Fifth-order Boundary Value Problems Research Articles

The Effect of Combining the Adomian Decomposition and Homotopy Perturbation Methods on Solving Fifth-Order Boundary Value Problems

This paper considers fifth-order boundary value problems with two-point boundary conditions. For the investigation, first, the Adomian decomposition method (ADM) was applied to the equation and then the homotopy perturbation method (HPM) was used to continue the solution. In addition, this combined method was applied to solve two linear and nonlinear experiments. Then the numerical results obtained by other methods were compared. Furthermore, the convergence of the methods was analyzed. In most of the articles, the HPM is used for equations with initial conditions and the ADM is used for equations with initial and boundary conditions. In this study, a combination of two methods (the ADM and HPM) was used to solve equations with boundary conditions. The results of this combined method are remarkable.

A Comparative Study of Approximate Solution of Fifth-order Boundary-value Problems by Applying HaarWavelets Technique and Galerkin Method with Quartic B-splines

A Comparative Study of Approximate Solution of Fifth-order Boundary-value Problems by Applying HaarWavelets Technique and Galerkin Method with Quartic B-splines

DEVELOPMENT AND IMPLEMENTATION OF A TENTH-ORDER HYBRID BLOCK METHOD FOR SOLVING FIFTH-ORDER BOUNDARY VALUE PROBLEMS

A hybrid convergent method of tenth-order is presented in this work for directly solving fifth-order boundary value problems in ordinary differential equations. A unique direct block approach is obtained by combining multiple Finite Difference Formulas which are derived via the collocation technique. The proposed method is fully analyzed and the existence and uniqueness of the discrete solution is established. Different numerical examples are considered and the results are compared with those provided by existing works in the literature. The comparison shows the good performance of the present method over some cited works in the literature, confirming the competitiveness and superiority of the new numerical integrator.

Existence of solution for a nonlinear fifth-order three-point boundary value problem

Existence of solution for a nonlinear fifth-order three-point boundary value problem

Boundary Value Method for Numerically Solving Fifth-order Boundary Value Problems

Boundary Value Method for Numerically Solving Fifth-order Boundary Value Problems

Spectral Solutions for Fifth-Order Boundary Value Problems Using Generalized Jacobi Operational Matrix of Derivatives

The main objective of this paper is to develop new spectral solutions for linear and nonlinear fifth-order boundary value problems. For this aim, a new operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The key idea for obtaining the suggested spectral solutions is to convert the fifth-order linear and nonlinear boundary value problems into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. A theoretical error bound is derived and the convergence analysis of the proposed expansion is proved. Numerical tests are presented aiming to ascertain the high efficiency and accuracy of the suggested algorithms.

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.

An Enhanced Quartic B-spline Method for a Class of Non-linear Fifth-Order Boundary Value Problems

In this paper, we apply quartic B-splines properly to study a new approximation method for numerical solutions and numerical derivatives for a class of non-linear fifth-order boundary value problems. Their analytic solutions and any-order derivatives are well approximated with $${O(h^{6})}$$ errors. Numerical tests are performed and numerical results show that our new method is very practical and effective.

A Comparison on Solutions of Fifth-Order Boundary- Value Problems

A fast and accurate numerical scheme for the solution of fifth-order boundary-value problems has been investigated in this work. We apply the reproducing kernel method (RKM) for solving this problem. The analytical results of the equations have been acquired in terms of convergent series with easily computable components. We compare our results with the numerical methods: B-spline method, decomposition method, variational iteration method, Sinc-Galerkin method and homotopy perturbation method. The comparison of the results with exact ones is made to confirm the validity and efficiency.

Septic B-Spline Solution of Fifth-Order Boundary Value Problems

A numerical method based on septic B-spline function is presented for the solution of linear and nonlinear fifth-order boundary value problems. The method is fourth order convergent. We use the quesilinearization technique to reduce the nonlinear problems to linear problems and use B-spline collocation method, which leads to a seven nonzero bands linear system. Illustrative example is included to demonstrate the validity and applicability of the proposed techniques.

