The Difference Schemes For The Fourth Order Parabolic Equation With Different Boundary Conditions

In this paper,fourth order parabolic equations with different boundary conditions are studied.According to different boundary conditions,they are divided into six categories.This thesis focuses on the fourth order parabolic equations with the third Dirichlet boundary conditions,the first Neumann boundary conditions and the third Neumann boundary conditions.The priori estimation for continuous problems,the discrete finite difference scheme,and the uniqueness and convergence analysis of the difference scheme will be given,respectively.This paper is divided into three parts.In the first part,the fourth order parabolic equations with the third Dirichlet boundary conditions are discussed.By using the method of energy analysis,the priori estimate in the semi norm of H1 is obtained.Further,the priori estimate in the maximum norm has been obtained.Using the value of boundary,we construct a quartic interpolation polynomial.With the help of this polynomial,the approximation of uxxxx(x1,tk)is obtained.The approximation of uxxxx(xm-1,tk)can also be obtained by the similar way.As the coefficients in the discrete form of uxxxx(x1,tk)is2/3,2/3ut(x1,tk)+1/3ut(x0,tk)is used to approximate ut(x1,tk)to ensure the consistency of discrete coefficients.By using the method of energy analysis,at the points of x1 and xm-1 the spatial step size h is put together with the truncation error term,then the rest part is amplified by the infinite module.It can be obtained that the convergence order is O(h2+T2)in H1 semi-norm and maximum norm.In the second part,the fourth order parabolic equations with the first Neumann boundary conditions are discussed.By using the method of energy analysis,the priori estimate in H1 semi-norm is obtained.Using the value of boundary,we construct a quartic interpolation polynomial.With the help of this polynomial,an approximation of uxxxx(x0,tk)can be obtained.The approximation of uxxxx(x1,tk),uxxxx(xm-1,tk),uxxxx(xm,tk)can also be obtained by the similar way.With the help of energy analysis,the difference scheme is proved to be convergent in H1 semi-norm and the convergence order is O(h2+T2).In the third part,the fourth-order parabolic equations with the third Neumann boundary conditions are discussed.Introducing a new function v(x,t)=uxx(x,t),the original equation is reduced to a system of two lower order differential equations.Then the method of energy analysis is used to obtain a priori estimate in the infinite norm.The approximation of vxx(x0,tk)and vxxx(xm,tk)can be established and the truncation error is O(h2).By using the method of energy analysis,we make fourth inner product with both sides of original equation,the convergence order O(h2+T2)of the difference schemes can be obtained in the L2 norm.