The Numerical Investigations of Non-Polynomial Spline for Solving Fractional Differential Equations

We present a crossing approach based on the new construction of non-polynomial spline function to investigate the numerical solution of the fractional differential equations. We find the accuracy of the spline method and to presenting the completion of non-polynomial spline two examples for problems are used. To clarify, we present the numerical computations that can be used to solve difficult problems while the results are found and got to be in good error estimation with comparing exact 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

An elegant operational matrix based on harmonic numbers: Effective solutions for linear and nonlinear fourth-order two point boundary value problems

This paper analyzes the solution of fourth-order linear and nonlinear two point boundary value problems. The suggested method is quite innovative and it is completely different from all previous methods used for solving such kind of boundary value problems. The method is based on employing an elegant operational matrix of derivatives expressed in terms of the well-known harmonic numbers. Two algorithms are presented and implemented for obtaining new approximate solutions of linear and nonlinear fourth-order boundary value problems. The two algorithms rely on employing the new introduced operational matrix for reducing the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. For this purpose, the two spectral methods namely, Petrov-Galerkin and collocation methods are applied. Some illustrative examples are considered aiming to ascertain the wide applicability, validity, and efficiency of the two proposed algorithms. The obtained numerical results are satisfactory and the approximate solutions are very close to the analytical solutions and they are more accurate than those obtained by some other existing techniques in literature

Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms

Solution of Eighth Order Boundary Value Problems Using Differential Transformation Technique

ABSTRACT In this paper we have applied the Differential Transform Method (DTM) for solving eighth-order boundary value problems. The analytical and numerical results of the equations have been obtained in terms of convergent series with the easily computable components. Three examples are considered for the numerical illustrate and implementation of this method. Numerical Comparisons with respect to the analytical solutions have been considered . It is observed that the method is an alternative and efficient for finding the approximate solutions of the boundary values problems