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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 \(u ( x,t ) \) 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 \(C^{5+\alpha }\), \(0<\alpha <1\), the nonhomogeneous term given in the heat equation is from the Hölder space \(C_{x,t}^{3+\alpha ,\frac{3+\alpha }{2}}\), the boundary functions are from \(C^{\frac{5+\alpha }{2}}\), and between the initial and boundary conditions the conjugation conditions up to second order \((q=0,1,2)\) are satisfied. When the initial function belongs to \(C^{7+\alpha }\), the nonhomogeneous term is from \(C_{x,t}^{5+\alpha ,\frac{5+\alpha }{2}}\), the boundary functions are from \(C^{\frac{7+\alpha }{2}}\) and 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 given four point and six point difference problems converge to the exact value of \(\frac{\partial u}{\partial x}\) on the grids of order \(O ( h^{2}+\tau ) \) and \(O ( h^{2}+\tau ^{2} ) \), 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.

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