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Objective: The main purpose of this work is to present a fourth-order fitted mesh scheme for solving the semilinear singularly perturbed reaction-diffusion problem to produce more accurate solutions., Results: Quasilinearization technique is used to linearize the semilinear term. The scheme is formulated with discretizing the solution domain piecewise uniformly and then replacing the differential equation by finite difference approximations. This gives the system of difference algebraic equations and is solved by the Thomas algorithm. Convergence analysis are investigated using solution bound and the truncation error bound. Numerical illustrations are investigated to support the theoretical results and the method's applicability. The method produces a more accurate solution than some existing methods in the literature., (© 2023. The Author(s).)

The solutions of two parameters singularly perturbed boundary value problems typically exhibit two boundary layers. Because of the presence of these layers standard numerical methods fail to give accurate approximations. This paper introduces a numerical treatment of a class of two parameters singularly perturbed boundary value problems whose solution exhibits boundary layer phenomena. A graded mesh is considered to resolve the boundary layers and collocation method with cubic B-splines on the graded mesh is proposed and analyzed. The proposed method leads to a tri-diagonal linear system of equations. The stability and parameters uniform convergence of the present method are examined. To verify the theoretical estimates and efficiency of the method several known test problems in the literature are considered. Comparisons to some existing results are made to show the better efficiency of the proposed method. Summing up:•The present method is found to be stable and parameters uniform convergent and the numerical results support the theoretical findings.•Experimental results show that the present method approximates the solution very well and has a rate of convergence of order two in the maximum norm.•Experimental results show that cubic B-spline collocation method on graded mesh is more efficient than cubic B-spline collocation method on Shishkin mesh and some other existing methods in the literature., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2023 The Authors. Published by Elsevier B.V.)

Objective: The paper is focused on developing and analyzing a uniformly convergent numerical scheme for a singularly perturbed reaction-diffusion problem with a negative shift. The solution of such problem exhibits strong boundary layers at the two ends of the domain due to the influence of the perturbation parameter, and the term with negative shift causes interior layer. The rapidly changing behavior of the solution in the layers brings significant difficulties in solving the problem analytically. We have treated the problem by proposing a numerical scheme using the implicit Euler method in the temporal direction and a fitted tension spline method in the spatial direction with uniform meshes., Result: Stability and uniform error estimates are investigated for the developed numerical scheme. The theoretical finding is demonstrated by numerical examples. It is obtained that the developed numerical scheme is uniformly convergent of order one in time and order two in space., (© 2023. The Author(s).)

This paper deals with the numerical treatment of a singularly perturbed unsteady non-linear Burger-Huxley problem. Due to the simultaneous presence of a singular perturbation parameter and non-linearity in the problem applying classical numerical methods to solve this problem on a uniform mesh are unable to provide oscillation-free results unless they are applied with very fine meshes inside the region. Thus, to resolve this issue, a uniformly convergent computational scheme is proposed. The scheme is formulated:•First, the non-linear singularly perturbed problem is linearized using the Newton-Raphson-Kantorovich quasilinearization technique.•The resulting linear singularly perturbed problem is semi-discretized in time using the implicit Euler method to yield a system of singularly perturbed ordinary differential equations in space.•Finally, the system of singularly perturbed ordinary differential equations are solved using fitted exponential cubic spline method.The stability and uniform convergence of the proposed scheme are investigated. The scheme is stable and ε - uniformly convergent with first order in time and second order in space directions. To validate the applicability of the proposed scheme several test examples are considered. The obtained numerical results depict that the proposed scheme provides more accurate results than some methods available in the literature., Competing Interests: The authors declare that they have no conflicts of interest., (© 2022 The Authors.)

Objectives: An accurate exponentially fitted numerical method is developed to solve the singularly perturbed time lag problem. The solution to the problem exhibits a boundary layer as the perturbation parameter approaches zero. A priori bounds and properties of the continuous solution are discussed., Result: The backward-Euler method is applied in the time direction and the higher order finite difference method is employed for the spatial derivative approximation. An exponential fitting factor is induced on the difference scheme for stabilizing the computed solution. Using the comparison principle, the stability of the method is examined and analyzed. It is proved that the method converges uniformly with linear order of convergence. To validate the theoretical findings and analysis, two test examples are given. Comparison is made with the results available in the literature. The proposed method has better accuracy than the schemes in the literature., (© 2022. The Author(s).)

This paper deals with the computational method for a class of second-order singularly perturbed parabolic differential equations with discontinuous coefficients involving large negative shift. The formulated method comprises the implicit Euler and the cubic-spline in compression methods for time and spatial dimensions, respectively. Intensive numerical experimentation has been done on some model examples and the results are tabulated. The results depict that the present method is more accurate than some methods existing in the literature. Further, the layer behavior of the solutions is presented using graphs and observed to agree with the existing theories. Finally, error analysis of the scheme is done and observed that the proposed method is parameter uniform convergent with the order of convergence ( Δ t + h 2 ) ., Competing Interests: The authors declare no conflict of interest., (© 2022 The Author(s).)

Objectives: The main aim of this paper is to develop a linear B-spline finite element method for solving generalized diffusion equations with delay. The linear B-spline basis function is used to discretize the space variable. The time discretization process is based on Crank-Nicolson. The benefit of the scheme is that the numerical solution is obtained as a smooth piecewise continuous function which empowers one to find an approximate solution at any desired position in the domain., Result: Sufficient and necessary conditions for the numerical method to be asymptotically stable are derived. The convergence of the numerical method is studied. Some numerical experiments are performed to verify the applicability of the numerical method., (© 2022. The Author(s).)

