scholarly journals A New Operational Matrices-Based Spectral Method for Multi-Order Fractional Problems

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1471 ◽  
Author(s):  
M. Hamid ◽  
Oi Mean Foong ◽  
Muhammad Usman ◽  
Ilyas Khan ◽  
Wei Wang
Keyword(s):  
Applied Sciences ◽  
Collocation Method ◽  
Computational Method ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Physical Problems ◽  
Nonlinear Algebraic Equations ◽  
New Type ◽  
Error Bound Analysis ◽  
Bound Analysis

The operational matrices-based computational algorithms are the promising tools to tackle the problems of non-integer derivatives and gained a substantial devotion among the scientific community. Here, an accurate and efficient computational scheme based on another new type of polynomial with the help of collocation method (CM) is presented for different nonlinear multi-order fractional differentials (NMOFDEs) and Bagley–Torvik (BT) equations. The methods are proposed utilizing some new operational matrices of derivatives using Chelyshkov polynomials (CPs) through Caputo’s sense. Two different ways are adopted to construct the approximated (AOM) and exact (EOM) operational matrices of derivatives for integer and non-integer orders and used to propose an algorithm. The understudy problems have been transformed to an equivalent nonlinear algebraic equations system and solved by means of collocation method. The proposed computational method is authenticated through convergence and error-bound analysis. The exactness and effectiveness of said method are shown on some fractional order physical problems. The attained outcomes are endorsing that the recommended method is really accurate, reliable and efficient and could be used as suitable tool to attain the solutions for a variety of the non-integer order differential equations arising in applied sciences.

scholarly journals A Nonclassical Radau Collocation Method for Nonlinear Initial-Value Problems with Applications to Lane-Emden Type Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Mohammad Maleki ◽  
M. Tavassoli Kajani ◽  
I. Hashim ◽  
A. Kilicman ◽  
K. A. M. Atan
Keyword(s):  
Orthogonal Polynomials ◽  
Numerical Method ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Algebraic Equations ◽  
Initial Value ◽  
Weighted Interpolation ◽  
Collocation Points ◽  
Present Solution ◽  
Nonlinear Algebraic Equations

We propose a numerical method for solving nonlinear initial-value problems of Lane-Emden type. The method is based upon nonclassical Gauss-Radau collocation points, and weighted interpolation. Nonclassical orthogonal polynomials, nonclassical Radau points and weighted interpolation are introduced on arbitrary intervals. Then they are utilized to reduce the computation of nonlinear initial-value problems to a system of nonlinear algebraic equations. We also present the comparison of this work with some well-known results and show that the present solution is very accurate.

scholarly journals Numerical solution of two-dimensional nonlinear fractional order reaction-advection-diffusion equation by using collocation method

2021 ◽  
Vol 29 (2) ◽  
pp. 211-230
Author(s):  
Manpal Singh ◽  
S. Das ◽  
Rajeev ◽  
E-M. Craciun
Keyword(s):  
Numerical Solution ◽  
Collocation Method ◽  
Operational Matrix ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Existing Problems ◽  
Advection Diffusion Equation ◽  
Error Analyses ◽  
Nonlinear Algebraic Equations ◽  
Accurate Numerical Solution

Abstract In this article, two-dimensional nonlinear and multi-term time fractional diffusion equations are solved numerically by collocation method, which is used with the help of Lucas operational matrix. In the proposed method solutions of the problems are expressed in terms of Lucas polynomial as basis function. To determine the unknowns, the residual, initial and boundary conditions are collocated at the chosen points, which produce a system of nonlinear algebraic equations those have been solved numerically. The concerned method provides the highly accurate numerical solution. The accuracy of the approximate solution of the problem can be increased by expanding the terms of the polynomial. The accuracy and efficiency of the concerned method have been authenticated through the error analyses with some existing problems whose solutions are already known.

scholarly journals Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
H. Jafari ◽  
S. Nemati ◽  
R. M. Ganji
Keyword(s):  
Differential Equations ◽  
Chebyshev Polynomials ◽  
Original Problem ◽  
High Accuracy ◽  
Operational Matrices ◽  
Variable Order ◽  
Algebraic Equations ◽  
Numerical Tests ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations

AbstractIn this research, we study a general class of variable order integro-differential equations (VO-IDEs). We propose a numerical scheme based on the shifted fifth-kind Chebyshev polynomials (SFKCPs). First, in this scheme, we expand the unknown function and its derivatives in terms of the SFKCPs. To carry out the proposed scheme, we calculate the operational matrices depending on the SFKCPs to find an approximate solution of the original problem. These matrices, together with the collocation points, are used to transform the original problem to form a system of linear or nonlinear algebraic equations. We discuss the convergence of the method and then give an estimation of the error. We end by solving numerical tests, which show the high accuracy of our results.

