Implicit methods for the first derivative of the solution to heat equation

Abstract We propose special difference problems of the four point scheme and the six point symmetric implicit scheme (Crank and Nicolson) for the first partial derivative of the solution u(x,t) $u ( x,t )$ of the first type boundary value problem for a one dimensional heat equation with respect to the spatial variable x. A four point implicit difference problem is proposed under the assumption that the initial function belongs to the Hölder space C5+α $C^{5+\alpha }$, 0<α<1 $0<\alpha <1$, the nonhomogeneous term given in the heat equation is from the Hölder space Cx,t3+α,3+α2 $C_{x,t}^{3+\alpha ,\frac{3+\alpha }{2}}$, the boundary functions are from C5+α2 $C^{\frac{5+\alpha }{2}}$, and between the initial and boundary conditions the conjugation conditions up to second order (q=0,1,2) $(q=0,1,2)$ are satisfied. When the initial function belongs to C7+α $C^{7+\alpha }$, the nonhomogeneous term is from Cx,t5+α,5+α2 $C_{x,t}^{5+\alpha ,\frac{5+\alpha }{2}}$, the boundary functions are from C7+α2 $C^{\frac{7+\alpha }{2}}$ and the conjugation conditions up to third order ( q=0,1,2,3 $q=0,1,2,3$) are satisfied, a six point implicit difference problem is given. It is proven that the solution of the given four point and six point difference problems converge to the exact value of ∂u∂x $\frac{\partial u}{\partial x}$ on the grids of order O(h2+τ) $O ( h^{2}+\tau )$ and O(h2+τ2) $O ( h^{2}+\tau ^{2} )$, respectively, where h is the step size in spatial variable x and τ is the step size in time. Theoretical results are justified by numerical examples