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This study presents an (ε, μ)−uniform numerical method for a two-parameter singularly perturbed time-delayed parabolic problems. The proposed approach is based on a fitted operator finite difference method. The Crank–Nicolson method is used on a uniform mesh to discretize the time variables initially. Subsequently, the resulting semi-discrete scheme is further discretized in space using Simpson's 1/3rd rule. Finally, the finite difference approximation of the first derivatives is applied. The method is unique in that it is not dependent on delay terms, asymptotic expansions, or fitted meshes. The fitting factor's value, which is used to account for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second-order accurate and uniformly convergent. The proposed method's applicability is validated by three model examples, which yielded more accurate results than some other methods found in the literature.

ABSTRACTIn this study, to solve the singularly perturbed delay convection–diffusion–reaction problem, we proposed a hybrid numerical scheme that converges uniformly. Parabolic right boundary layer outcomes from the presence of the small perturbation parameter. To grip this layer behaviour, the problem is solved by Bakhvalov–Shishkin mesh for spatial domain discretization and uniform mesh for temporal domain discretization. A hybrid scheme consisting of a non-polynomial spline scheme for fine mesh and a midpoint upwind scheme for coarse mesh is used to discretize the spatial derivative, while an implicit Euler scheme is used to discretize the time derivative. To make computed solutions more accurate and increase rate of convergence of the scheme, we applied Richardson extrapolation technique. The stability and convergence of the scheme are established. The scheme has a second order of convergence in the discrete supreme norm and is parametric uniformly convergent. The scheme's application is demonstrated through two test problems.

In this paper, a numerical scheme for a class of singularly perturbed delay parabolic convection–diffusion problems having Dirichlet boundary conditions is proposed. When the perturbation parameter tends to zero, the solution to these problems exhibits a boundary layer at the right side of the domain. The classical methods fail to provide good accuracy for these problems. To overcome this drawback, we proposed the Crank–Nicolson finite difference scheme to discretize in temporal direction on a uniform mesh and a nonstandard midpoint upwind scheme in spatial direction on a piecewise uniform Shishkin mesh. The resulting scheme has been shown to be uniformly stable and convergent with respect to perturbation parameter and mesh size. The scheme converges uniformly with the order of convergence OΔt2+N−1lnN2, where Δt is mesh size in temporal discretization and N is number of mesh elements in spatial discretization. To validate theoretical results, numerical experiments have been carried out and presented in the form of tables and graphs.

The numerical solution of time fractional parabolic differential equations with singular perturbations and delay is the subject of this article. An arbitrarily small perturbation parameter ɛ(0

An efficient numerical technique for a singularly perturbed parabolic reaction–diffusion problem with Robin type boundary conditions is presented in this work. The governing problem is discretized using the implicit Euler technique in time direction and a hybrid numerical technique that comprises a central finite difference method in the outer region and a cubic spline in compression method in the boundary layer regions in space direction. We use Shishkin-type meshes in the space domain to resolve the layers. The Robin type boundary conditions are handled using the second-order method. The totally discretized problem is well examined for stability and convergence. The use Bakhvalov–Shishkin and Vulanović–Shishkin meshes yields a more efficient and second-order convergence whereas Shishkin mesh produces almost second-order convergent solutions. Two test examples are computed. The current method has been compared to other methods in the scientific community.

The existence of boundary layers in the solutions of two-parameter singularly perturbed boundary value problems makes classical numerical methods insufficient in providing accurate approximations. Consequently, the development of layer-adapted mesh methods that achieve parameter uniform convergence and are specifically designed to accurately handle these layers becomes highly significant. The aim of this paper is to construct and examine a numerical approach for obtaining approximate solutions to a specific class of two-parameter singularly perturbed second order boundary value problems whose solutions exhibit boundary layers at both ends of the domain. The problem is discretized by employing a modified backward finite difference method on a mesh that is graded. To validate the theoretical findings, well known test problems from the existing literature are utilized. Moreover, the efficiency of the proposed method is demonstrated by comparing it to other existing methods in the literature. The stability and uniform convergence with respect to the parameters of the proposed method have been verified, revealing that it attains second-order convergence in the maximum norm. The numerical outcomes illustrate that the proposed method offers remarkably accurate approximations of the solution. Theoretical findings are in agreement with the experimental results.

Abstract 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.

In this work, nonstandard finite difference method is presented for the numerical solution of time-fractional singularly perturbed convection–diffusion problems with a delay in time. The time-fractional derivative is considered in the Caputo sense and discretized using Crank–Nicholson technique. Then, a nonstandard finite difference scheme is constructed on a uniform mesh discretization along the spatial direction. The parameter-uniform convergence of the proposed method is proved rigorously and shown to be ɛ-uniform convergent with order of convergence O((Δt)2) along the temporal domain and M−1 along the spatial domain. Finally, the proposed scheme is validated using model examples and the computational results are in agreement with the theoretical expectation.