New operational matrices approach for optimal control based on modified Chebyshev polynomials

2021 ◽  
Vol 2 (2) ◽  
pp. 68-78
Author(s):  
Anam Alwan Salih ◽  
Suha SHIHAB
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Chebyshev Polynomials ◽  
Optimization Technique ◽  
Computational Method ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Quadratic Optimal Control ◽  
Chebyshev Orthogonal Polynomial

The purpose of this paper is to introduce interesting modified Chebyshev orthogonal polynomial. Then, their new operational matrices of derivative and integration or modified Chebyshev polynomials of the first kind are introduced with explicit formulas. A direct computational method for solving a special class of optimal control problem, named, the quadratic optimal control problem is proposed using the obtained operational matrices. More precisely, this method is based on a state parameterization scheme, which gives an accurate approximation of the exact solution by utilizing a small number of unknown coefficients with the aid of modified Chebyshev polynomials. In addition, the constraint is reduced to some algebraic equations and the original optimal control problem reduces to optimization technique, which can be solved easily, and the approximate value of the performance index is calculated. Moreover, special attention is presented to discuss the convergence analysis and an upper bound of the error for the presented approximate solution is derived. Finally, some important illustrative examples of obtained results are shown and proved that powerful method in a simple way to get an optimal control of the considered.

Numerical Solution of Nonlinear Volterra-Fredholm Integral Equations Using Haar Wavelet Collocation Method

2017 ◽  
Vol 18 ◽  
pp. 50-59 ◽  
Author(s):  
S.C. Shiralashetti ◽  
R.A. Mundewadi
Keyword(s):  
Integral Equation ◽  
Error Analysis ◽  
Numerical Solution ◽  
Integral Equations ◽  
Collocation Method ◽  
Haar Wavelet ◽  
Fredholm Integral Equations ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Wavelet Collocation Method

In this paper, we present a numerical solution of nonlinear Volterra-Fredholm integral equations using Haar wavelet collocation method. Properties of Haar wavelet and its operational matrices are utilized to convert the integral equation into a system of algebraic equations, solving these equations using MATLAB to compute the Haar coefficients. The numerical results are compared with exact and existing method through error analysis, which shows the efficiency of the technique.

scholarly journals Some Algorithms for Solving Third-Order Boundary Value Problems Using Novel Operational Matrices of Generalized Jacobi Polynomials

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
W. M. Abd-Elhameed
Keyword(s):  
Boundary Value Problems ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Third Order ◽  
Generalized Jacobi Polynomials ◽  
Nonlinear Algebraic Equations ◽  
Derivatives Of

The main aim of this research article is to develop two new algorithms for handling linear and nonlinear third-order boundary value problems. For this purpose, a novel operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The suggested algorithms are built on utilizing the Galerkin and collocation spectral methods. Moreover, the principle idea behind these algorithms is based on converting the boundary value problems governed by their boundary conditions into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. We support our algorithms by a careful investigation of the convergence analysis of the suggested nonsymmetric generalized Jacobi expansion. Some illustrative examples are given for the sake of indicating the high accuracy and efficiency of the two proposed algorithms.

A computational method for time fractional partial integro-differential equations

2020 ◽  
Vol 26 (2) ◽  
pp. 315-323
Author(s):  
M. Mojahedfar ◽  
Abolfazl Tari ◽  
S. Shahmorad
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Error Estimation ◽  
Initial Conditions ◽  
Computational Method ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Numerical Examples

AbstractIn this paper, a class of time fractional partial integro-differential equations (FPIDEs) with initial conditions is studied. Some operational matrices are used to reduce a FPIDE problem to a system of algebraic equations with special properties. The resulted system is solved to give an approximate solution to the problem. Error estimation is also discussed for the approximate solution. Finally, some numerical examples are given to show the accuracy of the proposed method.

scholarly journals Collocation Method Based on Genocchi Operational Matrix for Solving Generalized Fractional Pantograph Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Abdulnasir Isah ◽  
Chang Phang ◽  
Piau Phang
Keyword(s):  
Fractional Derivative ◽  
Collocation Method ◽  
Upper Bound ◽  
Present Method ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Mathematical Tool ◽  
Algebraic Equations ◽  
Genocchi Polynomials

An effective collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations with initial and boundary conditions is presented. Using the properties of Genocchi polynomials, we derive a new Genocchi delay operational matrix which we used together with the Genocchi operational matrix of fractional derivative to approach the problems. The error upper bound for the Genocchi operational matrix of fractional derivative is also shown. Collocation method based on these operational matrices is applied to reduce the generalized fractional pantograph equations to a system of algebraic equations. The comparison of the numerical results with some existing methods shows that the present method is an excellent mathematical tool for finding the numerical solutions of generalized fractional pantograph equations.

scholarly journals New Wavelets Collocation Method for Solving Second-Order Multipoint Boundary Value Problems Using Chebyshev Polynomials of Third and Fourth Kinds

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
W. M. Abd-Elhameed ◽  
E. H. Doha ◽  
Y. H. Youssri
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Algebraic Equations ◽  
Chebyshev Wavelets ◽  
Multipoint Boundary ◽  
Nonlinear Algebraic Equations ◽  
Multipoint Boundary Value Problems ◽  
Fourth Kind

This paper is concerned with introducing two wavelets collocation algorithms for solving linear and nonlinear multipoint boundary value problems. The principal idea for obtaining spectral numerical solutions for such equations is employing third- and fourth-kind Chebyshev wavelets along with the spectral collocation method to transform the differential equation with its boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients which can be efficiently solved. Convergence analysis and some specific numerical examples are discussed to demonstrate the validity and applicability of the proposed algorithms. The obtained numerical results are comparing favorably with the analytical known solutions.

Sign in / Sign up

Export Citation Format

Share Document