SOLUTIONS OF FRATIONAL EMDEN-FOWLER EQUATIONS BY HOMOTOPY ANALYSIS METHOD

In this paper, we have solved the singular initial value problems of fractional Emden-Fowler type equations by using the homotopy analysis method. The approximate analytical solution of this type equations are obtained

Comparing numerical methods for the solutions of systems of ordinary differential equations

AbstractIn this article, we implement a relatively new numerical technique, the Adomian decomposition method, for solving linear and nonlinear systems of ordinary differential equations. The method in applied mathematics can be an effective procedure to obtain analytic and approximate solutions for different types of operator equations. In this scheme, the solution takes the form of a convergent power series with easily computable components. This paper will present a numerical comparison between the Adomian decomposition and a conventional method such as the fourth-order Runge-Kutta method for solving systems of ordinary differential equations. The numerical results demonstrate that the new method is quite accurate and readily implemented

A note on the existence of positive solutions of singular initial-value problem for second order differential equations

We are interested in the existence of positive solutions to initial-value problems for second-order nonlinear singular differential equations. Existence of solutions is proven under conditions which are directly applicable and considerably weaker than previously known conditions

Accurate approximate solution of classes of boundary value problems using modified differential transform method

In this paper, a numerical scheme so-called modified differential transformation method (MDTM) based on differential transformation method (DTM), Laplace transform and Pad´e approximation will be used to obtain accurate approximate solution for a class of boundary value problems (BVP’s). The MDTM is employed as an alternative technique to overcome some difficulties in the behavior of the solution and to be valid for a large region. The numerical results obtained demonstrate the applicability and validity of this technique. Numerical comparison is made with existing exact solution

NUMERICAL SOLUTIONS OF SINGULAR NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS USING SAID-BALL POLYNOMIALS

In this article, the collocation method based on Said-Ball polynomials have been used to solve the singular nonlinear ordinary differential equations of various orders numerically. An operational matrix forms of these ordinary differential equations are obtained from Said-Ball polynomial with variated relations of solution and different derivatives. The presented method reduces the given problem to a system of nonlinear algebraic equations, which removed the singularity of ordinary differential equations. Resulting system is solved using Newton\u27s iteration method to get the coefficients of Said-Ball polynomials. We obtained approximate solutions of the problem under study. Numerical results have been obtained and compared with exact and other works. The presented method gives impressive solutions, that show the accuracy and reliability of the proposed method

The numerical solution of singular initial value problems using Chebyshev wavelet collocation method

AbstractWavelet analysis is a recently developed mathematical tool for many problems. In this paper, an efficient and new numerical method is proposed for the numerical solution of singular initial value problems, which is based on collocation points with Chebyshev wavelet. The present method is developed using the Chebyshev wavelet and its operational matrices to obtain higher accuracy. It has been shown here that the present method can be easily implemented and the results obtained are most accurate. Hence the present method has a clear advantage over the classical methods. Numerical order of convergence of the proposed method is calculated. The results show the better accuracy of the proposed method, which is justified through the illustrative examples

The Taylor Matrix Method for Approximate Solution of Lane-Emden Equation with index-n

Abstract: Many problems in mathematical physics can be formulated as an equation of Lane-Emden type. There are many methods for the solution of this equation. One of these methods is the Taylor matrix method. The only types of nonlinear equations that this method has been applied so far are the Riccati and Abel equations. In this study, an algorithm based on the Taylor matrix method is proposed and applied to the nonlinear Lane-Emden equation with index-n. An example is also given