Department of Mathematics

Permanent URI for this collectionhttp://localhost:4000/handle/123456789/1920

Browse

Now showing 1 - 20 of 45

A collocation method for singularly perturbed differential-difference turning point problems exhibiting boundary/interior layers
(Taylor & Francis, 2018-06) Kumar, Devendra
In this article, a collocation method for the problems where the second-order derivative is multiplied by a small perturbation parameter ϵ, the coefficient of the convection term vanishes at a point within the domain of interest, and the shift δ is of o(ε) is proposed. Similar boundary value problems are encountered while simulating several real-life processes, for instance, first exit time problem in the modelling of neuronal variability. The presence of turning point results into twin boundary layers or an interior layer in the solution of the problem under consideration. A rigorous analysis is carried out and it has been shown theoretically that the numerical solution generated by the method converges uniformly to the solution of the continuous problem with respect to the singular perturbation parameter. The effect of the small shift on the boundary/interior layer(s) has also been observed. Several numerical examples are presented to support the theoretical analysis developed in this article.
Comparative study of singularly perturbed two-point BVPs via: Fitted-mesh finite difference method, B-spline collocation method and finite element method
(Elsevier, 2008-10) Kumar, Devendra
The objective of this paper is to present a comparative study of fitted-mesh finite difference method, B-spline collocation method and finite element method for general singularly perturbed two-point boundary value problems. Due to the small parameter , the boundary layer arises. We have taken a piecewise-uniform fitted-mesh to resolve the boundary layer and we have shown that fitted-mesh finite difference method has -uniform first order convergence, B-spline collocation method has almost second order -uniform convergence and Ritz–Galerkin method
A computational method for singularly perturbed nonlinear differential-difference equations with small shift
(Elsevier, 2010-09) Kumar, Devendra
This paper is devoted to the numerical study of the boundary value problems for nonlinear singularly perturbed differential-difference equations with small delay. Quasilinearization process is used to linearize the nonlinear differential equation. After applying the quasilinearization process to the nonlinear problem, a sequence of linearized problems is obtained. To obtain parameter-uniform convergence, a piecewise-uniform mesh is used, which is dense in the boundary layer region and coarse in the outer region. The parameter-uniform convergence analysis of the method has been discussed. The method has shown to have almost second-order parameter-uniform convergence. The effect of small shift on the boundary layer(s) has also been discussed. To demonstrate the performance of the proposed scheme two examples have been carried out. The maximum absolute errors and uniform rates of convergence have been presented in the form of the tables.
An effective numerical approach for two parameter time-delayed singularly perturbed problems
(Springer, 2022-10) Kumar, Devendra
A numerical scheme for the two-parameter singularly perturbed parabolic initial-boundary-value problems with a delay in time is considered. The solution to these problems exhibits twin boundary layers near the endpoints of the spatial domain. An appropriate piecewise-uniform mesh is constructed to resolve these layers. First, the given problem is semi-discretized in the temporal direction by employing the Crank–Nicolson scheme resulting in a system of ordinary differential equations at each time level. Then, to solve these systems, B-spline basis functions with the piecewise-uniform mesh leading to a tri-diagonal system of algebraic equations are used. The tri-diagonal system of algebraic equations is solved using the Thomas algorithm. Through rigorous analysis, we have shown that the scheme is second-order accurate in time and almost second-order accurate in space. Four test problems are solved to validate the theoretical results.
An efficient numerical technique for two-parameter singularly perturbed problems having discontinuity in convection coefficient and source term
(Springer, 2023-01) Kumar, Devendra
We devise a spline-based numerical technique for a class of two-parameter singularly perturbed problems having discontinuous convection and source terms. The problem is discretized using the Crank–Nicolson formula in the temporal direction, and the trigonometric -spline basis functions are used in the spatial direction. The presence of perturbation parameters and the discontinuous convection/source terms result in the interior and boundary layers in the solution to the problem. Our primary focus is to resolve these layers and develop a uniformly convergent scheme. Initially, the proposed method gives almost first and second-order convergence in the spatial and temporal directions, respectively. Then, to improve the accuracy in the spatial direction, we have used the Richardson extrapolation technique. Two numerical examples are taken to demonstrate the layer phenomenon and confirm the theoretical proofs. It is evident from the tables that the Richardson extrapolation technique increases the accuracy from one to two in the spatial direction.
