Reproducing Kernel Methods For Solving Proportional Delay Differential Equations

Proportional delay diferential equations are widely used in areas such as me-chanics, physics, biology, control science. Ever in the study of this kind of equation,most scholars focus on discussing the existence and stability of the solution whilethe discussion of numerical methods is very little. Therefore, to seek numericalalgorithm of such equations is particularly important.On the basis of previous work, this paper applied the method of reproducingkernel spaces, gave the algorithms for solving nonlinear infnite delay diferentialequations with proportional delay. First of all, it introduced two reproducing kernelspace and applied the good properties of reproducing kernel function to constructa convergence of the iterative sequence in reproducing kernel space. Afterwards,it get the approximate solution of the equation through the form of a truncatedseries. It proved that the error between the approximation solution and exact solu-tion is monotone decreasing and proved the validity of this algorithm through somenumerical examples. The innovative methods of this article is that it proved the ap-proximate solution converging to exact solution of the equation through proving theboundedness of iterative sequence which made the numerical results more accurate.