Fitted Operator Finite Difference Method Research Articles

Gaussian quadrature method with exponential fitting factor for two-parameter singularly perturbed parabolic problem

The parabolic convection-diffusion-reaction problem is examined in this work, where the diffusion and convection terms are multiplied by two small parameters, respectively. The proposed approach is based on a fitted operator finite difference method. The Crank-Nicolson method on uniform mesh is utilized to discretize the time variables in the first step. Two-point Gaussian quadrature rule is used for further discretizing these semi-discrete problems in space, and the second order interpolation of the first derivatives is utilized. The fitting factor’s value, which accounts 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 two examples, which yielded more accurate results than some other methods found in the literatures.

Numerical integration method for two-parameter singularly perturbed time delay parabolic problem

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.

The robust numerical schemes for two-dimensional elliptical singularly perturbed problems with space shifts

ABSTRACT This article focuses on the investigation of two-dimensional elliptic singularly perturbed problems that incorporate positive and negative shifts, the solution of this class of problems may demonstrate regular/parabolic/degenerate or interior boundary layers. The goal of this article is to establish the development of numerical techniques for two-dimensional elliptic singularly perturbed problems with positive and negative shifts having regular boundary layers. The three numerical schemes are proposed to estimate the solution of this class of problems based on the fitted operator and fitted mesh finite-difference methods. The fitted operator finite difference method is analysed for convergence. The effect of shift terms on the solution behaviour is demonstrated through numerical experiments. The paper concludes by providing several numerical results that demonstrate the performance of proposed numerical schemes.

A nonstandard fitted operator finite difference method for two-parameter singularly perturbed time-delay parabolic problems

In this article, a class of singularly perturbed time-delay two-parameter second-order parabolic problems are considered. The presence of the two small parameters attached to the derivatives causes the solution of the given problem to exhibit boundary layer(s). We have developed a uniformly convergent nonstandard fitted operator finite difference method (NSFOFDM) to solve the considered problems. The Crank-Nicolson scheme with a uniform mesh is used for the discretization of the time derivative, while for the spatial discretization, we have applied a fitted operator finite difference method following the nonstandard methodology of Mickens. Moreover, the solution bounds of the governing equation are shown by asymptotic analysis. The convergence of the proposed numerical scheme is investigated using truncation error and the barrier function approach. The study shows that our proposed scheme is uniformly convergent independent of the perturbation parameters, quadratically in time, and linearly in space. Numerical experiments are carried out, and the results are presented in tables and graphically.

Fitted Operator Average Finite Difference Method for Singularly Perturbed Delay Parabolic Reaction Diffusion Problems with Non-Local Boundary Conditions

This paper deals with numerical solution of singularly perturbed delay parabolic reaction diffusion problem having large delay on the spatial variable with non-local boundary condition. The solution of the problem exhibits parabolic boundary layer on both sides of the spatial domain and interior layer is also created. Introducing a fitting parameter into asymptotic solution and applying average finite difference approximation, a fitted operator finite difference method is developed for solving the problem under consideration. To treat the non-local boundary condition, Simpson's rule is applied. The stability and $\varepsilon$ uniform convergence analysis has been carried out. To validate the applicability of the scheme, numerical examples are presented and solved for different values of the perturbation parameter $\varepsilon$ and mesh sizes. 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 shown to be second order Uniformly convergent in both direction, and it also improves the results of the methods existing in the literature.

An optimal fitted numerical scheme for solving singularly perturbed parabolic problems with large negative shift and integral boundary condition

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.

A New Parameter-Uniform Discretization of Semilinear Singularly Perturbed Problems

In this paper, we present a numerical approach to solving singularly perturbed semilinear convection-diffusion problems. The nonlinear part of the problem is linearized via the quasilinearization technique. We then design and implement a fitted operator finite difference method to solve the sequence of linear singularly perturbed problems that emerges from the quasilinearization process. We carry out a rigorous analysis to attest to the convergence of the proposed procedure and notice that the method is first-order uniformly convergent. Some numerical evaluations are implemented on model examples to confirm the proposed theoretical results and to show the efficiency of the method.

