Department of Mathematics

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Author: search.filters.author.Kumar, DevendraHas files: No

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    A parameter-uniform implicit scheme for two-parameter singularly perturbed parabolic problems
    (2023) Kumar, Devendra
    A parameter-uniform implicit approach for two-parameter singularly perturbed boundary valueproblems is constructed. On the solution derivatives, sharp limits are presented. The solution is additionallydivided into regular and singular components, limiting thederivatives of these components utilized in theconvergence analysis. In the temporal direction, the system of ordinary differential equations produced by theCrank-Nicolson scheme on a uniform mesh is further discretized in the spatial direction by employing a finitedifference technique on a selected Shishkin mesh. Through a rigorous analysis, we establish the theoreticalresults for two cases: Case I.ε1/ε22→0 asε2→0, and Case II.ε22/ε1→0 asε1→0, showing that thetechnique is convergent regardless of the magnitude of theε1, ε2parameters. The order of accuracy in Case Iand II are shown to beO((∆t)2+N−1(lnN)2) andO((∆t)2+N−2(lnN)2), respectively. Two examples arepresented to verify the theoretical results
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    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.
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    Trigonometric B-spline based ε-uniform scheme for singularly perturbed problems with Robin boundary conditions
    (Taylor & Francis, 2022-07) Kumar, Devendra
    In this paper, a non-polynomial-based trigonometric cubic B-spline collocation method is developed to solve the reaction-diffusion singularly perturbed problems with Robin boundary conditions. These problems are more tedious to solve than those with Dirichlet and Neumann boundary conditions. The parameter ε in the differential equation results in a rapid change in the solution over a small region. A piecewise uniform mesh is constructed to handle this difficulty. Also, a modification of the proposed mesh is suggested to improve the accuracy of the numerical results by introducing a change in the transition parameter. Through rigorous analysis, it has been shown that the method is almost second-order uniformly convergent. The performance and theoretical findings of the proposed scheme are validated through numerical experiments presented for two test problems. The accuracy of the method is measured in the discrete maximum norm. The tabular results demonstrate that the newly added mesh produces better results.
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    Three-dimensional Haar wavelet method for singularly perturbed elliptic boundary value problems on non-uniform meshes
    (Springer, 2022-05) Kumar, Devendra
    Recently, the two-dimensional elliptic singularly perturbed boundary value problems have received attention. These problems have not been much explored numerically. A highly accurate numerical scheme on different non-uniform meshes is suggested to solve such problems. In particular, the Haar wavelet method on a special type of non-uniform mesh and Shishkin mesh is proposed; because of the use of the block pulse function, it is easy to derive and handle the operator matrices. Also, it has been shown that Shiskin mesh provides better results. The use of the piecewise-uniform Shishkin mesh with the Haar wavelet scheme contains a novelty in itself. Through rigorous analysis, the method is shown as first-order convergent in L2-norm. The theoretical results are confirmed by computational results obtained in the maximum norm and L2-norm for two test problems. From the comparative results provided in tables, it is worth noting that the Shishkin mesh is an excellent choice to solve these problems instead of a particular type of non-uniform mesh for the proposed scheme. This can be justified because the grid points defined by the Shishkin mesh are adequately distributed in the layer and non-layer regions.
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    A wavelet-based novel approximation to investigate the sensitivities of various path-independent binary options
    (Wiley, 2022-04) Kumar, Devendra
    We present a novel idea of a wavelet-based approximation technique using a multi-resolution analysis to investigate the sensitivities of various path-independent binary options under the Black–Scholes environment. The final value problem is transformed into a dimensionless initial value problem; also, to avoid the large truncation error, the infinite domain is truncated into a finite domain. A noteworthy observation is that the proposed Haar wavelet scheme is effective and easy to implement to analyze the different physical and numerical aspects of the options' Greeks. It explicitly provides the numerical approximation of all the derivatives of the solution function. Also, the non-smooth payoff functions are approximated well with the Haar wavelet approximation technique of estimating the spiked functions, so there is no need to deal with the discontinuity separately. We prove the consistency and stability of the proposed method and show that the proposed method is the first- and second-order accurate in the temporal and spatial directions, respectively. A variety of test examples conclusively demonstrates the computational proficiency and the theoretical results of the proposed scheme. Different attributes of the Greeks of distinct binary options are analyzed graphically. The motivational work of the study of various Greeks of different binary options significantly impacts the hedging strategies used by different financial institutes.