An efficient parameter uniform spline-based technique for singularly perturbed weakly coupled reaction-diffusion systems
(Authero, 2022-07) Kumar, Devendra
A parameter-uniform numerical scheme for a system of weakly coupled singularly perturbed reaction diffusion equations of arbitrary size with appropriate boundary conditions is investigated. More precisely, quadratic B-spline basis functions with an exponentially graded mesh are used to solve a ` × ` system whose solution exhibits parabolic (or exponential) boundary layers at both endpoints of the domain. A suitable mesh generating function is used to generate the exponentially graded mesh. The decomposition of the solution into regular and singular components is obtained to provide error estimates. A convergence analysis is addressed, which shows a uniform convergence of the second order. To validate the theoretical findings, two test problems are solved numerically
Fitted mesh B-spline collocation method for singularly perturbed differential–difference equations with small delay
(Elsevier, 2008-10) Kumar, Devendra
This paper deals with the singularly perturbed boundary value problem for a linear second order differential–difference equation of the convection–diffusion type with small delay parameter of whose solution has a boundary layer. The fitted mesh technique is employed to generate a piecewise-uniform mesh, condensed in the neighborhood of the boundary layers. B-spline collocation method is used with fitted mesh. Parameter-uniform convergence analysis of the method is discussed. The method is shown to have almost second order parameter-uniform convergence. The effect of small delay on boundary layer has also been discussed. Several examples are considered to demonstrate the performance of the proposed scheme and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution.
Fitted Mesh Method for a Class of Singularly Perturbed Differential-Difference Equations
(Global Science Press, 2015) Kumar, Devendra
This paper deals with a more general class of singularly perturbed boundary value problem for a differential-difference equations with small shifts. In particular, the numerical study for the problems where second order derivative is multiplied by a small parameter ε and the shifts depend on the small parameter ε has been considered. The fitted-mesh technique is employed to generate a piecewise-uniform mesh, condensed in the neighborhood of the boundary layer. The cubic B-spline basis functions with fitted-mesh are considered in the procedure which yield a tridiagonal system which can be solved efficiently by using any well-known algorithm. The stability and parameter-uniform convergence analysis of the proposed method have been discussed. The method has been shown to have almost second-order parameter-uniform convergence. The effect of small parameters on the boundary layer has also been discussed. To demonstrate the performance of the proposed scheme, several numerical experiments have been carried out.
Geometric mesh FDM for self-adjoint singular perturbation boundary value problems
(Elsevier, 2007-07) Kumar, Devendra
A numerical method based on finite difference method with variable mesh is given for second order singularly perturbed self-adjoint two point boundary value problems. The original problem is reduced to its normal form and the reduced problem is solved by FDM taking variable mesh(geometric mesh). The maximum absolute errors , for different values of parameter ϵ, number of points N, and the mesh ratio r, for three examples have been given in tables to support the efficiency of the method.
Haar-wavelet based approximation for pricing American options under linear complementarity formulations
(Wiley, 2020-10) Kumar, Devendra
In this manuscript, we present a novel and highly accurate wavelet-based approximation technique to explore the sensitivities and value of American options diagnosed by linear complementarity problems. For a detailed analysis of such financially relevant problems, we transform the actual final value problem into a dimensionless initial value problem. To avoid the unacceptable large truncation error, the unbounded domain is trimmed into a bounded domain. A remarkable observation is that to investigate the various physical and numerical aspects of the options' sensitivities; the proposed scheme is efficient as it explicitly provides the numerical approximation of all the derivatives of the solution function. The multiresolution technique of the wavelets and the convergence of the proposed wavelet scheme are comprehensively analyzed. The wavelet analysis is accompanied by illustrative examples to demonstrate the proficiency and robustness of the present method coupled with graphical representations. It has been shown that the present method is efficient to solve free boundary problems. It is worthy to note that the highly accurate and promising computational results are enough to confirm the performance of the proposed method. The simulated results of options' Greeks analyzed and discussed have vast applications in different financial institutes and trading markets
A highly accurate algorithm for retrieving the predicted behavior of problems with piecewise-smooth initial data
(Elsevier, 2022-03) Kumar, Devendra
A numerical scheme is constructed for the second-order parabolic partial differential equation with piecewise smooth initial data. The scheme comprises an orthogonal spline collocation strategy with the Rannacher time-marching. The irregular behavior of the underlying initial conditions of such differential equations results in inaccurate approximations due to the quantization error. For such problems, even the A-stable Crank-Nicolson scheme yields only first-order convergence in the temporal direction, with oscillations near the discontinuity. Applying mathematical perspective to dampen these oscillations, we present a highly accurate orthogonal spline collocation method with a smooth but straightforward time-marching scheme that significantly improves the convergence order. Through rigorous analysis, the present conjunctive scheme's convergence in the spatial direction is shown fourth-order (in and -norms) and third-order (in -norm), and it is shown second-order in the temporal direction. The performance and robustness of the contributed scheme are conclusively demonstrated with two test examples.
