In physics and mathematics,the heat equation is a partial differential equation that describes how the distribution of some quantity(such as heat)evolves over time in a solid medium.In many practical problems,the solution of the problem is often reduced to solve nonlinear partial differential equations.The Allen-Cahn and Cahn-Hilliard equations studied in this paper are mainly used to describe the inverse boundary motion of the junction solid,the complex phase separation and coarsening phenomena in crystalline solids,respectively.These two types of equations are widely used in materials science and fluid mechanics.The Fisher equation is an important class of nonlinear heat conduction equation,which has important applications in heat conduction,combustion theory,science,ecology and other fields.The gas dynamics equation is a mathematical expression based on physical laws such as conservation of mass,conservation of momentum,conservation of energy,etc.,and plays an important role in many industries.Traditional numerical methods are used to solve the above the three types of equations,which has certain advantages but the accuracy needs to be improved,so it is particularly necessary to find a high-precision numerical method.In view of the advantages of the barycentric lagrange interpolation method,which does not require meshing,simple procedures and fast operation speed,this paper uses the method to solve three heat conduction equations with initial and boundary conditions,and further does convergence analysis.The variables in the space domain and the time domain are all discretized by using Chebyshev nodes.The unknown function and its partial derivative are discretized by interpolation function and differential matrix.The initial and boundary conditions are applied by the substitution method,and then solved.The specific calculation examples are as follows:(1)The Allen-Cahn and Cahn-Hilliard equations are solved and the numerical solution images and energy curves are plotted.Meanwhile the property that the total free energys of the equations decay with the increase of time is verified.(2)The Fisher equation is solved and the numerical solution graphs and errorgraphs are plotted,and the error values obtained by this method are compared with the error values calculated by other literatures.Meanwhile the changes of three kinds of error values are also compared under different interpolation nodes.The comparison of the error values in both cases proves the validity and practicability of the barycentric lagrange interpolation collocation method.(3)The gas dynamics equation is solved and the corresponding numerical solution images,analytical solution images and error images are plotted.Meanwhile the changes of the three types of error values are also compared under different interpolation nodes.High agreement between numerical and analytical solution images and the significant variation of three error values can show the high accuracy and effectiveness of the barycentric lagrange interpolation collocation method. |