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    Rannacher time-marching with orthogonal spline collocation method for retrieving the discontinuous behavior of hedging parameters
    (Elsevier, 2022-08) Kumar, Devendra
    In this paper, we extensively study the orthogonal spline collocation method known as the spline collocation at Gauss points with Rannacher’s time-marching scheme for free boundary value option pricing problems. Such financial problems commonly feature non-smooth payoff functions that cause inaccuracies in approximating the solution and its derivatives. As a result, unlike the problems with the smooth initial data, the quadratic convergence is not realized by the Crank-Nicolson time-stepping scheme for these problems. Furthermore, the non-smoothness in the initial condition leads to severe degradation in the convergence rates and spurious oscillations near the discontinuity. The rationale is that classical schemes strongly rely on the smoothness of the initial data. A rigorous time-marching scheme referred to as Rannacher time-stepping scheme is introduced for the American option’s price diagnosed by a linear complementarity problem to smoothen the data. Moreover, with careful analysis, second and fourth orders of convergence are established for the present scheme in temporal and spatial directions, respectively. The numerical results for three test problems are presented in tables and graphs to validate the theory. These results show that the present scheme achieves higher accuracy and sufficiently restores the expected behavior.
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    Two-dimensional Haar wavelet based approximation technique to study the sensitivities of the price of an option
    (Wiley, 2020-12) Kumar, Devendra
    In the present work, a two-dimensional Haar wavelet method is proposed to study the sensitivities of the price of an option. The method is appropriate to study these sensitivities as it explicitly gives the values of all the derivatives of the solution. A Black–Scholes model for European style options is considered to analyze the physical and numerical aspects of the put and the call option Greeks. We use the concept of coordinate transformation to make the Black–Scholes equation dimensionless and to resolve the obstacle in approximating the Greeks having non-smoothness at the strike price. The infinite spatial domain is truncated into the finite domain to avoid large truncation errors. Through rigorous analysis, the method is shown first-order accurate in the L2−norm. The numerical computations performed to approximate the option price and various Greeks, like delta, theta, gamma, and so forth, confirm the theoretical results in L2−norm. The relative errors and the maximum absolute errors are also presented. The motivational work of option Greeks analysis may leave a significant impact on financial institutes; it helps them to manage the risk by setting the portfolio's new value and to estimate the probability of losing money.
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    A uniformly convergent quadratic -spline based scheme for singularly perturbed degenerate parabolic problems
    (Elsevier, 2022-05) Kumar, Devendra
    In this article, a numerical scheme is developed to solve singularly perturbed convection–diffusion type degenerate parabolic problems. The degenerative nature of the problem is due to the coefficient of the convection term. As the perturbation parameter approaches zero, the solution to this problem exhibits a parabolic boundary layer in the neighborhood of the left end side of the domain. The problem is semi-discretized using the Crank–Nicolson scheme, and then the quadratic spline basis functions are used to discretize the semi-discrete problem. A priori bounds for the solution (and its derivatives) of the continuous problem are given, which are necessary to analyze the error. A rigorous error analysis shows that the proposed method is boundary layer resolving and second-order parameter uniformly convergent. Some numerical experiments have been devised to support the theoretical findings and the effectiveness of the proposed scheme.
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    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.
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    Wavelet-based approximation for two-parameter singularly perturbed problems with Robin boundary conditions
    (Springer, 2021-03) Kumar, Devendra
    In this article, we present a highly-accurate wavelet-based approximation to study and analyze the physical and numerical aspects of two-parameter singularly perturbed problems with Robin boundary conditions. To explore the swiftly changing behavior of such problems, we have used a special type of non-uniform mesh known as Shishkin mesh. Using Shishkin mesh with the Haar wavelet scheme contains a novelty in itself. We comprehensively explain an approach to solve the Robin boundary conditions involving the proposed Haar wavelet scheme. Through rigorous analysis, the order of convergence of the present scheme is shown quadratic and linear in the spatial and temporal directions, respectively. The robustness and proficiency of the contributed scheme are conclusively demonstrated with three test examples. Irrespective of the problem’s geometry, the proposed method is highly accurate and very economical.