An implicit scheme for singularly perturbed parabolic problem with retarded terms arising in computational neuroscience
(Wiley, 2018-04) Kumar, Devendra
A class of time-dependent singularly perturbed convection-diffusion problems with retarded terms arising in computational neuroscience is considered. In particular, a numerical scheme for the parabolic convection-diffusion problem where the second-order derivative with respect to the spatial direction is multiplied by a small perturbation parameter urn:x-wiley:0749159X:media:num22269:num22269-math-0001 and the shifts urn:x-wiley:0749159X:media:num22269:num22269-math-0002 are of urn:x-wiley:0749159X:media:num22269:num22269-math-0003 is constructed. The Taylor series expansion is used to tackle the shift terms. The continuous problem is semidiscretized using the Crank-Nicolson finite difference method in the temporal direction and the resulting set of ordinary differential equations is discretized using a midpoint upwind finite difference scheme on an appropriate piecewise uniform mesh, which is dense in the boundary layer region. It is shown that the proposed numerical scheme is second-order accurate in time and almost first-order accurate in space with respect to the perturbation parameter urn:x-wiley:0749159X:media:num22269:num22269-math-0004. To validate the computational results and efficiency of the method some numerical examples are encountered and the numerical results are compared with some existing results. It is observed that the numerical approximations are fairly good irrespective of the size of the delay and the advance till they are of urn:x-wiley:0749159X:media:num22269:num22269-math-0005. The effect of the shifts on the boundary layer has also been observed.
A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation
(De Gruyter, 2022-10) Kumar, Devendra
In this article, we present a novel numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ (BBMB) equation using Atangana Baleanu Caputo (ABC) derivative. First, we apply a linearization technique to deal with the generalized non-linear expression, and then the Crank–Nicolson finite difference formula is used in the temporal direction. A reliable numerical technique is applied to discretize the time-fractional ABC derivative, and the central difference formulae are used to approximate the derivatives in the spatial direction. The method is shown unconditionally stable and second-order convergent in both directions through the Fourier analysis. The numerical results of two test problems are analyzed to validate the theoretical results.
A non-linear single step explicit scheme for non-linear two-point singularly perturbed boundary value problems via initial value technique
(Elsevier, 2008-08) Kumar, Devendra
In this paper, a method based on initial value technique is proposed for solving non-linear two-point singularly perturbed boundary value problems for second order ordinary differential equations (ODEs) with a boundary layer at one (either left or right) end. The original singularly perturbed boundary value problem is reduced to an initial value problem approximated by its outer solution (asymptotic approximation). The new initial value problem is solved by proposed non-linear single step explicit scheme followed the idea given in [F.D. Van Niekerk, Non-linear one-step methods for initial value problems, Comput. Math. Appl. 13 (1987) 367–371]. The proposed scheme has been shown to be of order two. To demonstrate the applicability of the proposed scheme several (linear and non-linear) problems have been solved. It is observed that the present scheme approximate the exact solution very well.
