Order Linear Boundary Value Problems Research Articles

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.

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

Numerical Solutions for Solving Special Tenth Order Linear Boundary Value Problems using Legendre Galerkin method

Numerical Solutions for Solving Special Tenth Order Linear Boundary Value Problems using Legendre Galerkin method

SOLVABILITY OF A CLASS OF N -TH ORDER LINEAR FOCAL PROBLEMS

This paper presents a recursive method which yields necessary and sufficient conditions for the existence of solutions of a class of n-th order linear focal boundary value problems in the interior of a given interval, in the form of integral inequalities. Some results on the sign of the derivatives of the Green functions of n-th order linear focal boundary value problems will also be provided.

OCTIC B-SPLINE COLLOCATION SOLUTION WITH NON-UNIFORM LENGTH FOR EIGHTH ORDER LINEAR DIFFERENTIAL EQUATION

Presentation of Numerical solution for eighth order linear boundary value problem using Octic B-spline collocation method with non-uniform length is the subject of this paper. In this approach recursive form of B-spline function is used as basis in collocation method. Numerical examples are considered to show the advantage of recursive of B-spline function particularly in non-fixing the length of subintervals.

Nonpolynomial spline technique for the solution of ninth order boundary value problems

In this paper, a nonpolynomial spline technique is applied to solve the ninth order linear special case boundary value problems. The end conditions are derived to complete the definition of a spline. Three examples are numerically illustrated to check the efficiency of the method. The comparative analysis shows that the proposed technique gives better results than the homotopy perturbation method and the modified variational iteration method.

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.

Solutions of Seventh Order Boundary Value Problems Using Ninth Degree Spline Functions and Comparison with Eighth Degree Spline Solutions

In this article, we develop numerical method by constructing ninth degree spline function using extended cubic spline Bickley’s method to find the approximate solution of seventh order linear boundary value problems at different step lengths. The approximate solution is compared with the solution obtained by eighth degree splines and exact solution. It has been observed that the approximate solution is an excellent agreement with exact solution. Low absolute error indicates that our numerical method is effective for solving high order linear boundary value problems.

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.

Solutions for linear conservation laws with gradient constraint

We consider variational inequality solutions with prescribed gradient constraints for first order linear boundary value problems. For operators with coefficients only in L^2 , we show the existence and uniqueness of the solution by using a combination of parabolic regularization with a penalization in the nonlinear diffusion coefficient. We also prove the continuous dependence of the solution with respect to the data, as well as, in a coercive case, the asymptotic stabilization as time t \rightarrow + \infty towards the stationary solution. In a particular situation, motivated by the transported sandpile problem, we give sufficient conditions for the equivalence of the first order problem with gradient constraint with a two obstacles problem, the obstacles being the signed distances to the boundary. This equivalence, in special conditions, illustrates also the possible stabilization of the solution in finite time.

A new spectral method for a class of linear boundary value problems

In this paper we use Bernoulli polynomials to derive a new spectral method to find the numerical solutions of second order linear boundary value problems. The proposed method has been applied to several BVPs with Dirichlet, Neumann and mixed types boundary conditions. Numerical examples provide favorable comparisons with other existing methods.

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.

Solvability ofNth Order Linear Boundary Value Problems

This paper presents a method that provides necessary and sufficient conditions for the existence of solutions ofnth order linear boundary value problems. The method is based on the recursive application of a linear integral operator to some functions and the comparison of the result with these same functions. The recursive comparison yields sequences of bounds of extremes that converge to the exact values of the extremes of the BVP for which a solution exists.

Numerical Solution of Tenth Order Boundary Value Problems by Galerkin Method with Septic B-splines

ABSTRACT In this paper, we present a finite element method involving Galerkin method with septic B-splines as basis functions to solve a general tenth order boundary value problem. The basis functions are redefined into a new set of basis functions which vanish on the boundary where the given set of boundary conditions are prescribed. The proposed method was applied to solve several examples of tenth order linear and nonlinear boundary value problems. The solution of a nonlinear boundary value problem has been obtained as the limit of a sequence of solution of linear boundary value problems generated by quasilinearization technique. The obtained numerical results are compared with the exact solutions available in the literature.

Numerical Solution of Eighth Order Boundary Value Problems by Galerkin Method with Septic B-splines

A finite element method involving Galerkin method with septic B-splines as basis functions has been solved the eighth order boundary value problems The basis functions are redefined into a new set of basis functions which vanish at the boundary where all types of boundary conditions are prescribed. The proposed method was applied to solve several examples of eighth order linear and nonlinear boundary value problems. To test the efficiency of the proposed method, obtained numerical results are compared with exact solutions available in the literature.

Numerical Solution of Fourth Order Boundary Value Problems by Petrov-Galerkin Method with Cubic B-splines as basis Functions and Quintic B-Splines as Weight Functions

paper deals with a finite element method involving Petrov-Galerkin method with cubic B-splines as basis functions and quintic 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. Keywords-Galerkin method, Cubic B-spline, Quintic B-spline, Fourth order boundary value problem, Absolute error.

A High Accuracy Finite Difference Method for Linear Fifth Order Boundary Value Prolbems

In this paper, some high accuracy finite difference schemes for solving the fifth order linear boundary value problem and the derivatives of the solutions are presented. Some appropriate examples are employed to evaluate the performance of the proposed element. The numerical results show that finite difference method can obtain high accuracy solutions.

Numerical Solutions of Sixth Order Linear and Nonlinear Boundary Value Problems

The aim of paper is to find the numerical solutions of sixth order linear and nonlinear differential equations with two point boundary conditions. The well known Galerkin method with Bernstein and modified Legendre polynomials as basis functions is exploited. In this method, the basis functions are transformed into a new set of basis functions, which satisfy the homogeneous form of Dirichlet boundary conditions. A rigorous matrix formulation is derived for solving the sixth order BVPs. Several numerical examples are considered to verify the efficiency and implementation of the proposed method. The numerical results are compared with both the exact solutions and the results of the other methods available in the literature. The comparison shows that the performance of the present method is more efficient and yields better results.

Ultraspherical Integration Method for Solving Beam Bending Boundary Value Problem

In this paper,we applied successive ultraspherical integration matrices, which are used for the numerical solution of fourth order linear boundary value problem arising in bending of a rectangular beam on elastic foundation. This method is used to approximate for the highest-order derivative and generating approximations to the lower-order derivatives through integration of the highest-order derivative. We can then use the produced equations in the form of algebraic system and hence it is converted to nonlinear programming. Numerical examples illustratedthe accuracy and efficiency of the proposed method