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Numerical Method for Solving a Class of Nonlinear Differential Equations

Author: WangYang
Tutor: MeHuanMin
School: Harbin Normal University
Course: Computational Mathematics
Keywords: Differential Equations Boundary conditions Reproducing kernel space Exact solution Algorithm
CLC: O241.8
Type: Master's thesis
Year: 2010
Downloads: 68
Quote: 0
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Abstract


The formation and development of the differential equations is closely related to the development and mechanics, astronomy, physics and other disciplines in the automatic control, electronics device design, the trajectory calculation, aircraft and missile flight stability, chemical the reaction stability research and other fields have a wide range of applications. these problems can be reduced to find the solution of differential equations, however, many of the differential equation is not the exact solution, but only to get the exact solutions of a higher degree of approximate solution It should also be pointed out that, used to describe the physical process of differential equations, as well as from the experimental determination of the initial conditions is approximate when the gradual improvement of the theory of differential equations, it can accurately describe the basic law of the changes of things In fact, as long as the corresponding differential equations are listed, understand the equation, differential equations will become the most viable branch of mathematics, seeking to solve these practical background and theoretical value of the differential effective numerical methods change becoming more and more important. In this paper, the reproducing kernel space numerical analysis theories and techniques to solve the following nonlinear differential equations numerically solving the problem. First of all, under the premise of the existence and uniqueness of the solution of the problem in reproducing kernel space W5 [0 1] New algorithm for solving a class of fourth-order ordinary differential equations with boundary conditions, the exact expression of the exact solutions of the form of series proved that the calculation error of the numerical solution according to the number of nodes increases strictly monotonically decreasing and has been satisfied with the numerical results, numerical examples verify the effectiveness of this method. tiny discuss with initial and boundary conditions in the reproducing kernel space W (3,3) (omega) the use of search nonlinear hyperbolic telegraph equation constructed orthonormal basis, the unknown function of the equation in this group based on the Fourier series expansion, to obtain the exact expression of the exact solutions of the form of a series of nonlinear terms Search tiny way to get the exact solution of the numerical approximation solution to prove the good nature of the truncation error of the approximate solution is monotonically decreasing with the increase of nodes, with satisfactory numerical results. Finally, in another reproducing kernel space W3 [0 1] for a class of nonlinear Neumman boundary value problem, using the method of the structure of the nonlinear iterative sequence to obtain the exact expression of the exact solutions of the form of series and satisfactory numerical results.

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CLC: > Mathematical sciences and chemical > Mathematics > Computational Mathematics > Numerical Analysis > The numerical solution of differential equations, integral equations
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