scholarly journals On a boundary value problem arising in elastic deflection theory

2006 ◽  
Vol 74 (3) ◽  
pp. 337-345 ◽  
Author(s):  
Xiuqin Wang
Keyword(s):  
Approximate Solution ◽  
Boundary Value Problem ◽  
Linear Problem ◽  
Boundary Value ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Point Boundary ◽  
Nonresonance Condition ◽  
Deflection Theory

In this paper, a finite-difference method for the determination of an approximate solution of a fourth-order two-point boundary value problem is presented under the nonresonance condition. The solution of this linear problem can be used to find approximate solutions of a broad range of nonlinear problems in applications.

scholarly journals A Hybrid Perturbation-Galerkin Method for Differential Equations Containing a Parameter

1989 ◽  
Vol 42 (11S) ◽  
pp. S69-S77 ◽  
Author(s):  
James F. Geer ◽  
Carl M. Andersen
Keyword(s):  
Asymptotic Expansion ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Perturbation Method ◽  
Galerkin Method ◽  
Hybrid Method ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Point Boundary ◽  
Research Application

A two-step hybrid perturbation-Galerkin method to solve a variety of differential equations which involve a parameter is presented and discussed. The method consists of: (1) the use of a perturbation method to determine the asymptotic expansion of the solution about one or more values of the parameter; and (2) the use of some of the perturbation coefficient functions as trial functions in the classical Bubnov-Galerkin method. This hybrid method has the potential of overcoming some of the drawbacks of the perturbation method and the Bubnov-Galerkin method when they are applied by themselves, while combining some of the good features of both. The proposed method is illustrated first with a simple linear two-point boundary value problem and is then applied to a nonlinear two-point boundary value problem in lubrication theory. The results obtained from the hybrid method are compared with approximate solutions obtained by purely numerical methods. Some general features of the method, as well as some special tips for its implementation, are discussed. A survey of some current research application areas is presented and its degree of applicability to broader problem areas is discussed.

scholarly journals Successive Approximation Technique in the Study of a Nonlinear Fractional Boundary Value Problem

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 724
Author(s):  
Kateryna Marynets
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Approximation Technique ◽  
Fractional Boundary Value Problem ◽  
Point Boundary ◽  
Fractional Differential ◽  
Fractional Boundary Value Problems

We studied one essentially nonlinear two–point boundary value problem for a system of fractional differential equations. An original parametrization technique and a dichotomy-type approach led to investigation of solutions of two “model”-type fractional boundary value problems, containing some artificially introduced parameters. The approximate solutions of these problems were constructed analytically, while the numerical values of the parameters were determined as solutions of the so-called “bifurcation” equations.

scholarly journals APPROXIMATE SOLUTION OF AN INITIAL-BOUNDARY VALUE PROBLEM FOR AN NONHOMOGENEOUS SECOND-ORDER DIFFERENTIAL EQUATION OF MIXED TYPE WITH TWO DEGENERATE LINES

2020 ◽  
pp. 81-98
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Mixed Type ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Second Order ◽  
Initial Boundary Value

This work is devoted to the study of an approximate solution of the initial-boundary value problem for the second order mixed type nonhomogeneous differential equation with two degenerate lines. Similar equations have many different applications, for example, boundary value problems for mixed type equations are applicable in various fields of the natural sciences: in problems of laser physics, in magneto hydrodynamics, in the theory of infinitesimal bindings of surfaces, in the theory of shells, in predicting the groundwater level, in plasma modeling, and in mathematical biology. In this paper, based on the idea of A.N. Tikhonov, the conditional correctness of the problem, namely, uniqueness and conditional stability theorems are proved, as well as approximate solutions that are stable on the set of correctness are constructed. In obtaining an apriori estimate of the solution of the equation, we used the logarithmic convexity method and the results of the spectral problem considered by S.G. Pyatkov. The results of the numerical solutions and the approximate solutions of the original problem were presented in the form of tables. The regularization parameter is determined from the minimum estimate of the norm of the difference between exact and approximate solutions.

scholarly journals Construction of a global solution for the one dimensional singularly-perturbed boundary value problem

2017 ◽  
Vol 8 (1-2) ◽  
pp. 52
Author(s):  
Samir Karasuljic ◽  
Enes Duvnjakovic ◽  
Vedad Pasic ◽  
Elvis Barakovic
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Uniform Convergence ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Singularly Perturbed ◽  
One Dimensional ◽  
The One ◽  
Theoretical Results

We consider an approximate solution for the one--dimensional semilinear singularly--perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Green's function. We present an \(\varepsilon\)--uniform convergence of such gained the approximate solutions, in the maximum norm of the order \(\mathcal{O}\left(N^{-1}\right)\) on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has \(\varepsilon\)--uniform convergence, but now of order \(\mathcal{O}\left(\ln^2N/N^2\right)\) on \([0,1]\). In the end a numerical experiment is presented to confirm previously shown theoretical results.

scholarly journals Existence of Solutions of a Discrete Fourth-Order Boundary Value Problem

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
Ruyun Ma ◽  
Chenghua Gao ◽  
Yongkui Chang
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Linear Problem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Linear Eigenvalue Problem ◽  
Existence And Multiplicity ◽  
Nonresonance Condition ◽  
Two Parameter

Leta,bbe two integers withb-a≥5and let𝕋2={a+2,a+3,…,b-2}. We show the existence of solutions for nonlinear fourth-order discrete boundary value problemΔ4u(t-2)=f(t,u(t),Δ2u(t-1)),t∈𝕋2,u(a+1)=u(b-1)=Δ2u(a)=Δ2u(b-2)=0under a nonresonance condition involving two-parameter linear eigenvalue problem. We also study the existence and multiplicity of solutions of nonlinear perturbation of a resonant linear problem.

scholarly journals APPROXIMATE SOLUTION OF NONLINEAR MULTI-POINT BOUNDARY VALUE PROBLEM ON THE HALF-LINE

2012 ◽  
Vol 17 (2) ◽  
pp. 190-202 ◽  
Author(s):  
Jing Niu ◽  
Ying Zhen Lin ◽  
Chi Ping Zhang
Keyword(s):  
Approximate Solution ◽  
Boundary Value Problem ◽  
Reproducing Kernel ◽  
Boundary Value ◽  
Half Line ◽  
Point Boundary ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Uniformly Convergence ◽  
Reproducing Kernel Function

In this work, we construct a novel weighted reproducing kernel space and give the expression of reproducing kernel function skillfully. Based on the orthogonal basis established in the reproducing kernel space, an efficient algorithm is provided to solve the nonlinear multi-point boundary value problem on the half-line. Uniformly convergence of the approximate solution and convergence estimation of our algorithm are studied. Numerical results show our method has high accuracy and efficiency.

Sign in / Sign up

Export Citation Format

Share Document