Application Of Variable Limit Integral Method In Solving Several Kinds Of Differential Equations

Partial differential equations can describe many natural phenomena and are widely used in electromagnetics,thermodynamics,fluid mechanics and other fields.However,it is difficult to solve the analytical solutions to partial differential equations,which shows the importance of numerical methods of practical applications.The main work of this paper is to provide a new numerical solution method for a class of two-dimensional steady-state constant coefficient partial differential equations,a class of unsteady variable coefficient partial differential equations,and a class of nonlinear unsteady partial differential equations(the Kdv equation).Firstly,the one-dimensional and two-dimensional Helmholtz equations are solved based on the variable integration method,and the discrete scheme with the tridiagonal matrix is constructed.For the one-dimensional Helmholtz equation,the influence of the variable limit factor on the error is studied.Through the error estimation,it is proved that the truncation error of the discrete scheme reaches the second order.Numerical examples show that the error of the discrete scheme is low when the variable limit factor and the step length are equal.At the same time,the application of different wave numbers in the twodimensional Helmholtz equation is explored.Through numerical examples,it is found that the numerical schemes have good accuracy under different wave numbers.Secondly,the variable limit integration method is applied to the numerical solution of the one-dimensional unsteady convection-diffusion equation with variable coefficients.Combined with the binary six-point Lagrange interpolation,an implicit discrete scheme with mixed accuracy is constructed.Fourier analysis proves that the method is unconditionally stable.After prior estimation,the discrete scheme converges in norm,truncation error of the corresponding discrete scheme is given by error estimation.At the same time,numerical examples show that the accuracy is improved compared with numerical methods such as GEL,(S)AGE and(D)AGE.Finally,based on the variable integration method and three-point Lagrange interpolation,the fully discrete scheme of the two-layer fluid Kd V equation is constructed,and the feasibility of the method is verified by numerical examples.