In this article, a uniformly convergent numerical scheme is developed to solve a singularly perturbed convection-diffusion equation with a small delay having a boundary layer along the left side. A priori bounds of continuous solution and its derivatives are discussed. To solve the problem, the Crank–Nicolson scheme in the time direction and the exponentially fitted finite difference scheme in the space direction are used. The stability of the method is analyzed. It is proved that the developed scheme converges uniformly with first order in space and second order in time. To validate the applicability of the theoretical finding of the developed scheme, numerical experiments are carried out by considering two test examples.

A fitted tension spline numerical scheme for a singularly perturbed parabolic problem (SPPP) with time delay is proposed. The presence of a small parameter ε as a multiple of the diffusion term leads to the suddenly changing behaviors of the solution in the boundary layer region. This results in a challenging duty to solve the problem analytically. Classical numerical methods cause spurious nonphysical oscillations unless an unacceptable number of mesh points is considered, which requires a large computational cost. To overcome this drawback, a numerical method comprising the backward Euler scheme in the time direction and the fitted spline scheme in the space direction on uniform meshes is proposed. To establish the stability and uniform convergence of the proposed method, an extensive amount of analysis is carried out. Three numerical examples are considered to validate the efficiency and applicability of the proposed scheme. It is proved that the proposed scheme is uniformly convergent of order one in both space and time. Further, the boundary layer behaviors of the solutions are given graphically.

This study presents families of the fourth-order Runge–Kutta methods for solving a quadratic Riccati differential equation. From these families, the England version is more efficient than other fourth-order Runge–Kutta methods and practically well-suited for solving initial value problems in general and quadratic Riccati differential equation in particular. The stability analysis of the present method is well-established. In order to verify the accuracy, we compared the numerical solutions obtained using the England version of fourth-order Runge–Kutta method with the recently published works reported in the literature. Several counter examples are solved using the present methods to demonstrate their reliability and efficiency.

Abstract 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.

This work presents a numerical solution to singularly perturbed Robin-type parabolic convection–diffusion problems. A hybrid method that combines the central difference scheme in the inner region and the midpoint of the upwind scheme in the outer region on a Shishkin mesh is utilized in the space direction, and an implicit Euler technique is used in the time direction. The analysis of convergence demonstrates that the present method converges with O(N−2ln2N+Δt). The theoretical findings are supported by two numerical examples. The upwind method proposed in the literature is improved by the present method.

This paper presents a uniformly convergent numerical scheme for singularly perturbed fractional order convection–diffusion equations with variable coefficients. First, the time-fractional derivative is considered in the Caputo sense and treated using the implicit Euler method. Then, a uniformly convergent numerical scheme based on the nonstandard finite difference method is developed. The technique is proved rigorously for parameter-uniform convergence. By numerical experimentation, it is also validated that the computational results agree with theoretical results.

In this study, we consider singularly perturbed large negative shift parabolic reaction–diffusion with integral boundary condition. The continuous solution's properties are discussed. On a non-uniform Shishkin mesh, the spatial derivative is discretized using the tension spline method, and the temporal derivative is discretized using the Crank–Nicolson method. In order to handle the integral boundary condition, Simpson's rule is used. After conducting an error analysis, it was determined that the method was uniformly convergent. To support the theoretical findings, numerical examples are taken into account and solved for different values of the perturbation parameter and mesh sizes. It is proved to be uniformly convergent to almost second order.

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.

In this paper, fitted operator finite difference methods are presented for two-parameter singularly perturbed one-dimensional parabolic partial differential equations with a delay in the time variable. Such boundary value problems are frequently encountered in the spatial diffusion of reactants and in control systems. To approximate the solution of the problem, we consider the Crank–Nicolson method for time discretization and the non-standard scheme for space discretization on uniform meshes. We prove that the proposed numerical method is uniformly convergent, having convergence of order two in time and one in space. Further, we present numerical results for two test problems in support of our theoretical findings and to demonstrate that the non-standard scheme is more effective than the hybrid scheme comprising central difference, upwind, and mid-point.

