In this paper, we address a class of boundary layer originated singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions having large time delay; for the higher order numerical analysis. The boundary layer originated nonuniform mesh, is generated by adaptive moving mesh approach based on equidistribution principle. The mesh is obtained by solving a nonlinear system; involving the discrete problem and the discrete nonlinear mesh generating functions. We provide the theoretical mesh structure. The difference scheme for Robin type boundary conditions having normalized flux; uses a combination of space and time discretizations so that the order of convergence in space is higher than the usual linear accurate upwind discretization in space. The present theoretical convergence analysis requires less regularity of the solution and proves to be exactly second order parameter uniform accurate in space. Numerical experiments highly validate the theoretical findings and also show that the present approach is better than the well-known upwind discretizations. 

Parameter uniform higher order numerical treatment for singularly perturbed Robin type parabolic reaction diffusion multiple scale problems with large delay in time

This article deals with designing and analyzing a higher order stable numerical analysis for the time‐fractional Kuramoto–Sivashinsky (K‐S) equation, which is a fourth‐order non‐linear equation. The fractional derivative of order 
 present in the considered problem is taken into Caputo sense and approximated using the 
 scheme. In space direction, the discretization process uses quintic 
‐spline functions to approximate the derivatives and the solution of the problem. The present approach is unconditionally stable and is convergent with rate of accuracy 
, where 
 and 
 denote the space and time step sizes, respectively. We have also noted that the linearized version of the K‐S equation leads the rate of accuracy to 
. The present approach is also highly effective for the time‐fractional Burgers' equation. We have shown that the present approach provides better accuracy than the 
 scheme with the same computational cost for several linear/non‐linear problems, with classical as well as fractional time derivatives.

A higher order stable numerical approximation for time‐fractional non‐linear Kuramoto–Sivashinsky equation based on quintic B$$ \mathfrak{B} $$‐spline

In this paper, a robust adaptive grid method is developed for solving first-order nonlinear singularly perturbed Fredholm integro-differential equations (SPFIDEs). Firstly such SPFIDEs are discretized by the backward Euler formula for differential part and the composite numerical quadrature rule for integral part. Then both a prior and an a posterior error analysis in the maximum norm are derived. Based on the prior error bound and the mesh equidistribution principle, it is proved that there exists a mesh gives optimal first-order convergence which is robust with respect to the perturbation parameter. Finally, the posterior error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Numerical results are given to illustrate our theoretical result.

A robust adaptive grid method for first-order nonlinear singularly perturbed Fredholm integro-differential equations

This paper presents a novel spectral algorithm for the numerical solution of multi-dimensional fractional-order telegraph equations, a critical model used to capture the combined effects of diffusion and wave propagation. The core innovation of this work is the application of Jacobi-Romanovski polynomials as the basis functions for spectral discretization. These polynomials offer unique advantages, including the ability to handle nonstandard domains and boundary conditions, making them particularly suitable for partial differential equation (PDE) applications. A comprehensive error analysis is conducted, providing deep insights into the convergence rates and factors affecting the accuracy of the numerical solutions. Extensive numerical experiments further demonstrate the superior performance of the proposed spectral algorithm in solving a wide range of multi-dimensional fractional-order telegraph equation models. The results show a significant improvement in accuracy and computational efficiency compared to traditional numerical methods, such as finite difference or finite element techniques. This research advances the field of computational science by offering a robust, efficient, and versatile numerical framework for the precise solution of complex multi-dimensional PDEs.

pdf/robust-and-accurate-numerical-framework-for-multi-d6xi8shelj14.pdf

Robust and accurate numerical framework for multi-dimensional fractional-order telegraph equations using Jacobi/Jacobi-Romanovski spectral technique

Parameter uniform numerical method for a system of singularly perturbed parabolic convection-diffusion equations

Wavelet-based approximation with nonstandard finite difference scheme for singularly perturbed partial integrodifferential equations

Uniformly convergent scheme for fourth-order singularly perturbed convection-diffusion ODE

A robust numerical technique for weakly coupled system of parabolic singularly perturbed reaction–diffusion equations

The generalized time-fractional Fisher's equation is a substantial model for illustrating the system's dynamics. Studying effective numerical methods for this equation has considerable scientific importance and application value. In that direction, this paper presents designing and analyzing a high-order numerical scheme for the generalized time-fractional Fisher's equation. The time-fractional derivative is taken in the Caputo sense and approximated using Euler backward discretization. The quasilinearization technique is used to linearize the problem, and then a compact finite difference scheme is considered for discretizing the equation in space direction. Our numerical method is convergent of O k 2 − α + h 4 $$ O\left({k}^{2-\alpha }+{h}^4\right) $$ , where h $$ h $$ and k $$ k $$ are step sizes in spatial and temporal directions, respectively. Three problems are tested numerically by implementing the proposed technique, and the acquired results reveal that the proposed method is suitable for solving this problem. 

A high‐order numerical technique for generalized time‐fractional Fisher's equation

A robust higher-order numerical technique with graded and harmonic meshes for the time-fractional diffusion–advection–reaction equation

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