Flow of Frictional-Cohesive Solids in Converging Channels

1964 ◽  
Vol 31 (1) ◽  
pp. 5-11 ◽  
Author(s):  
A. W. Jenike
Keyword(s):  
Partial Differential Equations ◽  
Stress Field ◽  
Differential Equations ◽  
Radial Stress ◽  
Numerical Solutions ◽  
Yield Function ◽  
Hyperbolic Partial Differential Equations ◽  
Stress Fields ◽  
Special Importance ◽  
Partial Differential

Frictional-cohesive solids such as soil, ores, chemicals, sugar, flour are regarded as plastic and represented by the Jenike-Shield yield function [1] during steady flow. The stress-strain rate relations are based on isotropy, continuity, and a one-to-one dependence of density on the major pressure. In plane strain and in axial symmetry the stress field requires the solution of a system of two hyperbolic partial differential equations. The velocity field can then be computed by solving another system of two linear homogeneous partial differential equations of the hyperbolic type. In straight conical channels, a particular stress field called the “radial stress field” assumes a special importance because evidence has been presented elsewhere that all general fields tend to approach the radial stress fields in the vicinity of the vertex. Examples of numerical solutions of radial stress fields are given.

scholarly journals A generalized scheme based on shifted Jacobi polynomials for numerical simulation of coupled systems of multi-term fractional-order partial differential equations

2017 ◽  
Vol 20 (1) ◽  
pp. 11-29 ◽  
Author(s):  
Kamal Shah ◽  
Hammad Khalil ◽  
Rahmat Ali Khan
Keyword(s):  
Numerical Simulation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Jacobi Polynomials ◽  
Initial Conditions ◽  
Coupled Systems ◽  
Operational Matrices ◽  
Partial Differential ◽  
Shifted Jacobi Polynomials

Due to the increasing application of fractional calculus in engineering and biomedical processes, we analyze a new method for the numerical simulation of a large class of coupled systems of fractional-order partial differential equations. In this paper, we study shifted Jacobi polynomials in the case of two variables and develop some new operational matrices of fractional-order integrations as well as fractional-order differentiations. By the use of these operational matrices, we present a new and easy method for solving a generalized class of coupled systems of fractional-order partial differential equations subject to some initial conditions. We convert the system under consideration to a system of easily solvable algebraic equation without discretizing the system, and obtain a highly accurate solution. Also, the proposed method is compared with some other well-known differential transform methods. The proposed method is computer oriented. We useMatLabto perform the necessary calculation. The next two parts will appear soon.

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