In this study, a robust higher-order numerical method for solving singularly perturbed parabolic reaction-diffusion problems is presented. The Crank-Nicolson method is applied to discretize the time derivative on a uniform mesh. On a Shishkin mesh, the space derivative is discretized using a hybrid numerical method that combines the cubic spline in tension method for the boundary layer regions with the central difference method for the outer region. Theoretically, we proved that the proposed hybrid numerical method is second-order in the outer region and fourth-order in the boundary layer regions in the space direction. As a result of this, the proposed method produces an almost second-order rate of convergence in the time domain and a higher-order rate of convergence in the space domain. The newly developed method is numerically demonstrated to be uniformly convergent at higher-order, independent of the perturbation parameter. Three numerical examples are computed to support the theoretical results.

This paper presents a parameter-uniform numerical method to solve the time dependent singularly perturbed delay parabolic convection-diffusion problems. The solution to these problems displays a parabolic boundary layer if the perturbation parameter approaches zero. The retarded argument of the delay term made to coincide with a mesh point and the resulting singularly perturbed delay parabolic convection-diffusion problem is approximated using the implicit Euler method in temporal direction and extended cubic B-spline collocation in spatial orientation by introducing artificial viscosity both on uniform mesh. The proposed method is shown to be parameter uniform convergent, unconditionally stable, and linear order of accuracy. Furthermore, the obtained numerical results agreed with the theoretical results.

A uniformly convergent numerical scheme is proposed to solve a singularly perturbed convection-diffusion problem with a large time delay. The diffusion term of the problem is multiplied by a perturbation parameter, ε. For a small ε, the problem exhibits a boundary layer, which makes it challenging to solve it analytically or using standard numerical methods. As a result, the backward Euler scheme is applied in the temporal direction. Non-symmetric finite difference schemes are applied for approximating the first-order derivative terms, and a higher-order finite difference method is applied for approximating the second-order derivative term. Furthermore, an exponential fitting factor is computed and induced in the difference scheme to handle the effect of the small parameter. Using the discrete maximum principle, the stability of the scheme is examined and analyzed. The developed scheme is parameter-uniform with a linear order of convergence in both space and time. To examine the accuracy of the method, two model examples are considered. Further, the boundary layer behavior of the solutions is given graphically.

Abstract 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 ( $$\theta$$ 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.

Abstract 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.

The singularly perturbed parabolic convection–diffusion equations with integral boundary conditions and a large negative shift are studied in this paper. The implicit Euler method for the temporal direction and the exponentially fitted finite difference scheme for the spatial direction are applied to formulate a parameter-uniform numerical method. The Simpson’s integration rule is used to handle the integral boundary condition. The Richardson extrapolation technique is applied to enhance the order of convergence of the method. The stability and uniform convergence analysis of the proposed method are studied. It is shown that the method is uniformly convergent with a convergence order of two in both temporal and spatial direction after Richardson extrapolation. Two test examples are considered to verify the validity of the proposed numerical scheme. The obtained numerical results confirm the theoretical estimates. The proposed method provide more accurate results and a higher order of convergence than methods available in the literature.

Abstract 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.

In this article, we proposed and analyzed a numerical scheme for singularly perturbed differential equations with both spatial and temporal delays. The presence of the perturbation parameter exhibits strong boundary layers, and the large negative shift gives rise to a strong interior layer in the solution. The abruptly changing behaviors of the solution in the layers make it difficult to solve the problem analytically. Standard numerical methods do not give satisfactory results, unless a large mesh number is considered, which needs a massive computational cost. We treated such problem by proposing a numerical scheme using the implicit Euler method in the temporal variable and the nonstandard finite difference method in the spatial variable on uniform meshes. The stability and uniform convergence of the proposed scheme have been investigated and proved. To demonstrate the theoretical results, numerical experiments are carried out. From the theoretical and numerical results, we observed that the method is uniformly convergent of order one in time and of order two in space.

A numerical scheme is developed to solve a large time delay two-parameter singularly perturbed one-dimensional parabolic problem in a rectangular domain. Two small parameters multiply the convective and diffusive terms, which determine the width of the left and right lateral surface boundary layers. Uniform mesh and piece-wise uniform Shishkin mesh discretization are applied in time and spatial dimensions, respectively. The numerical scheme is formulated by using the Crank–Nicolson method on two consecutive time steps and the average central finite difference approximates in spatial derivatives. It is proved that the method is uniformly convergent, independent of the perturbation parameters, where the convection term is dominated by the diffusion term. The resulting scheme is almost second-order convergent in the spatial direction and second-order convergent in the temporal direction. Numerical experiments illustrate theoretical findings, and the method provides more accurate numerical solutions than prior literature.

