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This study is devoted to designing two hybrid computational algorithms to find approximate solutions for a class of singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters. In our approaches, the time discretization is first performed by the well-known Rothe method and Taylor series procedures, which reduce the underlying model problem into a sequence of boundary value problems (BVPs). Hence, a matrix collocation technique based on novel shifted Delannoy functions (SDFs) is employed to solve each BVP at each time step. We show that our proposed hybrid approximate techniques are uniformly convergent in order O(Δτs+M−12){\mathcal{O}}\left(\Delta {\tau }^{s}+{M}^{-\tfrac{1}{2}}) for s=1,2s=1,2, where Δτ\Delta \tau is the time step and MM is the number of SDFs used in the approximation. Numerical simulations are performed to clarify the good alignment between numerical and theoretical findings. The computational results are more accurate as compared with those of existing numerical values in the literature.

The aim of the present paper is to give a Chlodowsky type generalization of Szász operators defined by means of the Brenke type polynomials. We obtain convergence properties of our operators with the help of universal Korovkin-type property and also establish the order of convergence by using a classical approach, the second order modulus of continuity and Peetre’s K -functional. We also give a Voronoskaja type theorem. Furthermore, we study the convergence of these operators in a weighted space of functions on a positive semi-axis. Some graphical examples for the convergence of our operators and error estimation are also given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new analogue of Bernstein operators and we call it as ( p, q )-Bernstein operators which is a generalization of q -Bernstein operators. We also study approximation properties based on Korovkin’s type approximation theorem of ( p, q )-Bernstein operators and establish some direct theorems. Furthermore, we show comparisons and some illustrative graphics for the convergence of operators to a function. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new analogue of Bernstein–Stancu operators based on ( p, q )-integers which we call as ( p, q )-Bernstein–Stancu operators. We study approximation properties for these operators based on Korovkin’s type approximation theorem and also study some direct theorems. Furthermore, we give comparisons and some illustrative graphics for the convergence of operators to some function. [ABSTRACT FROM AUTHOR]