Order Nonlinear Boundary Value Problems Research Articles

Integral formulations for 1-D Biharmonic and Second Order Coupled Linear and Nonlinear Boundary Value Problems

Integral formulations based on a boundary-domain interpretation of the boundary element method (BEM) are applied to develop the numerical solutions of biharmonic and second order coupled linear and nonlinear boundary value problems.

Existence and iterative method for some fourth order nonlinear boundary value problems

This work is devoted to proving the existence of the iterative solution to a fourth order boundary value problem. By the use of a novel efficient method for nonlinear fourth order boundary value problem, the existence and uniqueness of solution for the problem is obtained. The monotony of iterations is also considered. Some examples are presented to illustrate the main results.

Bezier Polynomials with Applications

In this paper, we use the Galerkin technique for solving higher order linear and nonlinear boundary value problems (BVPs). The well-known Bezier polynomials are exploited as basis functions in the technique. To use the Bezier polynomials, we need to satisfy the corresponding homogeneous form of the boundary conditions and modification is thus needed. A rigorous matrix formulation is developed by the Galerkin method for linear and nonlinear systems and solved it using Bezier polynomials. The approximate solutions are compared to the exact solutions through tabular form. All problems are computed using the software MATHEMATICA.
 Dhaka Univ. J. Sci. 66(2): 157-162, 2018 (July)

An efficient technique to find semi-analytical solutions for higher order multi-point boundary value problems

A new semi-analytical algorithm is presented to solve general multi-point boundary value problems. This method can be applied on nth order linear, nonlinear, singular and nonsingular multi-point boundary value problems. Mathematical base of the method is presented; convergence of the method is proved. Also, the algorithm is applied to solve multi-point boundary value problems including nonlinear sixth-order, nonlinear singular second-order five-point boundary value problems, and a singularly perturbed boundary value problem. Comparison results show that the new method works more accurate than the other methods.

New cubic B-spline approximation for solving third order Emden–Flower type equations

In this article, the typical cubic B-spline collocation method equipped with new approximations for second and third order derivatives is employed to explore the numerical solution of a class of third order non-linear singular boundary value problems. The singularity is removed by means of L’Hospital’s Rule. The Taylor’s series expansion of the error term reveals that our new scheme is fifth order accurate. The proposed technique is tested on several third order Emden–Flower type equations and the numerical results are compared with those found in the current literature. It is found that our new approximation technique performs superior to the existing methods due to its simple implementation, straight forward interpolation and very less computational cost.

Generalized Galerkin Finite Element Formulation for the Numerical Solutions Of Second Order Nonlinear Boundary Value Problems

We use Galerkin finite element method (GFEM) to solve second order linear and nonlinear boundary value problems (BVPs). First we develop FEM formulation for a class of linear and nonlinear BVPs. Then we present convergence analysis of the method. Later, we give the solution of some nonlinear BVPs with Diritchlet, Neumann and Robin boundary conditions. All results are compared with the exact solution and sometimes with the results of the existing method to verify the convergence, stability and consistency of this method. The results are depicted graphically as well as in the tabular form.GANIT J. Bangladesh Math. Soc.Vol. 37 (2017) 147-159

Existence of solutions for [formula omitted]-order nonlinear differential boundary value problems by means of fixed point theorems

This paper is devoted to prove the existence of one or multiple solutions of a wide range of nonlinear differential boundary value problems.To this end, we obtain some new fixed point theorems for a class of integral operators. We follow the well-known Krasnoselskiĭ’s fixed point Theorem together with two fixed point results of Leggett–Williams type. After obtaining a general existence result for a one parameter family of nonlinear differential equations, are proved, as particular cases, existence results for second and fourth order nonlinear boundary value problems.

Continuous Genetic Algorithm : A Robust Method to Solve Higher Order Non-Linear Boundary Value Problem

Continuous Genetic Algorithm : A Robust Method to Solve Higher Order Non-Linear Boundary Value Problem

A Posteriori Error Estimates of Two-Grid Finite Element Methods for Nonlinear Elliptic Problems

In this article, we study the residual-based a posteriori error estimates of the two-grid finite element methods for the second order nonlinear elliptic boundary value problems. Computable upper and lower bounds on the error in the \(H^1\)-norm are established. Numerical experiments are also provided to illustrate the performance of the proposed estimators.

Positive solutions of superlinear and sublinear boundary value problems

We study the existence of positive solutions of second order nonlinear separated boundary value problems of superlinear as well as sublinear type without imposing monotonicity restrictions on the problem. The type of problem investigated cannot be analyzed using the linearization about the trivial solution because either it does not exist (the sublinear case) or is trivial (the superlinear case). The results follow from a known fixed point theorem by noticing that the concavity of the solutions provides an important condition for the applicability of the fixed point result.

Solvability of generalized Hammerstein integral equations on unbounded domains, with sign-changing kernels

In this work we study aHammerstein generalized integral equation u ( t ) = ∫ − ∞ + ∞ k ( t , s ) f ( s , u ( s ) , u ′ ( s ) , … , u ( m ) ( s ) ) d s , where k : R 2 → R is a W m , ∞ ( R 2 ) , m ∈ N , kernel function and f : R m + 2 → R is a L 1 -Carathéodory function. To the best of our knowledge, this paper is the first one to consider discontinuous nonlinearities with derivatives dependence, without monotone or asymptotic assumptions, on the whole real line. Our method is applied to a fourth order nonlinear boundary value problem, which models moderately large deflections of infinite nonlinear beams resting on elastic foundations under localized external loads.

