A new exponential Chebyshev operational matrix of derivatives for solving high-order ordinary differential equations in unbounded domains

2016 ◽  
Vol 7 (1) ◽  
pp. 19 ◽  
Author(s):  
Mohamed Ramadan ◽  
Kamal Raslan ◽  
Talaat El Danaf ◽  
Mohamed A. Abd Elsalam
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Operational Matrix ◽  
Variable Coefficients ◽  
High Order ◽  
Algebraic Equations ◽  
Linear Ordinary Differential Equations ◽  
The Given ◽  
Derivatives Of

The purpose of this paper is to investigate a new exponential Chebyshev (EC) operational matrix of derivatives. The new operational matrix of derivatives of the EC functions is derived and introduced for solving high-order linear ordinary differential equations with variable coefficients in unbounded domain using the collocation method. This method transforms the given differential equation and conditions to matrix equation with unknown EC coefficients. These matrices together with the collocation method are utilized to reduce the solution of high-order ordinary differential equations to the solution of a system of algebraic equations. The solution is obtained in terms of EC functions. Numerical examples are given to demonstrate the validity and applicability of the method. The obtained numerical results are compared with others existing methods and the exact solution where it shown to be very attractive with good accuracy.

Solving systems of high-order linear ordinary differential equations with variable coefficients by means of exponential Chebyshev collocation method

2017 ◽  
Vol 8 (1-2) ◽  
pp. 40 ◽  
Author(s):  
Mohamed Ramadan ◽  
Kamal Raslan ◽  
Talaat El Danaf ◽  
Mohamed A. Abd Elsalam
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Variable Coefficients ◽  
High Order ◽  
Matrix Equations ◽  
Algebraic Equations ◽  
Order Linear ◽  
Linear Ordinary Differential Equations ◽  
Linear Algebraic Equations

The purpose of this paper is to investigate the use of exponential Chebyshev (EC) collocation method for solving systems of high-order linear ordinary differential equations with variable coefficients with new scheme, using the EC collocation method in unbounded domains. The EC functions approach deals directly with infinite boundaries without singularities. The method transforms the system of differential equations and the given conditions to block matrix equations with unknown EC coefficients. By means of the obtained matrix equations, a new system of equations which corresponds to the system of linear algebraic equations is gained. Numerical examples are given to illustrative the validity and applicability of the method.

scholarly journals A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Linear Functional ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Order Differential Equations ◽  
The Given ◽  
Functional Argument ◽  
Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

scholarly journals On Conjugacy of High-Order Linear Ordinary Differential Equations

1994 ◽  
Vol 1 (1) ◽  
pp. 1-8
Author(s):  
T. Chanturia
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
High Order ◽  
Summable Function ◽  
Order Linear ◽  
Linear Ordinary Differential Equations

Abstract It is shown that the differential equation u (n) = p(t)u, where n ≥ 2 and p : [a, b] → ℝ is a summable function, is not conjugate in the segment [a, b], if for some l ∈ {1, . . . , n – 1}, α ∈]a, b[ and β ∈]α, b[ the inequalities hold.

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 Numerical solutions of multi-order fractional differential equations by Boubaker polynomials

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 226-230 ◽  
Author(s):  
A. Bolandtalat ◽  
E. Babolian ◽  
H. Jafari
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Boubaker Polynomials ◽  
Fractional Differential ◽  
The Given

AbstractIn this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on Boubaker polynomials. Using this operational matrix, the given problem is converted into a set of algebraic equations. Illustrative examples are are given to demonstrate the efficiency and simplicity of this technique.

scholarly journals Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 502
Author(s):  
Benjamin Zanger ◽  
Christian B. Mendl ◽  
Martin Schulz ◽  
Martin Schreiber
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Optimization Problems ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
High Order ◽  
Quantum Computers ◽  
Integration Methods ◽  
Linear Ordinary Differential Equations ◽  
Legendre Collocation Method

Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6th order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an "oracle" within quantum search algorithms for inverse problems.

scholarly journals Rational Chebyshev functions with new collocation points in semi-infinite domains for solving higher-order linear ordinary differential equations

2015 ◽  
Vol 11 (7) ◽  
pp. 5403-5410 ◽  
Author(s):  
Mohamed Abdel -Latif Ramadan
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Algebraic Equations ◽  
Infinite Domain ◽  
Order Linear ◽  
Linear Ordinary Differential Equations ◽  
Infinite Domains ◽  
Collocation Points ◽  
Rational Chebyshev

The purpose of this paper is to investigate the use of rational Chebyshev (RC) functions for solving higher-order linear ordinary differential equations with variable coefficients on a semi-infinite domain using new rational Chebyshev collocation points.  This method transforms the higher-order linear ordinary differential equations and the given conditions to matrix equations with unknown rational Chebyshev coefficients. These matrices together with the collocation method are utilized to reduce the solution of higher-order ordinary differential equations to the solution of a system of algebraic equations. The solution is obtained in terms of RC series. Numerical examples are given to demonstrate the validity and applicability of the method. The obtained numerical results are compared with others existing methods and the exact solution where it shown to be very attractive and maintains better accuracy.

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