scholarly journals Approximation Techniques for Solving Linear Systems of Volterra Integro-Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Ahmad Issa ◽  
Naji Qatanani ◽  
Adnan Daraghmeh
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Differential Equations ◽  
Linear Systems ◽  
Collocation Method ◽  
Wavelet Method ◽  
Numerical Examples ◽  
Approximation Techniques ◽  
Sinc Functions ◽  
Chebyshev Wavelet

In this paper, a collocation method using sinc functions and Chebyshev wavelet method is implemented to solve linear systems of Volterra integro-differential equations. To test the validity of these methods, two numerical examples with known exact solution are presented. Numerical results indicate that the convergence and accuracy of these methods are in good a agreement with the analytical solution. However, according to comparison of these methods, we conclude that the Chebyshev wavelet method provides more accurate results.

scholarly journals Solving Some Differential Equations Arising in Electric Engineering Using Hermite Polynomials

2020 ◽  
Vol 12 (4) ◽  
pp. 517-523
Author(s):  
G. Singh ◽  
I. Singh
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Hermite Polynomials ◽  
Electric Circuit ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Electric Engineering ◽  
Collocation Points ◽  
Linear Algebraic Equations

In this paper, a collocation method based on Hermite polynomials is presented for the numerical solution of the electric circuit equations arising in many branches of sciences and engineering. By using collocation points and Hermite polynomials, electric circuit equations are transformed into a system of linear algebraic equations with unknown Hermite coefficients. These unknown Hermite coefficients have been computed by solving such algebraic equations. To illustrate the accuracy of the proposed method some numerical examples are presented.

LEAST-SQUARES MESHFREE COLLOCATION METHOD

2012 ◽  
Vol 09 (02) ◽  
pp. 1240031 ◽  
Author(s):  
BO-NAN JIANG
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Least Squares ◽  
Collocation Method ◽  
Present Method ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
Dual Variables

A least-squares meshfree collocation method is presented. The method is based on the first-order differential equations in order to result in a better conditioned linear algebraic equations, and to obtain the primary variables (displacements) and the dual variables (stresses) simultaneously with the same accuracy. The moving least-squares approximation is employed to construct the shape functions. The sum of squared residuals of both differential equations and boundary conditions at nodal points is minimized. The present method does not require any background mesh and additional evaluation points, and thus is a truly meshfree method. Unlike other collocation methods, the present method does not involve derivative boundary conditions, therefore no stabilization terms are needed, and the resulting stiffness matrix is symmetric positive definite. Numerical examples show that the proposed method possesses an optimal rate of convergence for both primary and dual variables, if the nodes are uniformly distributed. However, the present method is sensitive to the choice of the influence length. Numerical examples include one-dimensional diffusion and convection-diffusion problems, two-dimensional Poisson equation and linear elasticity problems.

scholarly journals An Exact Solution of the Binary Singular Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Baiqing Sun ◽  
Kun Tang ◽  
Hongmei Zhang ◽  
Shan Xiong
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Reproducing Kernel ◽  
Applied Mathematics ◽  
Singular Problem ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Singularity Problem ◽  
Numerical Examples ◽  
Singular Coefficients

Singularity problem exists in various branches of applied mathematics. Such ordinary differential equations accompany singular coefficients. In this paper, by using the properties of reproducing kernel, the exact solution expressions of dual singular problem are given in the reproducing kernel space and studied, also for a class of singular problem. For the binary equation of singular points, I put it into the singular problem first, and then reuse some excellent properties which are applied to solve the method of solving differential equations for its exact solution expression of binary singular integral equation in reproducing kernel space, and then obtain its approximate solution through the evaluation of exact solutions. Numerical examples will show the effectiveness of this method.

scholarly journals Modified Chebyshev Collocation Method for Solving Differential Equations

CAUCHY ◽  
2015 ◽  
Vol 3 (4) ◽  
pp. 208
Author(s):  
M Ziaul Arif ◽  
Ahmad Kamsyakawuni ◽  
Ikhsanul Halikin
Keyword(s):  
Exact Solution ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Collocation Method ◽  
Difference Method ◽  
Numerical Scheme ◽  
Chebyshev Collocation Method ◽  
Chebyshev Collocation ◽  
Polynomial Collocation

This paper presents derivation of alternative numerical scheme for solving differential equations, which is modified Chebyshev (Vieta-Lucas Polynomial) collocation differentiation matrices. The Scheme of modified Chebyshev (Vieta-Lucas Polynomial) collocation method is applied to both Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) cases. Finally, the performance of the proposed method is compared with finite difference method and the exact solution of the example. It is shown that modified Chebyshev collocation method more effective and accurate than FDM for some example given.

