scholarly journals Toward the Approximate Solution for Fractional Order Nonlinear Mixed Derivative and Nonlocal Boundary Value Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Hammad Khalil ◽  
Mohammed Al-Smadi ◽  
Khaled Moaddy ◽  
Rahmat Ali Khan ◽  
Ishak Hashim
Keyword(s):  
Approximate Solution ◽  
Boundary Conditions ◽  
Operational Matrix ◽  
Nonlocal Boundary ◽  
Coupled System ◽  
Nonlocal Boundary Conditions ◽  
Mixed Derivative ◽  
Test Problems ◽  
Operational Matrices ◽  
Nonlinear Coupled System

The paper is devoted to the study of operational matrix method for approximating solution for nonlinear coupled system fractional differential equations. The main aim of this paper is to approximate solution for the problem under two different types of boundary conditions,m^-point nonlocal boundary conditions and mixed derivative boundary conditions. We develop some new operational matrices. These matrices are used along with some previously derived results to convert the problem under consideration into a system of easily solvable matrix equations. The convergence of the developed scheme is studied analytically and is conformed by solving some test problems.

scholarly journals A Novel Method for Solution of Fractional Order Two-Dimensional Nonlocal Heat Conduction Phenomena

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Hammad Khalil ◽  
Ishak Hashim ◽  
Waqar Ahmad Khan ◽  
Abuzar Ghaffari
Keyword(s):  
Heat Conduction ◽  
Fractional Order ◽  
Operational Matrix ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Test Problems ◽  
Operational Matrices ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Operational Matrix Method

In this paper, we have extended the operational matrix method for approximating the solution of the fractional-order two-dimensional elliptic partial differential equations (FPDEs) under nonlocal boundary conditions. We use a general Legendre polynomials basis and construct some new operational matrices of fractional order operations. These matrices are used to convert a sample nonlocal heat conduction phenomenon of fractional order to a structure of easily solvable algebraic equations. The solution of the algebraic structure is then used to approximate a solution of the heat conduction phenomena. The proposed method is applied to some test problems. The obtained results are compared with the available data in the literature and are found in good agreement.Dedicated to my father Mr. Sher Mumtaz, (1955-2021), who gave me the basic knowledege of mathematics.

A Meshfree Computational Approach Based on Multiple-Scale Pascal Polynomials for Numerical Solution of a 2D Elliptic Problem with Nonlocal Boundary Conditions

2019 ◽  
Vol 17 (10) ◽  
pp. 1950080 ◽  
Author(s):  
Ömer Oruç
Keyword(s):  
Boundary Conditions ◽  
Elliptic Problem ◽  
Numerical Solutions ◽  
Nonlocal Boundary ◽  
Meshfree Method ◽  
Multiple Scale ◽  
Nonlocal Boundary Conditions ◽  
Condition Numbers ◽  
Test Problems ◽  
Numerical Integrations

A two-dimensional (2D) elliptic problem with nonlocal boundary conditions on both regular and irregular domains is solved numerically by Pascal polynomial basis unified with multiple-scale technique. Very accurate numerical solutions and quite reasonable condition numbers are obtained with the proposed method which is also a truly meshfree method since difficult meshing processes or numerical integrations over domains are not needed for considered problems. Four test problems are solved to show the accuracy and efficiency of the proposed method. Also stability of the method is studied against large noise effect.

scholarly journals Numerical solution of a boundary value problem for a loaded parabolic equation with nonlocal boundary conditions

2020 ◽  
pp. 15-28
Author(s):  
В.М. Абдуллаев
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Test Problems ◽  
Method Of Solution ◽  
Numerical Method Of Solution

В работе с использованием метода прямых исследуется численное решение краевой задачи относительно нагруженного параболического уравнения с нелокальными краевыми условиями. Получены расчетные формулы и приводится алгоритм для решения задачи. Приведены результаты численного решения двух тестовых задач, иллюстрирующие эффективность предложенного подхода In the work, we propose a numerical method of solution to the boundary-value problem with respect to the loaded parabolic equation with nonlocal boundary conditions. We have obtained formulas and derived an algorithm for the solution of the problem. We provide the results of numerical solution to two test problems, which illustrates the efficiency of the approach proposed.

scholarly journals An Efficient Spectral Approximation for Solving Several Types of Parabolic PDEs with Nonlocal Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
E. Tohidi ◽  
A. Kılıçman
Keyword(s):  
Boundary Conditions ◽  
Operational Matrix ◽  
Nonlocal Boundary ◽  
Main Idea ◽  
Nonlocal Boundary Conditions ◽  
Algebraic Equations ◽  
Parabolic Pdes ◽  
Direct Collocation ◽  
Operational Matrix Of Differentiation ◽  
The Given

The problem of solving several types of one-dimensional parabolic partial differential equations (PDEs) subject to the given initial and nonlocal boundary conditions is considered. The main idea is based on direct collocation and transforming the considered PDEs into their associated algebraic equations. After approximating the solution in the Legendre matrix form, we use Legendre operational matrix of differentiation for representing the mentioned algebraic equations clearly. Three numerical illustrations are provided to show the accuracy of the presented scheme. High accurate results with respect to the Bernstein Tau technique and Sinc collocation method confirm this accuracy.

scholarly journals Extension of Operational Matrix Technique for the Solution of Nonlinear System of Caputo Fractional Differential Equations Subjected to Integral Type Boundary Constrains

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1154
Author(s):  
Hammad Khalil ◽  
Murad Khalil ◽  
Ishak Hashim ◽  
Praveen Agarwal
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Algebraic Structure ◽  
Operational Matrix ◽  
Coupled System ◽  
Operational Matrices ◽  
Integral Type ◽  
Nonlinear Coupled System ◽  
Fractional Differential

We extend the operational matrices technique to design a spectral solution of nonlinear fractional differential equations (FDEs). The derivative is considered in the Caputo sense. The coupled system of two FDEs is considered, subjected to more generalized integral type conditions. The basis of our approach is the most simple orthogonal polynomials. Several new matrices are derived that have strong applications in the development of computational scheme. The scheme presented in this article is able to convert nonlinear coupled system of FDEs to an equivalent S-lvester type algebraic equation. The solution of the algebraic structure is constructed by converting the system into a complex Schur form. After conversion, the solution of the resultant triangular system is obtained and transformed back to construct the solution of algebraic structure. The solution of the matrix equation is used to construct the solution of the related nonlinear system of FDEs. The convergence of the proposed method is investigated analytically and verified experimentally through a wide variety of test problems.

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