Solving Some High-order Singular Differential Equations

As well as known, a large number of problems such as physics, astronomy,and chemical kinetics could be summarized as differential equations. Most of thedemand equation is not the primary solution, So it is necessary to discuss thenumerical solution of differential equations.Finite difference method is the first method that computer simulation approach,Its expression is simple,and the mathematical concepts intuitive. This methodtransform the differential problems into the approximate numerical solution of al-gebraic problems directly. Classical finite difference method is used differences atnodes instead of derivative (numerical integration) or integral interpolation (numer-ical integration) to construct the difference scheme. Finite difference method is usedto solving the numerical solution of ordinary differential boundary value problems.Sometimes it is diffcult to solve ordinary differential initial value problem. Thetheory of reproducing kernel theory comes from the integral equation theory, har-monic and biharmonic boundary value function. Reproducing kernel space Wp[a,b]is a Hilbert space with reproducing kernel. First of all, according to the order of theequation to define a reproducing kernel space. Then defined inner product and normin the space . Homogenization the boundary of the original equation, and transformthe domain into [0,1] interval. Choose a countable dense subset from [0,1] interval,and introduce a linear operator . Construct a orthonormal basis in the reproducingkernel space Wp[0,1]. According to the reproducing kernel space theory and appli-cation of reproducing kernel function of the renewable it will construct the seriesexpressions of differential exact solution and approximate solution.For the singular boundary value problem, many research tools used for thenon-singular boundary value problems is no longer apply. This paper used thedifference method and the theory of reproducing kernel space to solve the fourth-order boundary singular value problem.With the numerical simulation the relevantexamples,both methods has been proved validity to the singular boundary value problem.