scholarly journals A Collocation Method Based on the Bernoulli Operational Matrix for Solving High-Order Linear Complex Differential Equations in a Rectangular Domain

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Faezeh Toutounian ◽  
Emran Tohidi ◽  
Stanford Shateyi
Keyword(s):  
Differential Equations ◽  
Matrix Equation ◽  
Operational Matrix ◽  
Bernoulli Polynomials ◽  
High Order ◽  
Linear Complex ◽  
Order Linear ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Complex Differential Equations

This paper contributes a new matrix method for the solution of high-order linear complex differential equations with variable coefficients in rectangular domains under the considered initial conditions. On the basis of the presented approach, the matrix forms of the Bernoulli polynomials and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown Bernoulli coefficients are determined and thus the approximate solutions are obtained. Also, an error analysis based on the use of the Bernoulli polynomials is provided under several mild conditions. To illustrate the efficiency of our method, some numerical examples are given.

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.

A shifted Legendre method for solving a population model and delay linear Volterra integro-differential equations

2017 ◽  
Vol 10 (07) ◽  
pp. 1750091 ◽  
Author(s):  
Şuayip Yüzbaşi
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Matrix Equation ◽  
Population Model ◽  
Estimation Method ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Integro Differential Equation ◽  
Linear Algebraic Equations ◽  
The Matrix

In this paper, we propose a collocation method to obtain the approximate solutions of a population model and the delay linear Volterra integro-differential equations. The method is based on the shifted Legendre polynomials. By using the required matrix operations and collocation points, the delay linear Fredholm integro-differential equation is transformed into a matrix equation. The matrix equation corresponds to a system of linear algebraic equations. Also, an error estimation method for method and improvement of solutions is presented by using the residual function. Applications of population model and general delay integro-differential equation are given. The obtained results are compared with the known results.

scholarly journals On the Growth of Solutions of Second Order Linear Complex Differential Equations whose Coefficients Satisfy Certain Conditions

2020 ◽  
Vol 17 (2) ◽  
pp. 0530
Author(s):  
Ayad Alkhalidy ◽  
Eman Hussein
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Infinite Order ◽  
Second Order ◽  
The Other ◽  
Nontrivial Solutions ◽  
Linear Complex ◽  
Order Linear ◽  
Complex Differential Equations ◽  
Growth Of Solutions

In this paper, we study the growth of solutions of the second order linear complex differential equations  insuring that any nontrivial solutions are of infinite order. It is assumed that the coefficients satisfy the extremal condition for Yang’s inequality and the extremal condition for Denjoy’s conjecture. The other condition is that one of the coefficients itself is a solution of the differential equation .

scholarly journals Haar Wavelet Collocation Method for Solving Riccati and Fractional Riccati Differential Equations

2016 ◽  
Vol 17 ◽  
pp. 46-56 ◽  
Author(s):  
S.C. Shiralashetti ◽  
A.B. Deshi
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Haar Wavelet ◽  
Algebraic Equations ◽  
Riccati Differential Equations ◽  
Wavelet Collocation Method ◽  
Linear Algebraic Equations ◽  
The Matrix

In this paper, numerical solutions of Riccati and fractional Riccati differential equations are obtained by the Haar wavelet collocation method. An operational matrix of integration based on the Haar wavelet is established, and the procedure for applying the matrix to solve these equations. The fundamental idea of Haar wavelet method is to convert the proposed differential equations into a group of non-linear algebraic equations. The accuracy of approximate solution can be further improved by increasing the level of resolution and an error analysis is computed. The examples are given to demonstrate the fast and flexibility of the method. The results obtained are in good agreement with the exact in comparison with existing ones and it is shown that the technique introduced here is robust, easy to apply and is not only enough accurate but also quite stable.

scholarly journals A matrix approach to solving hyperbolic partial differential equations using Bernoulli polynomials

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 993-1000 ◽  
Author(s):  
Bicer Erdem ◽  
Salih Yalcinbas
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Error Analysis ◽  
Matrix Equation ◽  
Bernoulli Polynomials ◽  
Hyperbolic Partial Differential Equations ◽  
Matrix Approach ◽  
Residual Function ◽  
Partial Differential ◽  
The Matrix

The present study considers the solutions of hyperbolic partial differential equations. For this, an approximate method based on Bernoulli polynomials is developed. This method transforms the equation into the matrix equation and the unknown of this equation is a Bernoulli coefficients matrix. To demostrate the validity and applicability of the method, an error analysis developed based on residual function. Also examples are presented to illustrate the accuracy of the method.

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