Solving Multi-Point Boundary Value Problems Using Sinc-Derivative Interpolation

In this paper, the Sinc-derivative collocation method is used to solve linear and nonlinear multi-point boundary value problems. This is done by interpolating the first derivative of the unknown variable via Sinc numerical methods and obtaining the desired solution through numerical integration of the interpolation and all higher order derivatives through successive differentiation of the interpolation. Non-homogeneous boundary conditions are reduced to homogeneous using suitable transformations. The efficiency and the accuracy of the method are tested using illustrative examples previously considered by other researchers who used different approaches. The results show the excellent performance of the Sinc-derivative collocation method.

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Solving Integro-Differential Boundary Value Problems Using Sinc-Derivative Collocation

Mathematics ◽

10.3390/math8091637 ◽

2020 ◽

Vol 8 (9) ◽

pp. 1637

Author(s):

Kenzu Abdella ◽

Glen Ross

Keyword(s):

Numerical Methods ◽

Boundary Conditions ◽

Boundary Value Problems ◽

Numerical Integration ◽

Collocation Method ◽

Boundary Value ◽

Numerical Differentiation ◽

Higher Order ◽

Second Order ◽

Collocation Approach

In this paper, the sinc-derivative collocation approach is used to solve second order integro-differential boundary value problems. While the derivative of the unknown variables is interpolated using sinc numerical methods, the desired solution and the integral terms are evaluated through numerical integration and all higher order derivatives are approximated through successive numerical differentiation. Suitable transformations are used to reduce non-homogeneous boundary conditions to homogeneous. Comparison of the proposed method with different approaches that were previously considered in the literature is carried out in order to test its accuracy and efficiency. The results show that the sinc-derivative collocation method performs well.

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Solving nonlinear third-order three-point boundary value problems by boundary shape functions methods

Advances in Difference Equations ◽

10.1186/s13662-021-03288-x ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Ji Lin ◽

Yuhui Zhang ◽

Chein-Shan Liu

Keyword(s):

Boundary Conditions ◽

Boundary Value Problems ◽

Initial Conditions ◽

Boundary Value ◽

Accurate Solution ◽

Point Boundary ◽

Third Order ◽

Boundary Shape ◽

Free Function ◽

New Algorithms

AbstractFor nonlinear third-order three-point boundary value problems (BVPs), we develop two algorithms to find solutions, which automatically satisfy the specified three-point boundary conditions. We construct a boundary shape function (BSF), which is designed to automatically satisfy the boundary conditions and can be employed to develop new algorithms by assigning two different roles of free function in the BSF. In the first algorithm, we let the free functions be complete functions and the BSFs be the new bases of the solution, which not only satisfy the boundary conditions automatically, but also can be used to find solution by a collocation technique. In the second algorithm, we let the BSF be the solution of the BVP and the free function be another new variable, such that we can transform the BVP to a corresponding initial value problem for the new variable, whose initial conditions are given arbitrarily and terminal values are determined by iterations; hence, we can quickly find very accurate solution of nonlinear third-order three-point BVP through a few iterations. Numerical examples confirm the performance of the new algorithms.

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On a Class of Two-Point Boundary Value Problems with Singular Boundary Conditions

Rocky Mountain Journal of Mathematics ◽

10.1216/rmjm/1181069345 ◽

2006 ◽

Vol 36 (6) ◽

pp. 1803-1818

Author(s):

Ravi P. Agarwal ◽

To Fu Ma

Keyword(s):

Boundary Conditions ◽

Boundary Value Problems ◽

Boundary Value ◽

Singular Boundary ◽

Point Boundary

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Application of Optimal Homotopy Asymptotic Method to Some Well-Known Linear and Nonlinear Two-Point Boundary Value Problems

International Journal of Differential Equations ◽

10.1155/2018/8725014 ◽

2018 ◽

Vol 2018 ◽

pp. 1-11

Author(s):

Muhammad Asim Khan ◽

Shafiq Ullah ◽

Norhashidah Hj. Mohd Ali

Keyword(s):

Boundary Value Problems ◽

Asymptotic Method ◽

Homotopy Perturbation Method ◽

Boundary Value ◽

Point Boundary ◽

Small Perturbations ◽

Homotopy Perturbation ◽

Optimal Homotopy Asymptotic Method ◽

Linear And Nonlinear ◽

Semianalytical Method

The objective of this paper is to obtain an approximate solution for some well-known linear and nonlinear two-point boundary value problems. For this purpose, a semianalytical method known as optimal homotopy asymptotic method (OHAM) is used. Moreover, optimal homotopy asymptotic method does not involve any discretization, linearization, or small perturbations and that is why it reduces the computations a lot. OHAM results show the effectiveness and reliability of OHAM for application to two-point boundary value problems. The obtained results are compared to the exact solutions and homotopy perturbation method (HPM).

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