Generalized Hammerstein Equations and Applications

In this paper the authors study the Hammerstein generalized integral equation $$\begin{aligned} u(t)=\int _{0}^{1}k(t,s)\text { }g(s)\text { }f(s,u(s),u^{\prime }(s),\dots ,u^{(m)}(s))\,ds, \end{aligned}$$ where \(k:[0,1]^{2}\rightarrow {\mathbb {R}}\) are kernel functions, \(m\ge 1\), \(g:[0,1] \rightarrow [0,\infty )\), and \(f:[0,1]\times {\mathbb {R}}^{m+1} \rightarrow [0,\infty )\) is a \(L^{\infty }-\)Caratheodory function. The existence of solutions of integral equations has been studied in concrete and abstract cases, by different methods and techniques. However, in the existing literature, the nonlinearity depends only on the unknown function. This paper is one of a very few to consider equations having discontinuous nonlinearities that depend on the derivatives of the unknown function and having discontinuous kernels functions that have discontinuities in the partial derivatives with respect to their first variable. Our approach is based on the Krasnosel’skiĭ–Guo compression/expansion theorem on cones and it can be applied to boundary value problems of arbitrary order \(n>m\). The last two sections of the paper contain an application to a third order nonlinear boundary value problem and a concrete example.

Numerical Solution of Fifth Order Boundary Value Problems by Petrov-Galerkin Method with Quintic B-splines as basis functions and Septic B-Splines as weight functions

This paper deals with a finite element method involving Petrov-Galerkin method with quartic B-splines as basis functions and sextic B-splines as weight functions to solve a general fourth order boundary value problem with a particular case of boundary conditions. The basis functions are redefined into a new set of basis functions which vanish on the boundary where the Dirichlet type of boundary conditions are prescribed. The weight functions are also redefined into a new set of weight functions which in number match with the number of redefined basis functions. The proposed method was applied to solve several examples of fourth order linear and nonlinear boundary value problems. The obtained numerical results were found to be in good agreement with the exact solutions available in the literature.

Intelligent computing to solve fifth-order boundary value problem arising in induction motor models

In this study, biologically inspired intelligent computing approached based on artificial neural networks (ANN) models optimized with efficient local search methods like sequential quadratic programming (SQP), interior point technique (IPT) and active set technique (AST) is designed to solve the higher order nonlinear boundary value problems arise in studies of induction motor. The mathematical modeling of the problem is formulated in an unsupervised manner with ANNs by using transfer function based on log-sigmoid, and the learning of parameters of ANNs is carried out with SQP, IPT and ASTs. The solutions obtained by proposed methods are compared with the reference state-of-the-art numerical results. Simulation studies show that the proposed methods are useful and effective for solving higher order stiff problem with boundary conditions. The strong motivation of this research work is to find the reliable approximate solution of fifth-order differential equation problems which are validated through strong statistical analysis.

New Analytic Technique for the Solution of Nth Order Nonlinear Two-point Boundary Value Problems

In this article, a new analytic technique based on Parker-Sochacki iteration is introduced for computing series solution of a general nonlinear two-point boundary value problems with Dirichlet and Neumann boundary conditions. For problems with or without analytic solution, we found out that this easy-to-implement method produced highly accurate results without linearization when compared with their closed form solutions.

The Numerical Solution of Third Order Boundary Value Problems Using Piecewise Bernstein polynomials by the Galerkin Weighted Residual Method

There are few numerical techniques available to solve third order boundary value problems. In this paper, we present Galerkin weighted residual method for constructing the numerical solution of third order linear and nonlinear boundary value problems with two point boundary conditions. The method is formulated as a rigorous matrix form. Several numerical examples of both linear and nonlinear boundary value problems in the literature are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods in the literature. Keywords: Galerkin method, Third order linear and nonlinear BVPs, Bernstein Polynomials.

Existence of symmetric positive solutions for a semipositone problem on time scales

This paper studies the existence of symmetric positive solutions for a second order nonlinear semipositone boundary value problem with integral boundary conditions by applying the Krasnoselskii fixed point theorem. Emphasis is put on the fact that the nonlinear term f may take negative value. An example is presented to demonstrate the application of our main result.

On second order nonlinear boundary value problems and the distributional Henstock-Kurzweil integral

In the present paper, we investigate the existence of solutions to second order nonlinear boundary value problems (BVPs) involving the distributional Henstock-Kurzweil integral. The present results in this article are generalizations of previous results in the literature.

Spectral method for solving high order nonlinear boundary value problems via operational matrices

A spectral method based on operational matrices of Bernstein polynomials using collocation method is elaborated and employed for solving nonlinear ordinary and partial differential equations with multi-point boundary conditions. First, properties of Bernstein polynomial, operational matrices of integration, differentiation and product are introduced and then utilized to reduce the given differential equation to the solution of a system of algebraic equations. This new approach provides a significant computational advantage by converting the given original problem to an equivalent integro-differential equation which implies all boundary condition. Approximate solution is achieved by expanding the desired function in terms of a Bernstein basis and employing operational matrices. Unknown coefficients are determined by collocation. The method is compared with modified Adomian decomposition method, Birkhoff-type interpolation method, reproducing kernel Hilbert space method, fixed point method, finite-difference Keller-box method, multilevel augmentation method and shooting method. Illustrative examples are included to demonstrate the high precision, validity and good performance of the new scheme even for solving nonlinear singular differential equations.

Numerical Approaches for Tenth and Twelfth Order Linear and Nonlinear Differential Equations

The aim of this paper is to solve the tenth and twelfth order linear and nonlinear boundary value problems numerically by the Galerkin weighted residual technique with two point boundary conditions. The well known Bernstein polynomials are exploited as basis functions in the technique and thus the basis functions are needed to modify into a new set of basis functions where the Dirichlet types of boundary conditions are satisfied. The method is developed as a rigorous matrix formulation. Numerical examples, available in the literature, are considered to implement the proposed technique. The comparison shows that the present method is more efficient and yields better results.