The objective of this research work is to develop and analyse a numerical scheme for solving singularly perturbed parabolic reaction-diffusion problems with large spatial delay. The presence of the small positive parameter on the term with the highest order of derivative exhibits two strong boundary layers in the solution of the problem, and the large delay term gives rise to a strong interior layer. The layers’ behavior makes it difficult to solve the problem analytically. To treat such a problem, we developed a numerical scheme using the Crank–Nicolson method in the time direction and the central difference method in the spatial direction via nonstandard finite difference methods on uniform meshes. Stability and convergence analyses for the obtained scheme have been established, which show that the developed numerical scheme is uniformly convergent. To confirm the theoretical analysis, model numerical examples are considered and demonstrated.

This article deals with the numerical treatment of a singularly perturbed unsteady Burger-Huxley equation. The equation is linearized using the Newton-Raphson-Kantorovich approximation method. The resulting linear equation is discretized using the implicit Euler method and an exponential spline method for time and space variables, respectively. Richardson's extrapolation technique is employed to increase the accuracy in the temporal direction. The stability and uniform convergence of the proposed scheme are investigated. The scheme is shown uniformly convergent with the order of convergence O(τ + ℓ2) and O(τ2 + ℓ2) before and after Richardson extrapolation, respectively. Several test examples are considered to validate the applicability and efficiency of the scheme. It is observed that the proposed scheme provides more accurate results than the methods available in the literature.

Abstract 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.

This paper deals with numerical solution of singularly perturbed parabolic partial differential equations with large negative shift on the spatial variable and integral boundary condition on the right side of the domain. For small values of perturbation parameter ɛ, solution of the problem exhibits an interior layer besides a parabolic boundary layers. The numerical methods combine the Crank–Nicolson difference scheme for the temporal direction on the uniform mesh and exponentially fitted operator finite difference method in spatial discretization. The integral boundary condition is treated using numerical integration techniques. The rate of convergence of the scheme is optimized by applying Richardson extrapolation technique. The stability and uniform convergence analysis has been carried out. To validate the applicability of the scheme, numerical examples are presented and the method is more accurate and shown to be second order uniformly convergent in both direction, and it also improves the results of the methods (Elango et al., 2021; Gobena and Duressa, 2021) existing in the literature.

This paper presents numerical treatments for a class of singularly perturbed parabolic partial differential equations with nonlocal boundary conditions. The problem has strong boundary layers at x = 0 and x = 1. The nonstandard finite difference method was developed to solve the considered problem in the spatial direction, and the implicit Euler method was proposed to solve the resulting system of IVPs in the temporal direction. The nonlocal boundary condition is approximated by Simpsons 13 rule. The stability and uniform convergence analysis of the scheme are studied. The developed scheme is second-order uniformly convergent in the spatial direction and first-order in the temporal direction. Two test examples are carried out to validate the applicability of the developed numerical scheme. The obtained numerical results reflect the theoretical estimate.

A numerical method for solving singularly perturbed delay differential equations with a layer or oscillatory behavior for which a small shift is introduced in the reaction term is presented. The stability and convergence of the method have been investigated. To demonstrate the efficiency of the method, two model examples have been presented. Numerical results obtained by the present scheme described that the finding of the present method is accurate than the findings of the previous studies.

Abstract 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 $$\theta$$ θ -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.

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 (Δt+h2).

In this work, a numerical method for the singularly perturbed parabolic convection–diffusion turning point problem with Robin boundary condition was developed. The solution to the considered problem has a boundary layer on the left side of the domain. The present method comprises an implicit trapezoidal method for time discretization on a uniform mesh and second-order central difference schemes for space discretization on a Shishkin mesh. The resultant scheme has been shown that the numerical approximation converges uniformly to the solution of the continuous problem regardless of the diffusion parameter. Finally, to validate the resultant scheme, numerical experiments were performed.

This paper deals with the numerical treatment of two-parametric singularly perturbed parabolic convection-diffusion problems. The scheme is developed through the Crank-Nicholson discretization method in the temporal dimension followed by fitting the B-spline collocation method in the spatial direction. The effect of the perturbation parameters on the solution profile of the problem is controlled by fitting a parameter. As a result, it has been observed that the method is a parameter-uniform convergent and its order of convergence is two. This is shown by the boundedness of the solution, its derivatives, and error estimation. The effectiveness of the proposed method is demonstrated by model numerical examples, and more accurate solutions are obtained as compared to previous findings available in the literature.

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.

Abstract 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.

A mathematical model to estimate transmission dynamics of COVID-19 is developed. A real data of confirmed cases for Ethiopia is used for parameter estimation via model fitting. Results showed that, the diseases free and endemic equilibrium points are found to be locally and globally asymptotically stable for Ro 1 respectively. The basic reproduction number is Ro = 1.5085. Optimal control analysis also showed that, combination of optimal preventive strategies such as public health education, personal protective measures and treatment of hospitalized cases are effective to significantly decrease the number of COVID-19 cases in different compartments of the model.

