Numerical Method For Solving A Class Of Nonlinear Differential Equations

The formation and development of differential equations are closely related with mechanics, astronomy, physics and other subjects. They are widely applied in automatic control, all kinds of designs of electronic devices, ballistic computation, the study on the stability of flights of aircrafts and missiles and the process of chemical reactions. All above problems can be transformed to solving differential equations. But, the exact solutions of many kinds of equations can not be obtained and we can only get the higher precision approximate solutions.It should also be pointed out that the differential equations describing the physical process and the initial conditions determined by tests are also approxi-mate. When the theory of the differential equations becomes by means of it. In fact, as long as the corresponding differential equations are presented and the meth-ods of solving them are obtained, the differential equations will became the most vital branch of mathematics. Thus, looking for the effective numerical methods for solving those differential equations which have theoretical value and actual back-ground is becoming increasingly important.In this paper, based on the theory and skills of numerical analysis in the re-producing kernel space, we give some numerical methods for solving the following classes of linear and nonlinear differential equations.First, In the premise of existence and unique of the solution, a new algorithm for solving a fourth-order ordinary differential equations with boundary value condition is obtained in the reproducing kernel space W5[0,1]. The representation of the exact solutions is obtained in the series form. By truncating the series, we obtain the approximate solutions.The numerical results are displayed to demonstrate the validity of the method.Second, in terms of by searching minimal value method, for solving nonlinear hyperbolic telegraph equation with an integral condition is discussed in the repro-ducing kernel space W(3,3) (Ω). An orthogonal basis is constructed. The unknown functions of differential systems are expanded into the Fourier series form on the orthogonal basis. The representation of the exact solution is obtained in the series form. The representation of the exact solutions is obtained in the series form by searching minimal value method for the nonlinear terms and satisfactory numerical results are also presented.Finally, a class of nonlinear Neumman function with boundary value conditions in the reproducing kernel space W3[0,1]. by the method of constructing a itera-tive sequence for the nonlinear terms and satisfactory numerical results are also presented.