A novel finite difference based numerical approach for Modified Atangana- Baleanu Caputo derivative
(AIMS Press, 2022-07) Kumar, Devendra
In this paper, a new approach is presented to investigate the time-fractional advection-dispersion equation that is extensively used to study transport processes. The present modified fractional derivative operator based on Atangana-Baleanu's definition of a derivative in the Caputo sense involves singular and non-local kernels. A numerical approximation of this new modified fractional operator is provided and applied to an advection-dispersion equation. Through Fourier analysis, it has been proved that the proposed scheme is unconditionally stable. Numerical examples are solved that validate the theoretical results presented in this paper and ensure the proficiency of the numerical scheme.
Numerical Simulation for Generalized Time-Fractional Burgers' Equation With Three Distinct Linearization Schemes
(ASME, 2023-03) Kumar, Devendra
In the present study, we examined the effectiveness of three linearization approaches for solving the time-fractional generalized Burgers' equation using a modified version of the fractional derivative by adopting the Atangana-Baleanu Caputo derivative. A stability analysis of the linearized time-fractional Burgers' difference equation was also presented. All linearization strategies used to solve the proposed nonlinear problem are unconditionally stable. To support the theory, two numerical examples are considered. Furthermore, numerical results demonstrate the comparison of linearization strategies and manifest the effectiveness of the proposed numerical scheme in three distinct ways.
Parameter independent scheme for singularly perturbed problems including a boundary turning point of multiplicity ≥ 1
(JAAC, 2022) Kumar, Devendra
A numerical scheme is developed for parabolic singularly perturbed boundary value problems, including multiple boundary turning points at the left endpoint of the spatial direction. The highest order derivative of these problems is multiplied by a small parameter , and when it is close to zero, the solution exhibits a parabolic type boundary layer near the left lateral surface of the domain of consideration. Thus, large oscillations appear when classical/standard numerical methods are used to solve the problem, and one cannot achieve the expected accuracy. Thus, the Crank-Nicolson scheme on a uniform mesh in the temporal direction and an upwind scheme on a Shishkin-type mesh in the spatial direction is constructed. The theoretical analysis shows that the method converges irrespective of the size of with accuracy . Three test examples are presented to verify that the computational results agree with the theoretical ones.
A parameter uniform method for singularly perturbed differential-difference equations with small shifts
(De Gruyter, 2013) Kumar, Devendra
This paper is devoted to the numerical study for a class of boundary value problems of singularly perturbed linear second-order differential-difference equations with small shifts (i. e., containing both terms having a negative shift and terms having a positive shift). In particular, the numerical study for the problems where second order derivative is multiplied by a small parameter e and the shifts depend on the small parameter have been considered. To obtain a parameter-uniform convergence, a piecewise-uniform mesh is constructed, which is dense in the boundary layer region and coarse in the outer region. The parameter-uniform convergence analysis of the method has been discussed. The method is shown to have almost second order parameter-uniform convergence. The effect of small shifts on boundary layers have also been discussed. To demonstrate the efficiency of the proposed scheme several examples having boundary layers have been carried out.
A parameter-uniform collocation scheme for singularly perturbed delay problems with integral boundary condition
(Springer, 2020-03) Kumar, Devendra
Based on the basis of B-spline functions an efficient numerical scheme on a piecewise-uniform mesh is suggested to approximate the solution of singularly perturbed problems with an integral boundary condition and having a delay of unit magnitude. For the small diffusion parameter ε, an interior layer and a boundary layer occur in the solution. Unlike most numerical schemes our scheme does not require the differentiation of the problem data (integral boundary condition). The parameter-uniform convergence (the second-order convergence except for a logarithmic factor) is confirmed by numerical computations of two test problems. As a variant double mesh principle is used to measure the accuracy of the method in terms of the maximum absolute error.
Parameter-uniform fitted operator B-spline collocation method for self-adjoint singularly perturbed two-point boundary value problems
(ETNA, 2008) Kumar, Devendra
In this paper, we develop a B-spline collocation method for the numerical solution of a self-adjoint singularly perturbed boundary value problem of the form We construct a fitting factor and use the B-spline collocation method, which leads to a tridiagonal linear system. The method is analyzed for parameter-uniform convergence. Several numerical examples are reported which demonstrate the efficiency of the proposed method.