The Cauchy problem for equations with operator coefficients; mixed boundary value problem for systems of differential equations and approximation methods for their solution

1963 ◽  
pp. 173-278 ◽  
Author(s):  
M. L. Višik
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Approximation Methods ◽  
Mixed Boundary Value Problem ◽  
Systems Of Differential Equations ◽  
The Cauchy Problem

Positive and maximal positive solutions of singular mixed boundary value problem

2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Ravi Agarwal ◽  
Donal O’Regan ◽  
Svatoslav Staněk
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Existence Results ◽  
Mixed Boundary Value Problem ◽  
Time Singularity ◽  
Strong Singularity

AbstractThe paper is concerned with existence results for positive solutions and maximal positive solutions of singular mixed boundary value problems. Nonlinearities h(t;x;y) in differential equations admit a time singularity at t=0 and/or at t=T and a strong singularity at x=0.

scholarly journals ON THE ALGORITHM FOR SOLVING OF A LINEAR BOUNDARY VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATION WITH PARAMETER

2020 ◽  
Vol 70 (2) ◽  
pp. 71-76
Author(s):  
N.B. Iskakova ◽  
◽  
Zh. Kubanychbekkyzy ◽  
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Linear Boundary ◽  
Existence Of A Solution ◽  
The Cauchy Problem ◽  
Linear Boundary Value Problem

A linear boundary value problem for a system of ordinary differential equations containing a parameter is considered on a bounded segment. For a fixed parameter value, the Cauchy problem for an ordinary differential equation is solved. Using the fundamental matrix of differential part and assuming uniqueness solvability of the Cauchy problem an origin boundary value problem is reduced to the system of linear algebraic equation with respect to unknown parameter. The existence of a solution to this system ensures the existence of a solution to the boundary value problem under study. The algorithm of finding of solution for initial problem is offered based on a construction and solving of the linear algebraic equation. The basic auxiliary problem of algorithm is: the Cauchy problem for ordinary differential equations. The numerical implementation of algorithm offered in the article uses the method of Runge-Kutta of fourth order to solve the Cauchy problem for ordinary differential equations.

scholarly journals Numerical implementation of solving a control problem for loaded differential equations with multi-point condition

2020 ◽  
Vol 100 (4) ◽  
pp. 81-91
Author(s):  
Zh.M. Kadirbayeva ◽  
◽  
A.D. Dzhumabaev ◽  
◽  
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Numerical Implementation ◽  
Linear Boundary ◽  
Equivalent Boundary ◽  
The Cauchy Problem ◽  
Point Condition ◽  
Linear Boundary Value Problem

A linear boundary value problem with a parameter for loaded differential equations with multi-point condition is considered. The method of parameterization is used for solving the considered problem. We offer an algorithm for solving a control problem for the system of loaded differential equations with multi-point condition. The linear boundary value problem with a parameter for loaded differential equations with multi-point condition by introducing additional parameters at the partition points is reduced to equivalent boundary value problem with parameters. The equivalent boundary value problem with parameters consists of the Cauchy problem for the system of ordinary differential equations with parameters, multi-point condition, and continuity conditions. The solution of the Cauchy problem for the system of ordinary differential equations with parameters is constructed using the fundamental matrix of differential equation. The system of linear algebraic equations concerning the parameters is composed by substituting the values of the corresponding points in the built solutions to the multi-point condition and continuity conditions. The numerical method for finding the solution of the problem is suggested, which based on the solving the constructed system and solving Cauchy problem on the subintervals by Adams method and Bulirsch-Stoer method. The proposed numerical implementation is illustrated by example.

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