Boundary shape function iterative method for nonlinear second-order boundary value problem with nonlinear boundary conditions

2021 ◽  
Author(s):  
Aimin Deng ◽  
Ji Lin ◽  
Chein-Shan Liu
Keyword(s):  
Iterative Method ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Shape Function ◽  
Boundary Value ◽  
Nonlinear Boundary Conditions ◽  
Second Order ◽  
Boundary Shape

Solving nonlinear boundary value problems by a boundary shape function method and a splitting and linearizing method

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Chein-Shan Liu ◽  
Essam R. El-Zahar ◽  
Chih-Wen Chang
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Shape Function ◽  
Boundary Value ◽  
Linear Equations ◽  
Accurate Solution ◽  
Iteration Step ◽  
Nonlinear Boundary Value Problems ◽  
Boundary Shape ◽  
Nonlinear Boundary Value

Abstract In the paper, we develop two novel iterative methods to determine the solution of a second-order nonlinear boundary value problem (BVP), which precisely satisfies the specified non-separable boundary conditions by taking advantage of the property of the corresponding boundary shape function (BSF). The first method based on the BSF can exactly transform the BVP to an initial value problem for the new variable with two given initial values, while two unknown terminal values are determined iteratively. By using the BSF in the second method, we derive the fractional powers exponential functions as the bases, which automatically satisfy the boundary conditions. A new splitting and linearizing technique is used to transform the nonlinear BVP into linear equations at each iteration step, which are solved to determine the expansion coefficients and then the solution is available. Upon adopting those two novel methods very accurate solution for the nonlinear BVP with non-separable boundary conditions can be found quickly. Several numerical examples are solved to assess the efficiency and accuracy of the proposed iterative algorithms, which are compared to the shooting method.

Solving second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative method

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Chein-Shan Liu ◽  
Jiang-Ren Chang
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Closed Form ◽  
Iterative Algorithm ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Boundary Conditions ◽  
Nonlinear Boundary Value Problem ◽  
Content Type ◽  
Nonlinear Boundary Value

Purpose The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method. Design/methodology/approach The authors introduce eigenfunctions as test functions, such that a weak-form integral equation is derived. By expanding the numerical solution in terms of the weighted eigenfunctions and using the orthogonality of eigenfunctions with respect to a weight function, and together with the non-separated/mixed boundary conditions, one can obtain the closed-form expansion coefficients with the aid of Drazin inversion formula. Findings When the authors develop the iterative algorithm, removing the time-varying terms as well as the nonlinear terms to the right-hand sides, to solve the nonlinear boundary value problem, it is convergent very fast and also provides very accurate numerical solutions. Research limitations/implications Basically, the authors’ strategy for the iterative numerical algorithm is putting the time-varying terms as well as the nonlinear terms on the right-hand sides. Practical implications Starting from an initial guess with zero value, the authors used the closed-form formula to quickly generate the new solution, until the convergence is satisfied. Originality/value Through the tests by six numerical experiments, the authors have demonstrated that the proposed iterative algorithm is applicable to the highly complex nonlinear boundary value problems with nonlinear boundary conditions. Because the coefficient matrix is set up outside the iterative loop, and due to the property of closed-form expansion coefficients, the presented iterative algorithm is very time saving and converges very fast.

scholarly journals TYPES AND MULTIPLICITY OF SOLUTIONS TO STURM–LIOUVILLE BOUNDARY VALUE PROBLEM

2015 ◽  
Vol 20 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Maria Dobkevich ◽  
Felix Sadyrbaev
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Second Order ◽  
Nonlinear Boundary Value Problems ◽  
Multiplicity Of Solutions ◽  
Different Types ◽  
Sturm Liouville

We consider the second-order nonlinear boundary value problems (BVPs) with Sturm–Liouville boundary conditions. We define types of solutions and show that if there exist solutions of different types then there exist intermediate solutions also.

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