Numerical Solution For A Class Of Age-dependent Capital System And Population Model

In this paper, we analyze the numerical solutions based on Bernstein polynomials and Chebyshev wavelets for the nonlinear age-dependent capital system and age-structured population model, respective-ly. Also the error of two methods are presented. Furthermore, We concern mean-square dissipativity of split-step backward Euler (SSBE) method and compensated split-step backward Euler(CSSBE) method for the nonlinear age-dependent capital system with fractional Brownian motion and Poisson jumps. This paper mainly discusses the following three parts:The first chapter, we present and analyze a numerical method based on Bernstein polynomials for the nonlinear age-dependent (vintage) capital system. The operational matrices of integration, differentiation, dual and product are introduced, which is utilized to simplify the age-dependent capital problem instead of the solution of algebraic equations. A numerical example is provided to illustrate the theoretical result.The second chapter, the numerical solution of nonlinear age-structured population models is obtained by using Chebyshev wavelets. The properties of the Chebyshev wavelets and Chebyshev polynomials are presented. This method transforms the integral equation into a matrix equation. Also we give an example to demonstrate the efficiency of the proposed method.The third chapter, we investigate the mean-square dissipativity of split-step backward Euler method and compensated split-step backward Euler method for the nonlinear age-dependent capital system with fractional Brownian motion and Poisson jumps. Under Lipschitz condition, we give the mean-square dissipativity of two methods, respectively.