Agorithm Research For Some Operator Equations

With the development of science and technology, more and more problems arise inengineering applications. Besides differential equations, integral equations and integral-differential equations with the various conditions of determining solution, new kind equa-tions are coming out constantly. Fractional order differential equation is a new type pre-senting from biochemistry, hydrogeology, image processing. In the reproducing kernelspace, we regard these equations as an unified operator equation, and discuss the uniformnumerical method for solving these equations. The reason that we select the reproduc-ing kernel space is that it has high unity to investigate these problems in the reproducingkernel space. The main results obtained in this thesis are summarized as follows:Firstly, in the practical application, different problems are limited by differentboundary value conditions. According to these various conditions, we correspondinglyconstruct reproducing kernel spaces, in which the reproducing kernel and the other func-tions satisfy the conditions. That is, once the reproducing kernel is fixed, the boundaryconditions are not considered, so it is easier to approximate problems. We construct suc-cessively the basal reproducing kernel space, the reproducing kernel space of satisfyingthree-point boundary value condition and the reproducing kernel space for solving partialdifferential equations, and give the concrete methods of obtaining the reproducing kernelfunctions. When computing the reproducing kernel for solving partial differential equa-tions, we don't use the traditional method that the reproducing kernel is obtained by theproduct reproducing kernel of two one-dimension reproducing kernel spaces, but deducefrom the expression of the inner product in the reproducing kernel space.Secondly, we give double projection method based on single projection method tosolve Fredholm integral equation of the third kind. The convergence of single projectionmethod is discontent, but the double projection method overcoming the deficiency ofsingle projection method. The single, double projection method are only suitable forlinear operator equation. Based on the double projection method, we present doubleprojection iteration method for solving the nonlinear operator equation and prove theconvergence of this method during solving nonlinear third-order three-point boundaryvalue problem. Thirdly, as we known, there is no reference about applying reproducing kernel the-ory to solve fractional order equations. The double projection method is expanded tosolve linear partial differential equations with fractional derivatives. Improved doubleprojection method is given with an example of space-time fractional advection-dispersionequation concretely.Finally, for nonlinear fractional order equations, Adomian decomposition methodis introduced into double projection method. On the condition that the original nonlinearproblem is invariable, ADM deals with the problem primely by Adomian polynomial. Thecombination method Adomian-double projection decomposition method not only simpli-fies the progress of double projection method, but also uses the advantage of obtainingsolution of fast convergence by ADM. Therefore, nonlinear fractional order equationscan be effectively solved by using advantages of these two method.In a word, we provide a single projection method for solving operator equationsbased on reproducing kernel theory and this paper takes the idea of single projectionmethod as the main line, develops double projection method, double projection iterationmethod, improved double projection method and Adomian-double projection decomposi-tion method and applies those methods to solve ordinary and partial differential equations.In order to verify the theory analysis and apply the abstract theory, the numerical exper-iments are given for different equations by corresponding method. The results show thatthose methods are practical.