scholarly journals Solution of Ordinary Differential Equation Using Green's Function & Sturm - Liouville Problem

2019 ◽  
pp. 241-244
Author(s):  
S. Angel Auxzaline Mary ◽  
T. Ramesh
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Initial Value Problem ◽  
Series Representation ◽  
Liouville Problem ◽  
Initial Value ◽  
Sturm Liouville

In this paper, we describe Green's function to determine the importance of this function, i.e. Boundary & Initial Value problem, Sturm-Liouville Problem. Along with the series representation of Green's Function.

The ODE- and BVP-Solvers With M&T Applications

2021 ◽  
pp. 166-190
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Value Problems ◽  
Initial Value Problem ◽  
Hydrodynamic Lubrication ◽  
Boundary Value ◽  
Final Part ◽  
Sliding Surface ◽  
Initial Value ◽  
Mass System

The chapter introduces solvers for solving initial and boundary value problems (IVP&BVP) of the ordinary differential equation (ODE). It begins with a description of the ODE solver commands applied to the initial value problem and presents the steps for solving the actual ODE. Further, the chapter presents the BVP-solver commands and steps for their usage. The solutions are presented through real examples. In the final part, the studied ODE and BVP commands are applied, mainly to problems oriented for mechanics and tribology (M&T). At the end of the chapter, applications to the M&T problems are presented; they illustrate how to solve IVP for the spring-mass system and particle falling, as well as BVP for a single clamped beam and hydrodynamic lubrication of a sliding surface covered with semicircular pores.

scholarly journals LINEAR DIFFERENTIAL EQUATION WITH ADDITIONAL CONDITIONS AND FORMULAE FOR GREEN'S FUNCTION

2011 ◽  
Vol 16 (3) ◽  
pp. 401-417 ◽  
Author(s):  
Svetlana Roman
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence Condition ◽  
Linear Ordinary Differential Equation ◽  
Linearly Independent ◽  
Linear Differential

In this paper, we investigate the m-order linear ordinary differential equation with m linearly independent additional conditions. We have found the solution to this problem and give the formula and the existence condition of Green's function. We compare two Green's functions for two such problems with different additional conditions and apply these results to the problems with nonlocal boundary conditions.

scholarly journals The Green's function for Caputo fractional boundary value problem with a convection term

2022 ◽  
Vol 7 (4) ◽  
pp. 4887-4897
Author(s):  
Youyu Wang ◽  
◽  
Xianfei Li ◽  
Yue Huang
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Green’S Function ◽  
Green's Function ◽  
Operator Theory ◽  
Fractional Boundary Value Problem ◽  
Convection Term ◽  
New Ideas ◽  
Sturm Liouville ◽  
Fractional Differential

<abstract><p>By using the operator theory, we establish the Green's function for Caputo fractional differential equation under Sturm-Liouville boundary conditions. The results are new, the method used in this paper will provide some new ideas for the study of this kind of problems and easy to be generalized to solving other problems.</p></abstract>

scholarly journals Solution of Inhomogeneous Fractional Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1944
Author(s):  
Tohru Morita ◽  
Ken-ichi Sato
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Green’S Function ◽  
Fractional Differential Equation ◽  
Green's Function ◽  
Nonstandard Analysis ◽  
Polynomial Coefficients ◽  
Particular Solution ◽  
Fractional Differential ◽  
Detailed Derivation

Discussions are presented by Morita and Sato in Mathematics 2017; 5, 62: 1–24, on the problem of obtaining the particular solution of an inhomogeneous ordinary differential equation with polynomial coefficients in terms of the Green’s function, in the framework of distribution theory. In the present paper, a compact recipe in nonstandard analysis is presented, which is applicable to an inhomogeneous ordinary and also fractional differential equation with polynomial coefficients. The recipe consists of three theorems, each of which provides the particular solution of a differential equation for an inhomogeneous term, satisfying one of three conditions. The detailed derivation of the applications of these theorems is given for a simple fractional differential equation and an ordinary differential equation.

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