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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.

This article focuses on an efficient and highly accurate approximate solver for a class of generalized singular boundary value problems (SBVPs) having nonlinearity and with two‐term fractional derivatives. The involved fractional derivative operators are given in the form of Liouville–Caputo. The developed algorithm for solving the generalized SBVPs consists of two main stages. The first stage is devoted to an iterative quasilinearization method (QLM) to conquer the (strong) nonlinearity of the governing SBVPs. Secondly, we employ the generalized Genocchi polynomials (GGPs) to treat the resulting sequence of linearized SBVPs numerically. An upper error estimate for the Genocchi series solution in the L2$$ {L}^2 $$ norm is obtained via a rigorous error analysis. The main benefit of the presented QLM‐GGPs procedure is that the required number of iteration in the first stage is within a few steps, and an accurate polynomial solution is obtained through computer implementations in the second stage. Three widely applicable test cases are investigated to observe the efficacy as well as the high‐order accuracy of the QLM‐GGPs algorithm. The comparable accuracy and robustness of the presented algorithm are validated by doing comparisons with the results of some well‐established available computational methods. It is apparently shown that the QLM‐GGPs algorithm provides a promising tool to solve strongly nonlinear SBVPs with two‐term fractional derivatives accurately and efficiently. [ABSTRACT FROM AUTHOR]

With the weight of α$$ \alpha $$‐Baskakov basis function, this article considers the approximation properties of the integral modification of exponential type operators. The moments are estimated, and the Voronovskaja‐type result is established in simultaneous approximation. Additionally, the power series method is used for estimating convergence and error estimation. For some functions, the convergence of these operators has also been graphically illustrated. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate Korteweg–De Vries equations (KDVEs) with third order under fractional Atangana–Baleanu–Caputo (ABC) derivative. We use an analytical method due to the Laplace transform and the Adomian decomposition technique. We establish the convergence analysis results for the considered problem by using nonlinear functional analysis. First of all, a general algorithm is developed for the computation of the required analytical approximate solution by using the mentioned method. By applying the developed algorithm, we then investigate the approximate solutions for some examples. The computed solution has also been presented graphically using Matlab. Laplace Adomian decomposition method (LADM) is a powerful full technique that is easy to implement. In addition, the said method has also a fast convergent quality. [ABSTRACT FROM AUTHOR]

This work is devoted to establish a numerical scheme for system of fractional-order differential equations (FODEs) with order 1 < ≤ 2. The scheme is established by using Bernstein polynomials (BPs). Based on the said materials, some operational matrices are formed. With the help of obtained operational matrices, the considered system is reduced to some algebraic system of equations. On using MATLAB-16, the system is then solved to get the required numerical solution for the proposed system. Several examples are treated with the help of the proposed method for numerical solutions. Further, error analysis is also recorded for different fractional orders and various scale levels. The mentioned results are displayed graphically. Comparison with exact solution at traditional order derivative is also given. It should be kept in mind that the proposed method does not require any kind of discretization or collocation. Also, there is no external parameter which controls the method. Due to these features, the proposed method is powerful and efficient for different classes of FODEs to compute their numerical solutions. The efficiency of the proposed method can be enhanced by increasing the scale level. [ABSTRACT FROM AUTHOR]