This review paper contains computational methods or solution methodologies for singularly perturbed differential difference equations with negative and/or positive shifts in a spatial variable. This survey limits its coverage to singular perturbation equations arising in the modeling of neuronal activity and the methods developed by numerous researchers between 2012 and 2022. The review covered singularly perturbed ordinary delay differential equations with small or large negative shift(s), singularly perturbed ordinary differential–differential equations with mixed shift(s), singularly perturbed delay partial differential equations with small or large negative shift(s) and singularly perturbed partial differential–difference equations of the mixed type. The main aim of this review is to find out what numerical and asymptotic methods were developed in the last ten years to solve such problems. Further, it aims to stimulate researchers to develop new robust methods for solving families of the problems under consideration.

In this study, we focus on the formulation and analysis of an exponentially fitted numerical scheme by decomposing the domain into subdomains to solve singularly perturbed differential equations with large negative shift. The solution of problem exhibits twin boundary layers due to the presence of the perturbation parameter and strong interior layer due to the large negative shift. The original domain is divided into six subdomains, such as two boundary layer regions, two interior (interfacing) layer regions, and two regular regions. Constructing an exponentially fitted numerical scheme on each boundary and interior layer subdomains and combining with the solutions on the regular subdomains, we obtain a second order ε-uniformly convergent numerical scheme. To demonstrate the theoretical results, numerical examples are provided and analyzed.

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.

A numerical treatment via a difference scheme constructed by the Crank–Nicolson scheme for the time derivative and cubic spline in tension for the spatial derivatives on a layer resolving nonuniform Bakhvalov-type mesh for a singularly perturbed unsteady-state initial-boundary-value problem with two small parameters is presented. Error analysis of the constructed scheme is discussed and shown to be parameter-uniformly convergent with second-order convergence. Numerical experimentation is taken to confirm the theoretical findings.

Abstract In this paper, accelerated fitted finite difference method for solving singularly perturbed delay differential equation with non-local boundary condition is considered. To treat the non-local boundary condition, Simpson’s rule is applied. The stability and parameter uniform convergence for the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter ε and mesh size h. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and ε-uniformly convergent for h ≥ ε where the classical numerical methods fails to give good result, and it also improves the results of the methods existing in the literature.

Abstract This study introduces a stable central difference method for solving second-order self-adjoint singularly perturbed boundary value problems. First, the solution domain is discretized. Then, the derivatives in the given boundary value problem are replaced by finite difference approximations and the numerical scheme that provides algebraic systems of equations is developed. The obtained system of algebraic equations is solved by Thomas algorithm. The consistency and stability that guarantee the convergence of the scheme are investigated. The established convergence of the scheme is further accelerated by applying the Richardson extrapolation which yields sixth order convergent. To validate the applicability of the method, two model examples are solved for different values of perturbation parameter ε and different mesh size h. The proposed method approximates the exact solution very well. Moreover, the present method is convergent and gives more accurate results than some existing numerical methods reported in the literature.

This study introduces a sixth order numerical method for solving Liénard second order nonlinear differential equations. First, the second order Liénard differential equation is transformed into a first order system of equations. Then, the given interval is discretized, and the method is formulated by using Newton's backward difference interpolation formula. The stability and convergence analysis is studied. Three model examples have been presented to demonstrate the reliability and efficiency of the method. The numerical results presented in the tables and graphs show that the present method approximates the exact solution very well.

A robust numerical scheme is proposed to solve singularly perturbed large time-delay parabolic convection–diffusion problems. For domain discretization, the backward-Euler method for the time derivative and Micken’s type discretization for the space derivatives are used. Using a Richardson extrapolation technique the ɛ-uniform accuracy of the method is improved to second order convergences. We present numerical examples to confirm the agreement of the numerical methods with the theoretical results. The proposed numerical scheme is parameter-uniform convergent, more accurate and robust.

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.

In this article, a numerical solution is proposed for singularly perturbed delay parabolic reaction-diffusion problem with mixed-type boundary conditions. The problem is discretized by the implicit Euler method on uniform mesh in time and extended cubic B-spline collocation method on a Shishkin mesh in space. The parameter-uniform convergence of the method is given, and it is shown to be ε-uniformly convergent of OΔt+N−2ln2N, where Δt and N denote the step size in time and number of mesh intervals in space, respectively. The proposed method gives accurate results by choosing suitable value of the free parameter λ. Some numerical results are carried out to support the theory.