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 Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 96 ◽  
Author(s):  
İbrahim Avcı ◽  
Nazim I. Mahmudov
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Main Idea ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Fractional Order Differential Equations ◽  
Fractional Differential ◽  
The Given

In this article, we propose a numerical method based on the fractional Taylor vector for solving multi-term fractional differential equations. The main idea of this method is to reduce the given problems to a set of algebraic equations by utilizing the fractional Taylor operational matrix of fractional integration. This system of equations can be solved efficiently. Some numerical examples are given to demonstrate the accuracy and applicability. The results show that the presented method is efficient and applicable.

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.

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 Numerical Method for 1-D Parabolic Equation with Nonlocal Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
M. Tadi ◽  
Miloje Radenkovic
Keyword(s):  
Parabolic Equation ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Coordinate Transformation ◽  
Nonlocal Boundary ◽  
Nonlinear Problems ◽  
Nonlocal Boundary Conditions ◽  
Local Method ◽  
One Dimensional ◽  
The Given

This paper is concerned with a local method for the solution of one-dimensional parabolic equation with nonlocal boundary conditions. The method uses a coordinate transformation. After the coordinate transformation, it is then possible to obtain exact solutions for the resulting equations in terms of the local variables. These exact solutions are in terms of constants of integration that are unknown. By imposing the given boundary conditions and smoothness requirements for the solution, it is possible to furnish a set of linearly independent conditions that can be used to solve for the constants of integration. A number of examples are used to study the applicability of the method. In particular, three nonlinear problems are used to show the novelty of the method.

scholarly journals A Langevin-Type q-Variant System of Nonlinear Fractional Integro-Difference Equations with Nonlocal Boundary Conditions

2022 ◽  
Vol 6 (1) ◽  
pp. 45
Author(s):  
Ravi P. Agarwal ◽  
Hana Al-Hutami ◽  
Bashir Ahmad
Keyword(s):  
Boundary Conditions ◽  
Difference Equations ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence And Uniqueness Results ◽  
New Class ◽  
Uniqueness Results ◽  
Multipoint Boundary Conditions ◽  
The Given ◽  
Variant System

We introduce a new class of boundary value problems consisting of a q-variant system of Langevin-type nonlinear coupled fractional integro-difference equations and nonlocal multipoint boundary conditions. We make use of standard fixed-point theorems to derive the existence and uniqueness results for the given problem. Illustrative examples for the obtained results are also presented.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

scholarly journals Multi-term fractional differential equations with nonlocal boundary conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 1519-1536
Author(s):  
Bashir Ahmad ◽  
Najla Alghamdi ◽  
Ahmed Alsaedi ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
The Given

AbstractWe introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.

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