In this paper, a second-order robust method for solving singularly perturbed Burgers' equation were presented. To find the numerical approximation, we apply the quasilinearization technique before formulation of the scheme. The obtained experimental results show that the presented method has better numerical accuracy and convergence as compared to some existing methods in the literature. Thus, the present method provides an accurate solution and efficient to solve the singularly perturbed Burgers' equation., Competing Interests: The authors declare no conflict of interest., (© 2022 The Author(s).)

In this study, a parameter-uniform numerical scheme is built and analyzed to treat a singularly perturbed parabolic differential equation involving large spatial delay. The solution of the considered problem has two strong boundary layers due to the effect of the perturbation parameter, and the large delay causes a strong interior layer. The behavior of the layers makes it difficult to solve such problem analytically. To treat the problem, we developed a numerical scheme using the weighted average ( θ -method) difference approximation on a uniform time mesh and the central difference method on a piece-wise uniform spatial mesh. We established the Stability and convergence analysis for the proposed scheme and obtained that the method is uniformly convergent of order two in the temporal direction and almost second order in the spatial direction. To validate the applicability of the proposed numerical scheme, two model examples are treated and confirmed with the theoretical findings., Competing Interests: Conflict of interestThe authors declared that this research work was conducted with out any potential conflict of interest., (© The Author(s) 2022.)

Objective: The main purpose of this paper is to present an accelerated nonstandard finite difference method for solving the singularly perturbed Burger-Huxley equation in order to produce more accurate solutions., Results: The quasilinearization technique is used to linearize the nonlinear term. A nonstandard methodology of Mickens procedure is used in the spatial direction and also within the first order temporal direction that construct the first-order finite difference approximation to solve the considered problem numerically. To accelerate the rate of convergence from first to second-order, the Richardson extrapolation technique is applied. Numerical experiments were conducted to support the theoretical results., (© 2021. The Author(s).)

Objectives: Numerical treatment of singularly perturbed parabolic delay differential equation is considered. Solution of the equation exhibits a boundary layer, which makes it difficult for numerical computation. Accurate numerical scheme is proposed using [Formula: see text]-method in time discretization and non-standard finite difference method in space discretization., Result: Stability and uniform convergence of the proposed scheme is investigated. The scheme is uniformly convergent with linear order of convergence before Richardson extrapolation and second order convergent after Richardson extrapolation. Numerical examples are considered to validate the theoretical findings., (© 2021. The Author(s).)

Singularly perturbed boundary value problems with negative shift parameter are special types of differential difference equations whose solution exhibits boundary layer behaviour. A simple but novel numerical method is developed to approximate the numerical solution of the problems of these types. The method gives accurate solutions for h ≥ ε in the inner region of the boundary layer where other classical numerical methods fail to give smooth solution. The present method is proved to be point-wise uniformly convergent with second order rate of convergence., Competing Interests: The authors declare no conflict of interest., (© 2021 The Author. Published by Elsevier Ltd.)

A parameter uniform numerical method is presented for solving singularly perturbed parabolic differential-difference equations with small shift arguments in the reaction terms arising in computational neuroscience. To approximate the terms with the shift arguments, Taylor's series expansion is used. The resulting singularly perturbed parabolic differential equation is solved by applying the implicit Euler method in temporal direction and extended cubic B-spline basis functions consisting of a free parameter λ for the resulting system of ordinary differential equations in the spatial direction. The proposed method is shown to be accurate of order O Δ t + h 2 ε + h by preserving an ε- uniform convergence. To demonstrate the applicability of the proposed method, two test examples are solved by the method and the numerical results are compared with some existing results. The obtained numerical results agreed with the theoretical results., (© 2020 John Wiley & Sons, Ltd.)

We consider a SEAIR epidemic model with Atangana-Baleanu fractional-order derivative. We approximate the solution of the model using the numerical scheme developed by Toufic and Atangana. The numerical simulation corresponding to several fractional orders shows that, as the fractional order reduces from 1, the spread of the endemic grows slower. Optimal control analysis and simulation show that the control strategy designed is operative in reducing the number of cases in different compartments. Moreover, simulating the optimal profile revealed that reducing the fractional-order from 1 leads to the need for quick starting of the application of the designed control strategy at the maximum possible level and maintaining it for the majority of the period of the pandemic., Competing Interests: Competing interestsThe authors declare that they have no competing interests., (© The Author(s) 2021.)

Analysis of mathematical models designed for COVID-19 results in several important outputs that may help stakeholders to answer disease control policy questions. A mathematical model for COVID-19 is developed and equilibrium points are shown to be locally and globally stable. Sensitivity analysis of the basic reproductive number (R 0 ) showed that the rate of transmission from asymptomatically infected cases to susceptible cases is the most sensitive parameter. Numerical simulation indicated that a 10% reduction of R 0 by reducing the most sensitive parameter results in a 24% reduction of the size of exposed cases. Optimal control analysis revealed that the optimal practice of combining all three (public health education, personal protective measure, and treating COVID-19 patients) intervention strategies or combination of any two of them leads to the required mitigation of transmission of the pandemic., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2020 The Author(s).)