The Reproducing Kernel Method Of Soving A Class Of Reaction-diffusion Equations

The formation and development of the differential equations are closely relatedwith astronomy, mechanics and other subjects. They are widely applied in ballisticcomputation, the study on the stability of ffights of aircrafts and missiles and so on.Meanwhile, with the appearance and development of amount of borderline sciencesuch as electromagnetic, hydromechanics, chemical kinetics, dynamical meteorology,ocean dynamics, a multitude of the differential equations have appeared. Withthe soaring development in science and technology, as one of an important studyproblems in the field of mathematics, the application of nonlinear problems becomesmore and more important in the field of control process, ecologic and economicsystem, circulation system of chemical industry and epidemiology.Being a class of important partial differential equations, nonlinear diffusionequations come from diffusion phenomenon, the theory of percolation, biochemistry.It can describe engineering and so on, and be applied very extensive. It is well knownthat it is not easy to obtain the exact solutions of nonlinear equations under severalconditions. therefor, how to find out approximate solutions with high precision isof far-reachingly current significance and practical use.In this paper, based on the theory and skills of numerical analysis in the re-producing Hilbert space, we solve a class of nonlinear reaction-diffusion equations.First, based on the relationship between completely continuous functions andabsolutely continuous ones, we construct the reproducing Hilbert space W(m,n)(D),and define its inner product and its norm. At the same time, we give the proofsof relative important theorems. By studying, we find that the separable functionshave good properties in W(m,n)(D) in the meaning of inner product and the norm.Second, in the above of fundamental research foundation, we redefine and re-construct the reproducing Hilbert space W[D]. According to the initial boundaryvalue condition of the model, we transform the nonlinear diffusion equations intoequal operator equations by constructing the bounded linear operator. After that,we decompose the reproducing kernel space W[D] into the direct sum of orthogonal complement subspaces, and construct the normal orthogonal base in the subspaces.In this way, we construct the normal orthogonal base of the reproducing Hilbertspace W[D],and obtain the series representation of the exact solutions.At last, we find out the representation of numerical solutions of nonlinear dif-fusion equations by searching the minimum-norm.We modify the approximate so-lutions in order to achieve higher computational precision.Meanwhile, we give theerror comparison between the exact solutions and the approximate ones, and thedata in this paper show that the algorithm is of high precision.The overlapped fig-ures of the exact solutions and the approximate ones visually validate the precisionand effciency of the algorithm.