Grid Approximation of Derivatives of the Solution of Heat Conduction Equation

In this study we propose special difference problems of the four point scheme and six point symmetric implicit scheme (Crank and Nicolson) for the approximation of first and second partial derivatives of the solution ( , ) u x t of the first type boundary value problem for one-dimensional heat conduction equation, with constant coefficients. A four point implicit difference problem is proposed for the approximation of u x   under the assumption that the initial function belongs to the Hölder space 5 , C  0 1,  the nonhomogeneous function given in the heat equation is from the Hölder space 3 3 , 2 , , x t C     the boundary functions are from 5 2 C  also between the initial and boundary functions the conjugation conditions up to second order (q  0,1,2) are satisfied. When the initial function belongs to 7 , C  the nonhomogeneous term is from 5 5 , 2 , , x t C     the boundary functions are from 7 2 , C  also the conjugation conditions up to third order (q  0,1,2,3) are satisfied, a six point implicit difference problem is given. It is proven that the solution of the constructed four and six point implicit difference problems converge to the exact value of u x   on the grids of order 2 O(h  ) and 2 2 O(h  ) respectively, where, h is the step size in spatial variable x and  is the step size in time variable t . Furthermore, boundary value problems and implicit difference problems are given to the first derivative of the solution with respect to time variable t , u t         and for the pure second derivative with respect to the spatial variable x . Also special implicit difference boundary value problem is proposed for the mixed second derivative of the solution, 2u xt    . When the initial function belongs to 8 , C  the heat source function given in the heat equation is from 6 6, 2 , , xtC     the boundary functions are from 8 C 2  Hölder spaces and between the initial and boundary function the conjugation conditions of orders 0,1, 2, 3, 4 q  are satisfied, it is proven that the solution of the proposed implicit difference schemes converge uniformly to the corresponding exact derivatives u t   , 2 2 u x   and 2u xt   on the grids of the order 2()Oh   . On the other hand, when the initial function belongs to 10 , C  the heat source function is from 8 8, 2 , , xtC     the boundary functions are from 10 2 C  Hölder spaces and between the initial and boundary functions the conjugation conditions of orders q  0,1,2,3,4,5 are satisfied, the constructed six-point symmetric (Crank-Nicolson) implicit difference boundary value problems converge with the order 2 2 O(h  ) to the corresponding exact derivatives u t   , 2 2 u x   and 2u xt   . Finally, in order to justify the theoretical results, several numerical examples are constructed and the obtained results are presented through tables and figures. Keywords: Finite difference method, Approximation of derivatives, Crank-Nicolson scheme, Uniform error, Heat equation.