scholarly journals Efficient Boundary Value Problem Solution for a Lane-Emden Equation

2010 ◽  
Vol 15 (4) ◽  
pp. 613-620 ◽  
Author(s):  
C. Harley ◽  
E. Momoniat
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Problem Solution ◽  
Emden Equation

scholarly journals Seminonlinear boundary value problems for nondegenerate differential-algebraic system

2019 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Differential Algebraic Equations ◽  
Problem Solution ◽  
Algebraic Equations ◽  
Arbitrary Continuous Function ◽  
Differential Algebraic System

In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of А.М. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

ESTIMATION OF A BOUNDARY VALUE PROBLEM SOLUTION WITH INITIAL JUMP FOR LINEAR DIFFERENTIAL EQUATION

2020 ◽  
Vol 69 (1) ◽  
pp. 168-173
Author(s):  
B. Sharip ◽  
◽  
А.Т. Yessimova ◽  
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Linear Differential Equation ◽  
Special Functions ◽  
Boundary Value ◽  
Analytical Formula ◽  
Singularly Perturbed ◽  
Problem Solution ◽  
Unperturbed Problem ◽  
Linear Differential

The paper considers a boundary value problem for a singularly perturbed linear differential equation with constant third-order coefficients. In this problem, a small parameter is indicated before the highest derivatives that are part of the differential equation and the boundary condition at t = 0.The fundamental system of solutions of a homogeneous singularly perturbed differential equation is constructed on the basis of asymptotic representations obtained for the roots of the corresponding characteristic equation. This system was used to construct the Cauchy function, special functions of boundary value problems, and also the Green function. With the help of these functions, an analytical formula is obtained for solving a singularly perturbed boundary value problem and it turns out that this solution has an initial zero-order jump at t = 0. It is proved that the solution to the considered singularly perturbed boundary value problem tends to the corresponding unperturbed problem obtained from it under .

scholarly journals Asymptotic Estimates of the Solution of a Restoration Problem with an Initial Jump

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Duisebek Nurgabyl
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Problem Solution ◽  
Asymptotic Estimates ◽  
Cauchy Function ◽  
Unperturbed Problem ◽  
Boundary Functions ◽  
Restoration Problem ◽  
Initial Jump

The asymptotic behavior of the solution of the singularly perturbed boundary value problemLεy=htλ,Liy+σiλ=ai,i=1,n+1̅is examined. The derivations prove that a unique pair(ỹt,λ̃ε,ε,λ̃ε)exists, in which componentsy(t,λ̃ε,ε)andλ̃(ε)satisfy the equationLεy=h(t)λand boundary value conditionsLiy+σiλ=ai,i=1,n+1̅. The issues of limit transfer of the perturbed problem solution to the unperturbed problem solution as a small parameter approaches zero and the existence of the initial jump phenomenon are studied. This research is conducted in two stages. In the first stage, the Cauchy function and boundary functions are introduced. Then, on the basis of the introduced Cauchy function and boundary functions, the solution of the restoration problemLεy=htλ,Liy+σiλ=ai,i=1,n+1̅is obtained from the position of the singularly perturbed problem with the initial jump. Through this process, the formula of the initial jump and the asymptotic estimates of the solution of the considered boundary value problem are identified.

scholarly journals SHAPE DIFFERENTIABILITY OF DRAG FUNCTIONAL AND BOUNDARY VALUE PROBLEM SOLUTIONS FOR FLUID MIXTURE EQUATIONS

2016 ◽  
pp. 41-56
Author(s):  
Nikolay Kucher ◽  
Nikolay Kucher ◽  
Aleksandra Zhalnina ◽  
Aleksandra Zhalnina
Keyword(s):  
Boundary Value Problem ◽  
Original Problem ◽  
Stokes Equations ◽  
Boundary Value ◽  
Equation System ◽  
The Body ◽  
Compressible Fluids ◽  
Problem Solution ◽  
Partial Differential ◽  
Fluid Mixture

Problems of optimal design of various elements of technical structures stimulate mathematical statements of new problems of continuum mechanics and hydrodynamics in particular. This study refers to problems of shape optimization of profiles in a flow of fluid or gas. The paper deals with properties of solutions and their functional of inhomogeneous boundary value problem for nonlinear composite type partial differential equation system, simulating the a mixture of viscous compressible fluids (gases) flowing around an obstacle. Methods of the theory of partial differential equations, functional analysis and, in particular, the results on the solvability of boundary value problems for transport and Stokes equations established the well-posedness of a linear boundary value problem with singular coefficients (the problem of the original problem solution difference). This result allowed to obtain the uniqueness theorem to determine the nature of solutions dependence on the shape of the flow range and to prove domain differentiability of the solutions considered. Domain differentiability of the solution functional reflecting the force of the obstacle resistance to the incident flow is proved. A formula to equate this derivative as a sum of two summands, one of which clearly depends on mapping setting the domain shape, and the other can be expressed in terms of the so-called adjoint state, depending only on the solution of the original problem in a non-deformed domain. The functional derivative formulas may be used as the basis for building a numerical algorithm for finding the optimal shape of the body in a flow of mixture of viscous compressible fluids.

Approximate Boundary Value Problems of a Deformed Flexible Closed Torso Shell with Excited Edges

2016 ◽  
Vol 08 (04) ◽  
pp. 1650051 ◽  
Author(s):  
I. V. Andrianov ◽  
V. I. Olevskyi ◽  
J. Awrejcewicz
Keyword(s):  
Exact Solution ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Asymptotic Series ◽  
Problem Solution ◽  
Rational Approximations ◽  
Two Dimensional ◽  
Artificial Parameter

A boundary value problem solution is presented to treatment the deformations of a closed flexible elastic torso shell having perturbations at its axial edges. A so-called artificial parameter technique is applied to obtain a solution in the form of a double asymptotic series further summed using two-dimensional fractional rational approximations. Convergence of the approximations to the exact solution is proven.

scholarly journals Tenth order boundary value problem solution existence by fixed point theorem

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Nicola Fabiano ◽  
Nebojša Nikolić ◽  
Thenmozhi Shanmugam ◽  
Stojan Radenović ◽  
Nada Čitaković
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Point Theorem ◽  
Boundary Value ◽  
Problem Solution ◽  
Solution Existence

scholarly journals An Inverse Problem Solution for Thermal Conductivity Reconstruction

2021 ◽  
Vol 20 ◽  
pp. 187-195
Author(s):  
Tchavdar T. Marinov ◽  
Rossitza S. Marinova
Keyword(s):  
Thermal Conductivity ◽  
Inverse Problem ◽  
Boundary Value Problem ◽  
Thermal Conductivity Coefficient ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Nonlinear Coefficient ◽  
Higher Order ◽  
Quadratic Functional ◽  
Problem Solution

This work deals with the inverse problem of reconstructing the thermal conductivity coefficient of the (2+1)D heat equation from over–posed data at the boundaries. The proposed solution uses a variational approach for identifying the coefficient. The inverse problem is reformulated as a higher–order elliptic boundary–value problem for minimization of a quadratic functional of the original equation. The resulting system consists of a well–posed fourth–order boundary–value problem for the temperature and an explicit equation for the unknown thermal conductivity coefficient. The existence and uniqueness of the resulting higher–order boundary–value problem are investigated. The unique solvability of the inverse coefficient problem is proven. The numerical algorithm is validated and applied to problems of reconstructing continuous nonlinear coefficient and discontinuous coefficients. Accurate and stable numerical solutions are obtained.

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