A new algorithm for solving differential equations

Hosseini, K.; Biazar, Jafar; Ansari, R.; Gholamin, P.

doi:10.1002/mma.1601

Cited by 11 publications

(8 citation statements)

References 15 publications

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“…Division Theorem : (see [9][10][11]) Suppose that P(ω,z) and Q(ω,z) are polynomials in the complex domain C[ω,z]; and P(ω,z) is irreducible in C[ω,z]. If Q(ω,z) vanishes at all zero points of P(ω,z), then there exists a polynomial G(ω,z) in C[ω,z] such that ( , ) = ( , ) ( , ).…”

Section: The First Integral Methodsmentioning

confidence: 99%

New Solutions for the Higher-Order Nonlinear Schrödinger Equation Using Integral Methods

Eldabe¹,

Moussa²,

–Shiekh³

et al. 2012

AJCAM

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In this paper, we use two integral methods, the first integral method and the direct integral method to study a higher-order nonlinear Schrödinger equation (NLSE). The application of the first integral method yield trigonometric function solutions and solitary wave solutions. Using the direct integration lead to shock wave solution and Jacobi elliptic function solutions. The direct integral method is more concise and direct than the first integral method.

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Section: The First Integral Methodsmentioning

confidence: 99%

New Solutions for the Higher-Order Nonlinear Schrödinger Equation Using Integral Methods

Eldabe¹,

Moussa²,

–Shiekh³

et al. 2012

AJCAM

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“…Dehghan and Shakeri [10] applied an exponential transformation to the Lane-Emden type equations to overcome the difficulty of a singular point at x D 0 and solved the resulting nonsingular problem by the variational iteration method. For other works placed to solve this class of nonlinear singular differential equations, see, for example, [6,[11][12][13][14][15][16][17][18][19][20].…”

Section: The Lane-emden-fowler Equationmentioning

confidence: 99%

“…Similarly, by substituting (16) and (17) in Eqs (13) and (14) and then by collocating them in the points i , i D 1, : : : , n, we obtain again (20) and (21). Finally, for better vision of the discretized problem in Eqs (13), (14), and (24), the 2n nonlinear algebraic equations are represented as…”

Section: Solution Of the Lane-emden-fowler Boundary Value Problemsmentioning

confidence: 99%

An efficient pseudospectral method for numerical solution of nonlinear singular initial and boundary value problems arising in astrophysics

Mehrpouya

2015

Math Methods in App Sciences

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In the manuscript, a pseudospectral method is developed for approximate and efficient solution of nonlinear singular Lane-Emden-Fowler initial and boundary value problems arising in astrophysics. In the proposed method, the Gauss pseudospectral method is utilized to reduce the problem to the solution of a system of algebraic equations. Furthermore, the Gauss pseudospectral method is developed for finding the first zero of the solution of this equation that gives the radius of the star, in which the numerous properties of the star such as mass, central pressure, and binding energy can be computed through their relations to this solution. The main advantage of the proposed method is that good results are obtained even by using a small number of discretization points and the rate of convergence is high. The accuracy and performance of the proposed method are examined by means of some numerical experiments.

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“…Several authors have investigated this equation. The interested reader can see . However, the approach in the current paper is different.…”

Section: Introductionmentioning

confidence: 99%

Four techniques based on the B‐spline expansion and the collocation approach for the numerical solution of the Lane–Emden equation

Lakestani

Dehghan

2013

Math Methods in App Sciences

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Four numerical techniques based on the linear B-spline functions are presented for the numerical solution of the Lane-Emden equation. Some properties of the B-spline functions are presented and are utilized to reduce the solution of the Lane-Emden equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new techniques.Several authors have investigated this equation. The interested reader can see [1,[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. However, the approach in the current paper is different. This paper deals with the numerical solution of the Lane-Emden equation [1] by means of the B-spline techniques where either the first or second derivatives of the solution are approximated by B-splines. In the current paper, we give four different numerical methods to solve the Lane-Emden equation (1.1). In the present methods, we reduce the problem to a set of algebraic equations by expanding the unknown function as the linear B-spline functions specially constructed on bounded intervals with unknown coefficients. Some properties of the linear B-spline function like the operational matrices of the derivative and integration are given. These matrices together with the B-spline functions are then utilized to evaluate the unknown coefficients.Also, there have been numerous applications of B-spline to the numerical solution of singular initial and/or boundary value problems. But we will not review them here. Also, we refer the interested reader to [26][27][28] for alternative approaches in the numerical solution of initial or boundary value problems [29][30][31][32][33][34][35][36][37][38].This article is organized as follows: In Section 2, we describe the formulation of the B-spline functions on OE0, 1, the operational matrices of the derivative and integration and some properties required for our subsequent development. In Section 3, the present techniques are proposed to approximate the solution of the given problem. As a result, a set of algebraic equations is formed, and a solution of the considered problem is introduced. In Section 4, we report our computational results and demonstrate the accuracy of the proposed numerical schemes by presenting several test problems. Section 5 completes this paper with a brief conclusion.

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A new algorithm for solving differential equations

Cited by 11 publications

References 15 publications

New Solutions for the Higher-Order Nonlinear Schrödinger Equation Using Integral Methods

New Solutions for the Higher-Order Nonlinear Schrödinger Equation Using Integral Methods

An efficient pseudospectral method for numerical solution of nonlinear singular initial and boundary value problems arising in astrophysics

Four techniques based on the B‐spline expansion and the collocation approach for the numerical solution of the Lane–Emden equation

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