Research On Inverse Problems Of The Heat Equation With Time-dependent Coefficient And The Integral Equation And Application Of The Variational Iteration Method

Starting from the backward heat conduction problem,this paper explores the reg-ularization method and numerical solution of ill-posed problems.The emphasis is on the regularization method of the inverse time problem of the nonhomogeneous heat conduction equation with time-dependent coefficient in two-dimensional domain,the regularization method of the first kind integral equation,and the numerical experiment method of the regularization solution.For the backward heat conduction problem with time-dependent coefficient in gen-eral domain,we use the method of logarithmic convexity to obtain the conditional sta-bility of the solution.For the backward heat conduction problem with time-dependent coefficient in two-dimensional circular domain,the formal representation of the solution is obtained by transformation,and two regularization methods are given respectively,and each method gives the stability and convergence of the modified solution,and the corresponding error estimates are obtained.Due to the serious ill posed nature of the inverse time problem,the stability effect of the regular solution depends not only on the error level,but also on the rounding error and truncation error,which makes the numerical calculation difficult.In order to find the calculation method of the inverse problem,this paper studies the variational iteration method,gives the convergence conclusion,and uses it to solve the first class boundary value problem and the second class boundary value problem with zero order term and only time-dependent unknown coefficient heat conduction equation,and obtains good convergence results in a small range.For the large-scale convergence problem,we introduce the piecewise variational iteration method,and compare it with the variational iteration method.By using the piecewise variational iteration method,we can solve the heat conduction parameter identification problem with nonlinear source term,even in a large scale,we can get better convergence results.In addition,this paper attempts to use the variational iteration method and the piecewise variational iteration method to solve the nonlinear backward heat conduction problem,and compares their approximation effects with numerical examples.At the end of this paper,we construct the regularization equation of the backward heat conduction problem,and use the variational iteration method to solve the regular-ization equation.The convergence of the variational iteration method ensures the fea-sibility of the method,the numerical examples verify the effectiveness of the method,and the calculation efficiency reflects the superiority of the method.We transform the first kind of Fredholm and Volterra integral equations into the corresponding second kind of integral equations,and obtain the stability of the modified solution.For the first kind of Fredholm integral equation,by using the variational iteration method and the selection of the optimal Lagrangian multiplier,we establish the iterative scheme of the modified equation,and under the allowable range of regularization parameters,we test the equation with disturbance observation data,and obtain satisfactory approxima-tion effect.For the first kind of Volterra integral equation,we establish the iteration sequence of the modified solution,and use numerical examples to test the effectiveness of the method,and compare the approximation effect of the modified solution under different Lagrangian multipliers.In this paper,we study the variational iteration method for the problem that the numerical test of the regularization method of backward problem is more difficult,and apply it to the numerical test of the quasi-reversibility method,and get better test results.At the same time,we also use it to test the regularization problem of the first kind of integral equation,and get satisfactory results,which proves that the variational iteration method is effective in studying some regularization problems.