Application of non-polynomial spline to the solution of fifth-order boundary value problems in induction motor

Abstract Non-polynomial spline functions of the form Span{1, x, x 2 , x 3 , x 4 , x 5 , cos ( kx ) + e x } , where k can be real or pure imaginary, are used to find the numerical solution of linear fifth-order boundary value problems. The order of convergence of the method is observed to be of O ( h 2 ) . A fifth order convergent method is defined with the help of improved end-conditions. Three examples are considered to show the reliability and efficiency of the method. The numerical results, obtained, endorse the improved order of convergence of the method.

A Numerical Solution to the Nonlinear Fifth Order Boundary Value Problems

The reproducing kernel space method is used to solve the nonlinear fifth-order boundary-value problems. The approach provides the solution in the form of a convergent series with easily computable components. The present method compared with the others methods, reveals that the present method is more effective and convenient.

Solving a Class of Singular Fifth-Order Boundary Value Problems Using Reproducing Kernel Hilbert Space Method

We use the reproducing kernel Hilbert space method to solve the fifth-order boundary value problems. The exact solution to the fifth-order boundary value problems is obtained in reproducing kernel space. The approximate solution is given by using an iterative method and the finite section method. The present method reveals to be more effective and convenient compared with the other methods.

Sextic spline collocation methods for nonlinear fifth-order boundary value problems

In this paper, two sextic-spline collocation methods are developed and analysed for approximating solutions of nonlinear fifth-order boundary-value problems. The first method uses a spline interpolant and the second one is based on a spline quasi-interpolant, which are constructed from sextic splines. They are both proved to be second-order convergent. Numerical results confirm the order of convergence predicted by the analysis. It has been observed that the methods developed in this paper are better than the others given in the literature.

A new method for solving a system of fifth-order obstacle boundary value problems

In this paper, we use the variation of parameters method for solving a system of fifth-order boundary value problems associated with obstacle problems. The results are compared with the exact solution over the whole domain and error estimates are calculated. An example is given to illustrate the efficiency and implementation of the variation of parameters method. Key words: Variation of parameters method, system of fifth-order obstacle boundary value problems.

Spectral Shifted Jacobi Tau and Collocation Methods for Solving Fifth-Order Boundary Value Problems

We have presented an efficient spectral algorithm based on shifted Jacobi tau method of linear fifth-order two-point boundary value problems (BVPs). An approach that is implementing the shifted Jacobi tau method in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of fifth-order differential equations with variable coefficients. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplify the problem. Shifted Jacobi collocation method is developed for solving nonlinear fifth-order BVPs. Numerical examples are performed to show the validity and applicability of the techniques. A comparison has been made with the existing results. The method is easy to implement and gives very accurate results.

An efficient computational method for linear fifth-order two-point boundary value problems

In this paper, we present a new algorithm to solve general linear fifth-order boundary value problems (BVPs) in the reproducing kernel space W 2 6 [ a , b ] . Representation of the exact solution is given in the reproducing kernel space. Its approximate solution is obtained by truncating the n -term of the exact solution. Some examples are displayed to demonstrate the computational efficiency of the method.

The numerical solution of fifth-order boundary value problems by the variational iteration method

This paper applies the variational iteration method to solve fifth-order boundary value problems; just one iteration results in highly accurate solutions. A comparison of the results with exact ones is made to confirm the validity and efficiency.

Approximate analytical solutions of fifth-order boundary value problems by the variational iteration method

The work presents a derivation of approximate analytical solutions of high-order boundary value problems using the variational iteration method. The solution procedure is simple and the obtained results are of high accuracy. Some examples are given to elucidate the effectiveness and convenience of the method.

Existence Theorems for a Fourth Order Boundary Value Problem

In this paper,we study the existence of a solution for fifth-order boundary value problem ଶ , ሻplacing certain restrictions on the nonlinear term , we prove the existence of at least one solution to the boundary value problem with the use of the lower and upper solution method and Schauder fixed-point theorem. The construction of lower or upper solution is also presented. Boundary value problems of very similar type are also considered.