A robust numerical scheme for singularly perturbed parabolic reaction-diffusion problems via the method of lines

In this paper, we consider one- and two-dimensional singularly perturbed parabolic reaction-diffusion problems. We propose a parameter-uniform numerical scheme to solve these problems. The continuous problem is first discretized in the space variable using a fitted operator finite difference method. The partial differential equation is thus transformed into a system of initial value problems which are then integrated in time with the Crank–Nicolson finite difference method. A convergence analysis shows that the scheme is second-order ε-uniform convergent in space and time. Richardson extrapolation of the space variable results in a fourth order ε-uniform convergence. Numerical experiments on two test examples confirm the theoretical findings.

Parameter uniform numerical methods for singularly perturbed delay parabolic differential equations with non-local boundary condition

The motive of this paper is, to develop accurate and parameter uniform numerical method for solving singularly perturbed delay parabolic differential equation with non-local boundary condition exhibiting parabolic boundary layers. Also, the delay term that occurs in the space variable gives rise to interior layer. Fitted operator finite difference method on uniform mesh that uses the procedures of method of line for spatial discretization and backward Euler method for the resulting system of initial value problems in temporal direction is considered. To treat the non-local boundary condition, Simpsons rule is applied. The stability and parameter uniform convergence for the proposed method are proved. To validate the applicability of the scheme, numerical examples are presented and solved for different values of the perturbation parameter. The method is shown to be accurate of O(h2 + △t) . Finally, conclusion of the work is included at the end.

Accelerated fitted operator finite difference method for singularly perturbed parabolic reaction-diffusion problems

This paper deals with the numerical treatment of singularly perturbed parabolic reaction-diffusion initial boundary value problems. Introducing a fitting parameter into the asymptotic solution and applying average finite difference approximation, a fitted operator finite difference method is developed for solving the problem. To accelerate the rate of convergence of the method, Richardson extrapolation technique is applied. The consistency and stability of the proposed method have been established very well to ensure the convergence of the method. Numerical experimentation is carried out on some model problems and both the results are presented in tables and graphs. The numerical results are compared with findings of some methods existing in the literature and found to be more accurate. Generally, the formulated method is consistent, stable, and more accurate than some methods existing in the literature for solving singularly perturbed parabolic reaction-diffusion initial boundary value problems.

Fitted Operator Finite Difference Method for Singularly Perturbed Parabolic Convection-Diffusion Type

In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson extrapolation method is applied on the time direction and the fitted operator finite difference method on the spatial direction is used, on the uniform grids. The stability and consistency of the method were established very well to guarantee the convergence of the method. Numerical experimentation is carried out on model examples, and the results are presented both in tables and graphs. Further, the present method gives a more accurate solution than some existing methods reported in the literature.

Fitted Numerical Scheme for Solving Singularly Perturbed Parabolic Delay Partial Differential Equations

In this paper, exponentially fitted finite difference scheme is developed for solving singularly perturbed parabolic delay partial differential equations having small delay on the spatial variable. The term with the delay is approximated using Taylor series approximation. The resulting singularly perturbed parabolic partial differential equation is treated using im- plicit Euler method in the temporal discretization with exponentially fitted operator finite difference method in the spatial discretization. The parameter uniform convergence analysis has been carried out with the order of convergence one. Test examples and numerical results are considered to validate the theoretical analysis of the scheme.

Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation

This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε , taking arbitrary value in the interval 0 , 1 . For small values of ε , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numerical test examples for different values of ε .

Treatment of Singularly Perturbed Differential Equations with Delay and Shift Using Haar Wavelet Collocation Method

An efficient Haar wavelet collocation method is proposed for the numerical solution of singularly perturbed differential equations with delay and shift. Taylor series (upto the first order) is used to convert the problem with delay and shift into a new problem without the delay and shift and then solved by Haar wavelet collocation method, which reduces the time and complexity of the system. Further, we apply the Haar wavelet collocation method directly to solve the problems. Also, we demonstrated several test examples to show the accuracy and efficiency of the Haar wavelet collocation method and compared our results with the finite difference and fitted operator finite difference method [11], [28].

Accelerated fitted operator finite difference method for singularly perturbed delay differential equations with non-local boundary condition

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.

