Barycentric Interpolation Collocation Method For Some Nonlinear Partial Differential Equations(Groups) And Its Applications

Nonlinear partial differential equation?s??NLPDE?s??have been widely applied in many fields,such as biochemistry,fluid mechanics,atmospheric science and finance.The mathematical models of many nonlinear problems in these fields can be attributed to the fixed solution of NLPDE?s?,such as the diffusion phenomenon in biochemistry,the population model in the biological field,and the option pricing model in the fi-nancial field,and so on.It is generally difficult to obtain the analytical solution of the NLPDE?s?.So the numerical method is generally used to solve them.Although the traditional numerical methods have shown their advantages in the process of solving NLPDE?s?,it is still of great significance to find a numerical method with high nu-merical accuracy and simple calculation.In this paper,we mainly solve two kinds of nonlinear diffusion equations of gener-alized Burgers-Huxley and Korteweg-de Vries-Burgers?KdVB?by barycentric interpo-lation collocation method,and extend the method to the solution of nonlinear coupled Burgers equations.Above all,the method is firstly applied to the numerical simulation of Verhulst type NLPDE population model and nonlinear Black-Scholes option pricing model in illiquid markets.In the population model,we study the distribution of popu-lation density for the different values of population mortality,and how the error norm of population density changes with the change of number of nodes,and compare and analyse the numerical results obtained by present method and the numerical results in the literature.In the Black-Scholes option pricing model,we study how the option price varies with the the change of underlying asset price and the range of the Delta of option price.The influence of volatility?₀and market liquidity parameter?on the rise and fall of option price is also analyzed and discussed.Some numerical examples and numerical experiments are given in this paper.The numerical results show the superiority of present method in terms of computation accuracy,program implemen-tation and computational efficiency.