Numerical Differentiation Based On Direct And Inverse Problems Of The Heat Conduction Equation And Its Numerical Implementation

Numerical differentiation problem, numerical derivative problem, aims to compute the derivatives approximately from the function values on discrete points. Generally, there are many methods such as the finite difference method, the mollification method, the integral equation method, etc.The numerical differentiation method based on direct and inverse problems of the heat conduction equation and its numerical implementation is studied in this thesis. Its main idea is converting the numerical differentiation problem into an inversion source problem of heat conduction equation: firstly, put the derivable function into an initial condition of heat conduction equation, and get the final value by calculation the direct problem; then use the final value data and the derivable function value as an additional condition, and obtain the numerical derivative by solving the inverse source problem of heat conduction equation.In the first chapter, we introduce the studied contents and purposes of this thesis.In the second chapter, the numerical differential method based on direct problems and source inversion of the heat conduction equation is studied. The well-posedness of the direct problem is proved and the conditional stability of numerical differentiation is derived. By homogeneous principle analysis we discuss the ill-posedness of the inverse source problem. To overcome the ill-posedness, using the superposition principle of linear equation and regularization technique we convert the numerical differentiation problem to a discrete functional optimization problem. From the necessary condition of the functional minimization problem we obtain a regularized linear algebraic equation, which is solved for numerical differential.In the third chapter, we give a 8-points high precision implicit difference scheme by Taylor expansion method, and analysis the stability condition of the scheme. Numerical examples show that the proposed finite difference scheme with high accuracy and good stability characteristics. At the same time, the implicit alternate iteration of finite difference method is introduced to solve two dimensional heat conduction equation.In the fourth chapter, some numerical results of the first and second order derivatives are given for one and two dimensional functions. Numerical results show that the proposed algorithm is efficient and have strong stability respect to data noise.