Study On Variable Limit Integral Method For The Approximate Solution Of Two Partial Differential Equations

Differential equations are important mathematical models for solving engineering physical problems.However,a number of partial differential equations are difficult to obtain analytical solutions with initial and boundary conditions,or their analytical solutions can’t be expressed with elementary functions.Therefore,numerical methods for solving partial differential equations have developed rapidly.This paper mainly focuses on the numerical study of Rosenau-Burgers equation and dissipative symmetric regularized long wave equation.Firstly,the intermediate variables are introduced to reduce the orders of original equations,and use the Variable Limit Integral method to construct its discrete schemes.Next,the well-posedness of the numerical schemes is proved.Finally,numerical experiments are used to verify the effectiveness of numerical schemes.On the one hand,we study the Rosenau-Burgers equation,which describes the problem of shallow water.Firstly,the intermediate variables are introduced to reduce the orders of original equation.Then we combine the Variable Limit Integral method with Taylor function method to discretize the equation in the space direction.At the same time,Crank-Nicolson difference method is used to discretize in the time direction.After that,we obtain two nonlinear implicit fully-discrete schemes with fourth order accuracy in space direction and second order accuracy in time direction.Next,the existence and uniqueness of the numerical solution and the energy conservation,convergence and stability of the discrete schemes are all proved in detail.Finally,the numerical experiments verify the effectiveness of numerical schemes with analyzing the energy property of numerical schemes and the errors between the numerical solution and the analytical solution.On the other hand,we study the dissipative symmetric regular long wave equation with damping terms.Firstly,the intermediate variables are introduced to reduce the orders of original equations.In the space direction,the Variable Limit Integral method and Taylor function method are combined for numerical discretization.The Crank-Nicholson difference method is used for numerical discretization in the time direction.Then we obtain two nonlinear implicit fully-discrete schemes with fourth-order accuracy in space direction and second-order accuracy in time direction.Next,the existence and uniqueness of the numerical solution and the energy conservation,convergence and stability of discrete schemes are derived.Numerical experiments simulate the dissipation law of soliton waves with different damping coefficients.It proves the effectiveness of the numerical schemes by analyzing the energy property of numerical schemes and the errors between the numerical solution and the analytical solution.