APPLICATION OF AN EFFECTIVE METHOD ON THE SYSTEM OF NONLINEAR FUZZY INTEGRO-DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 21 (2) ◽  
pp. 407-422
Author(s):  
ANGBEEN IQBAL ◽  
JAMSHAD AHMAD ◽  
QAZI MAHMOOD UL HASSAN
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Graphical Representation ◽  
Numerical Approach ◽  
Fuzzy Arithmetic ◽  
Transform Method ◽  
Numerical Examples ◽  
Homotopy Perturbation ◽  
Sumudu Transform ◽  
Fuzzy Parameter

In real world physical applications purpose, it is complicated to acquire an exact solution of fuzzy differential equations due to complexities in fuzzy arithmetic and therefore creating the need for the use of reliable and efficient techniques in the solution of fuzzy differential equations. The purpose of this research paper is to utilize the reliable analytic approach of homotopy perturbation Sumudu transform method for better understanding of systems of non-linear fuzzy integro-differential equations, while using the concept of fuzzy parameter in certain dynamical problems to remove the hurdles faced in numerical approach. These mathematical models are of great interest in engineering and physics. Some numerical examples are also given to demonstrate the efficiency and superiority of the method, followed by graphical representation of the comparison of exact and approximated solution by using Maple 2017

A New Semi-Analytical Solution of the Telegraph Equation with Integral Condition

2011 ◽  
Vol 66 (12) ◽  
pp. 760-768 ◽  
Author(s):  
S. Abbasbandy ◽  
H. Roohani Ghehsarehb
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Iterative Algorithm ◽  
Homotopy Analysis Method ◽  
Integral Condition ◽  
Telegraph Equation ◽  
Current Work ◽  
Analysis Method ◽  
Numerical Examples ◽  
Homotopy Analysis

In the current work, the telegraph equation in its general form and with an integral condition is investigated. Also the well-known homotopy analysis method (HAM) is applied and an interesting iterative algorithm is proposed for solving the problem in general form. Some numerical examples are given and compared with the exact solution to show the effectiveness of the proposed method.

scholarly journals New Algorithm for the Numerical Solutions of Nonlinear Third-Order Differential Equations Using Jacobi-Gauss Collocation Method

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
A. H. Bhrawy ◽  
W. M. Abd-Elhameed
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Spectral Method ◽  
Collocation Method ◽  
Numerical Solutions ◽  
Order Differential Equation ◽  
Third Order ◽  
Numerical Examples ◽  
Collocation Spectral Method ◽  
Gauss Points

A new algorithm for solving the general nonlinear third-order differential equation is developed by means of a shifted Jacobi-Gauss collocation spectral method. The shifted Jacobi-Gauss points are used as collocation nodes. Numerical examples are included to demonstrate the validity and applicability of the proposed algorithm, and some comparisons are made with the existing results. The method is easy to implement and yields very accurate results.

Controller Gain Design of 2-D Linear Systems Based on the Existing 1-D Analytical Solution

2013 ◽  
Vol 681 ◽  
pp. 55-59
Author(s):  
Wen Jeng Liu
Keyword(s):  
Analytical Solution ◽  
Linear Systems ◽  
State Feedback ◽  
Sufficient Conditions ◽  
Feedback Gain ◽  
Two Dimensional ◽  
Numerical Examples ◽  
One Dimensional ◽  
Controller Gain ◽  
Necessary And Sufficient

Abstract. A controller gain design problem of two-dimensional (2-D) linear systems is proposed in this paper. For one-dimensional (1-D) systems, the necessary and sufficient conditions have been established for the problem, and an analytical solution for the feedback gain is given by [1]. Based on the existing 1-D analytical solution, a 2-D state feedback controller gain can be designed to achieve the desired poles. Finally, two numerical examples are shown to exhibit the validity of the proposed approach.

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