A fitted operator method for a model arising in vascular tumor dynamics

In this paper, we consider a model for the population kinetics of human tumor cells in vitro, differentiated by phases of the cell division cycle and length of time within each phase. Since it is not easy to isolate the effects of cancer treatment on the cell cycle of human cancer lines, during the process of radiotherapy or chemotherapy, therefore, we include the spatial effects of cells in each phase and analyse the extended model. The extended model is not easy to solve analytically, because perturbation by cancer therapy causes the flow cytometric profile to change in relation to one another. Hence, making it difficult for the resulting model to be solved analytically. Thus, in [16] it is reported that the non-standard schemes are reliable and propagate sharp fronts accurately, even when the advection, reaction processes are highly dominant and the initial data are not smooth. As a result, we construct a fitted operator finite difference method (FOFDM) coupled with non-standard finite difference method (NSFDM) to solve the extended model. The FOFDM and NSFDM are analyzed for convergence and are seen that they are unconditionally stable and have the accuracy of O(∆t + (∆x)^2), where ∆t and ∆x denote time and space step-sizes, respectively. Some numerical results confirming theoretical observations are presented.

Uniformly convergent numerical scheme for singularly perturbed parabolic delay differential equations

This paper deals with numerical treatment of singularly perturbed parabolic differential difference equations having small shifts on the spatial variable. The considered problem contain small perturbation parameter (ε) multiplied on the diffusion term of the equation. For small values of ε the solution of the problem exhibits a boundary layer on the left or right side of the spatial domain depending on the sign of the convective term. The terms involving the shifts are approximated using Taylor’s series approximation. The resulting singularly perturbed parabolic partial differential equation is solved using implicit Euler method in the temporal discretization with exponentially fitted operator finite difference method in the spatial discretization. The uniform stability of the scheme investigated using comparison principle and discrete solution bound by constructing barrier function. Uniform convergence analysis has been carried out. The scheme gives second order convergence for the case ε >N−1and first order convergence for the case ε «N−1, whereNis number of mesh interval. Test examples and numerical results are considered for validating the theoretical analysis of the scheme.

A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer

AbstractThe objective of this paper is to construct and analyze a fitted operator finite difference method (FOFDM) for the family of time‐dependent singularly perturbed parabolic convection–diffusion problems. The solution to the problems we consider exhibits an interior layer due to the presence of a turning point. We first establish sharp bounds on the solution and its derivatives. Then, we discretize the time variable using the classical Euler method. This results in a system of singularly perturbed interior layer two‐point boundary value problems. We propose a FOFDM to solve the system above. Through a rigorous error analysis, we show that the scheme is uniformly convergent of order one with respect to both time and space variables. Moreover, we apply Richardson extrapolation to enhance the accuracy and the order of convergence of the proposed scheme. Numerical investigations are carried out to demonstrate the efficacy and robustness of the scheme.

A fitted operator finite difference method of lines for singularly perturbed parabolic convection–diffusion problems

We propose a uniformly convergent finite difference method to solve singularly perturbed time-dependent convection–diffusion problems in the framework of method of lines. The method uses the fitted operator finite difference method to discretize the spatial derivatives followed by the Crank–Nicolson method for the time derivative. Richardson extrapolation is performed in space to improve the accuracy of the method. We prove that the method is uniformly convergent with respect to the perturbation and the discretization parameters. We present numerical simulations to illustrate and confirm the theoretical results.

A robust fitted operator finite difference method for singularly perturbed problems whose solution has an interior layer

The objectives of this paper are to construct and study a fitted operator finite difference method for the class of singularly perturbed problems whose solution exhibits an interior layer due to the presence of a turning point. We first establish sharp bounds on the solution and its derivatives. Then in line with other fitted operator methods that are designed in various recent works in numerical singular perturbation theory by the first two authors, we propose a fitted operator finite difference method to solve this interior layer problem. This method is then analysed by making use of the bounds on the solutions that we derive. We show that the scheme is uniformly convergent of order one. We also apply Richardson extrapolation as the acceleration technique to improve the accuracy and the order of convergence of the scheme up to two. Numerical investigations are carried out to demonstrate the efficacy and robustness of the scheme.