Hikmet Caglar

B-SPLINE INTERPOLATION COMPARED WITH FINITE DIFFERENCE, FINITE ELEMENT AND FINITE VOLUME METHODS WHICH APPLIED TO TWO-POINT BOUNDARY VALUE PROBLEMS

Abstract This paper considers the B-spline interpolation and compares this method with finite difference, finite element and finite volume methods which applied to the two-point boundary value problem. - d d x p ( x ) d u d x = f ( x ) , a x b , u ( a ) = u ( b ) = 0 .

B-SPLINE SOLUTION OF SINGULAR BOUNDARY VALUE PROBLEMS

Homogeneous and non-homogenous singular boundary value problems (special case) are solved using B-splines. The original differential equation is modified at singular point then the boundary value problem is treated by using B-spline approximation. The method is tested on some model problems from the literature, and the numerical results are compared with exact solution.

B-spline method for solving Bratu's problem

In this paper, we propose a B-spline method for solving the one-dimensional Bratus problem. The numerical approximations to the exact solution are computed and then compared with other existing methods. The effectiveness and accuracy of the B-spline method is verified for different values of the parameter, below its critical value, where two solutions occur.

B-spline method for solving linear system of second-order boundary value problems

The B-spline method is used for the numerical solution of a linear system of second-order boundary value problems. Two examples are considered for the numerical illustration and the method is also compared with the method proposed by J. Lu [J. Lu, Variational iteration method for solving a nonlinear system of second-order boundary value problems, Comput. Math. Appl. 54 (2007) 1133-1138]. The comparison shows that the B-spline method is a more efficient and effective tool and yields better results.

Fifth-degree B-spline solution for a fourth-order parabolic partial differential equations

In this paper, we find the numerical solution of a fourth-order parabolic partial differential equation. A family of B-spline methods has been considered for the numerical solution of the problems. The results show that the present method is an applicable technique and approximates the exact solution very well.

Solution of fifth order boundary value problems by using local polynomial regression

Abstract In this paper, we present a novel method based on the local polynomial regression for solving of fifth order boundary value problems. The method is tested on numerical example to demonstrate its usefulness. The method presented in this paper is also compared with those developed by Siddiqi and Akram [Solution of fifth order boundary value problems using nonpolynomial spline technique, Appl. Math. Comput. 175 (2006) 1575–1581], as well and is observed to be better.

Fifth-degree B-spline solution for nonlinear fourth-order problems with separated boundary conditions

In this paper, we discussed a fifth-degree B-spline solution for the numerical solution to nonlinear fourth-order boundary value problems (BVPs) with separated boundary conditions. Two numerical examples are given to illustrate the efficiency and performance of the method. The method gives accurate results for both the linear and nonlinear cases.

Non-polynomial Spline Method for the Solution of Non-linear Burgers’ Equation

A non-polynomial cubic spline method is proposed in this paper to solve one-dimensional non-linear Burgers’ equation [Burger, A Mathematical Model Illustrating the Theory of Turbulence (1948); Rashidinia and Mohammadi, Int. J. Comp. Math. 85, 843–850 (2008)]. An example is solved to assess the accuracy of the method. The numerical results obtained by this way are compared with the exact solution to show the efficiency of the method.

Non-polynomial Spline Solution for a Fourth-Order Non-homogeneous Parabolic Partial Differential Equation with a Separated Boundary Condition

In this paper, a fourth-order non-homogeneous parabolic partial differential equation with initial and separated boundary conditions is solved by using a non-polynomial spline method. In the solution of the problem, finite difference discretization in time, and parametric quintic spline along the spatial coordinate have been carried out. The result shows that the applied method in this paper is an applicable technique and approximates the exact solution very well.