scholarly journals Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices of Bernstein Polynomials

2020 ◽  
Author(s):  
Kamal Shah ◽  
Thabet Abdeljawad ◽  
Hammad Khalil ◽  
Rahmat Ali Khan
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Bernstein Polynomials ◽  
Approximate Solutions ◽  
Operational Matrices

scholarly journals Minimax approximate solutions of linear boundary value problems

1979 ◽  
Vol 33 (145) ◽  
pp. 139-148 ◽  
Author(s):  
Darrell Schmidt ◽  
Kenneth L. Wiggins
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Linear Boundary

Quasilinearization and boundary value problems at resonance

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Kareem Alanazi ◽  
Meshal Alshammari ◽  
Paul Eloe
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Approximate Solutions ◽  
Monotonicity Condition ◽  
Uniqueness Of Solutions ◽  
Shift Method ◽  
At Resonance ◽  
Problem At Resonance

Abstract A quasilinearization algorithm is developed for boundary value problems at resonance. To do so, a standard monotonicity condition is assumed to obtain the uniqueness of solutions for the boundary value problem at resonance. Then the method of upper and lower solutions and the shift method are applied to obtain the existence of solutions. A quasilinearization algorithm is developed and sequences of approximate solutions are constructed, which converge monotonically and quadratically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.

Multidimensional Manifold Continuation for Adaptive Boundary-Value Problems

2020 ◽  
Vol 15 (5) ◽  
Author(s):  
Harry Dankowicz ◽  
Yuqing Wang ◽  
Frank Schilder ◽  
Michael E. Henderson
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Approximate Solutions ◽  
General Purpose ◽  
Solution Set ◽  
Benchmark Problems ◽  
Finite Dimensional ◽  
Predictor Corrector ◽  
Parameter Continuation ◽  
Autonomous Dynamical Systems

Abstract Parameter continuation of finitely parameterized, approximate solutions to integro-differential boundary-value problems typically involves regular adaptive updates to the number and meaning of the unknowns and/or the associated constraints. Different continuation steps produce solutions with different discretizations or to formally different sets of equations. Existing general-purpose, multidimensional continuation algorithms fail to account for such differences without significant additional coding and are therefore prone to redundant coverage of the set of solutions. We describe a new algorithm, implemented in the software package coco, which overcomes this problem by characterizing the solution set in an invariant, finite dimensional, projected geometry rather than in the space of unknowns corresponding to any particular discretization. It is in this geometry that distances between solutions and angles between tangent spaces are quantified and used to construct possible directions of outward expansion. A pointwise lift identifies such directions in the projected geometry with directions of continuation in the full set of unknowns, used by a nonlinear predictor-corrector algorithm to expand into uncharted parts of the solution set. Several benchmark problems from the analysis of periodic orbits in autonomous dynamical systems are used to illustrate the theory.

scholarly journals Numerical Solution of Higher Order Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Shahid S. Siddiqi ◽  
Muzammal Iftikhar
Keyword(s):  
Boundary Value Problems ◽  
Homotopy Analysis Method ◽  
Present Method ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Higher Order ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Linear And Nonlinear

The aim of this paper is to use the homotopy analysis method (HAM), an approximating technique for solving linear and nonlinear higher order boundary value problems. Using HAM, approximate solutions of seventh-, eighth-, and tenth-order boundary value problems are developed. This approach provides the solution in terms of a convergent series. Approximate results are given for several examples to illustrate the implementation and accuracy of the method. The results obtained from this method are compared with the exact solutions and other methods (Akram and Rehman (2013), Farajeyan and Maleki (2012), Geng and Li (2009), Golbabai and Javidi (2007), He (2007), Inc and Evans (2004), Lamnii et al. (2008), Siddiqi and Akram (2007), Siddiqi et al. (2012), Siddiqi et al. (2009), Siddiqi and Iftikhar (2013), Siddiqi and Twizell (1996), Siddiqi and Twizell (1998), Torvattanabun and Koonprasert (2010), and Kasi Viswanadham and Raju (2012)) revealing that the present method is more accurate.

Solution of Initial and Boundary Value Problems by an Effective Accurate Method

2017 ◽  
Vol 14 (06) ◽  
pp. 1750069 ◽  
Author(s):  
Mustafa Turkyilmazoglu
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Taylor Series Expansion ◽  
Approximate Solutions ◽  
Accurate Method ◽  
Linear Differential ◽  
Appl Math

The newly proposed analytic approximate solution method in the recent publications [Turkyilmazoglu, M. [2013] “Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane-Emden-Fowler type,” Appl. Math. Mod. 37, 7539–7548; Turkyilmazoglu, M. [2014] “An effective approach for numerical solutions of high-order Fredholm integro-differential equations,” Appl. Math. Comput. 227, 384–398; Turkyilmazoglu, M. [2015] “Parabolic partial differential equations with nonlocal initial and boundary values,” Int. J. Comput. Methods, doi: 10.1142/S0219876215500243] is extended in this paper to solve initial and boundary value problems governed by any order linear differential equations whose exact solutions are hard to obtain. Exact solutions are found from the method when the solutions are themselves polynomials. Better accuracies are achieved within the method by increasing the number of polynomials. Comparisons with some available methods show the ability of the proposed technique, even performing much better than the traditional Taylor series expansion.

Sign in / Sign up

Export Citation Format

Share Document