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This paper considers the difference operator Aht,x approximation of the following differential operator At,xu(t,x) = -a11(t,x)utt(t,x)-a22(t,x)uxx(t,x)+σu(t,x), defined over the region R+×R with the boundary condition u(0,x) = 0,x∈R. Here, the coefficients aii(t,x),i = 1,2 are continuously differentiable and satisfy the uniform ellipticity a112(t,x)+a222(t,x)≥δ>0, and σ > 0. It investigates the structure of the fractional spaces generated by this difference operator. Moreover, the positivity of Aht,x operator in difference analogy Ho¨lder spaces is proved. [ABSTRACT FROM AUTHOR]

A finite difference method for the approximate solution of the reverse multidimensional parabolic differential equation with a multipoint boundary condition and Neumann condition is applied. Stability, almost coercive stability, and coercive stability estimates for the solution of the first and second orders of accuracy difference schemes are obtained. The theoretical statements are supported by the numerical example. [ABSTRACT FROM AUTHOR]