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266 results on '"Mehra, Mani"'

1. Modified least squares method and a review of its applications in machine learning and fractional differential/integral equations

Author

Singh, Abhishek Kumar, Mehra, Mani, and Alikhanov, Anatoly A.

Subjects
Mathematics - Numerical Analysis
Abstract

The least squares method provides the best-fit curve by minimizing the total squares error. In this work, we provide the modified least squares method based on the fractional orthogonal polynomials that belong to the space $M_{n}^{\lambda} := \text{span}\{1,x^{\lambda},x^{2\lambda},\ldots,x^{n\lambda}\},~\lambda \in (0,2]$. Numerical experiments demonstrate how to solve different problems using the modified least squares method. Moreover, the results show the advantage of the modified least squares method compared to the classical least squares method. Furthermore, we discuss the various applications of the modified least squares method in the fields like fractional differential/integral equations and machine learning.

Published
2024

3. High order approximation to Caputo derivative on graded mesh and time-fractional diffusion equation for non-smooth solutions

Author

Kumari, Shweta, Singh, Abhishek Kumar, Mehandiratta, Vaibhav, and Mehra, Mani

Subjects
Mathematics - Numerical Analysis
Abstract

In this paper, a high-order approximation to Caputo-type time-fractional diffusion equations involving an initial-time singularity of the solution is proposed. At first, we employ a numerical algorithm based on the Lagrange polynomial interpolation to approximate the Caputo derivative on the non-uniform mesh. Then truncation error rate and the optimal grading constant of the approximation on a graded mesh are obtained as $\min\{4-\alpha,r\alpha\}$ and $\frac{4-\alpha}{\alpha}$, respectively, where $\alpha\in(0,1)$ is the order of fractional derivative and $r\geq 1$ is the mesh grading parameter. Using this new approximation, a difference scheme for the Caputo-type time-fractional diffusion equation on graded temporal mesh is formulated. The scheme proves to be uniquely solvable for general $r$. Then we derive the unconditional stability of the scheme on uniform mesh. The convergence of the scheme, in particular for $r=1$, is analyzed for non-smooth solutions and concluded for smooth solutions. Finally, the accuracy of the scheme is verified by analyzing the error through a few numerical examples., Comment: 18 pages, 2 figures and 7 tables

Published
2023

8. Existence and uniqueness of time-fractional diffusion equation on a metric star graph

Author

Mehandiratta, Vaibhav, Mehra, Mani, and Leugering, Gunter

Subjects
Mathematics - Analysis of PDEs
Abstract

In this paper, we study the time-fractional diffusion equation on a metric star graph. The existence and uniqueness of the weak solution are investigated and the proof is based on eigenfunction expansions. Some priori estimates and regularity results of the solution are proved., Comment: 17 pages manuscript including 1 figure

Published
2020

9. The Crank-Nicholson type compact difference scheme for a loaded time-fractional Hallaire's equation

Author

Alikhanov, Anatoly, Beshtokov, Murat, and Mehra, Mani

Subjects
Mathematics - Numerical Analysis
Abstract

In this paper, we study loaded modified diffusion equation (the Hallaire equation with the fractional derivative with respect to time). The compact finite difference scheme of Crank-Nicholson type of higher order is developed for approximating the stated problem on uniform grids. A priori estimates are obtained in difference and differential interpretations, from which there follow uniqueness, stability, and convergence of the solution of the difference problem to solution of the differential problem with the rate $\mathcal{O}(h^4+\tau^{2-\alpha}.)$ Proposed theoretical calculations are confirmed by numerical experiments on test problem., Comment: 10 pages, 2 tables

Published
2019

10. Data-Driven Discovery of Time Fractional Differential Equations

Author

Singh, Abhishek Kumar, Mehra, Mani, Alikhanov, Anatoly A., Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Groen, Derek, editor, de Mulatier, Clélia, editor, Paszynski, Maciej, editor, Krzhizhanovskaya, Valeria V., editor, Dongarra, Jack J., editor, and Sloot, Peter M. A., editor

Published
2022
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11. Compact finite difference method for pricing European and American options under jump-diffusion models

Author

Patel, Kuldip Singh and Mehra, Mani

Subjects
Quantitative Finance - Computational Finance
Abstract

In this article, a compact finite difference method is proposed for pricing European and American options under jump-diffusion models. Partial integro-differential equation and linear complementary problem governing European and American options respectively are discretized using Crank-Nicolson Leap-Frog scheme. In proposed compact finite difference method, the second derivative is approximated by the value of unknowns and their first derivative approximations which allow us to obtain a tri-diagonal system of linear equations for the fully discrete problem. Further, consistency and stability for the fully discrete problem are also proved. Since jump-diffusion models do not have smooth initial conditions, the smoothing operators are employed to ensure fourth-order convergence rate. Numerical illustrations for pricing European and American options under Merton jump-diffusion model are presented to validate the theoretical results., Comment: arXiv admin note: substantial text overlap with arXiv:1804.07534

Published
2018

12. Fourth order compact scheme for option pricing under Merton and Kou jump-diffusion models

Author

Patel, Kuldip Singh and Mehra, Mani

Subjects
Quantitative Finance - Computational Finance
Abstract

In this article, a three-time levels compact scheme is proposed to solve the partial integro-differential equation governing the option prices under jump-diffusion models. In the proposed compact scheme, the second derivative approximation of unknowns is approximated by the value of unknowns and their first derivative approximations which allow us to obtain a tri-diagonal system of linear equations for the fully discrete problem. Moreover, consistency and stability of the proposed compact scheme are proved. Due to the low regularity of typical initial conditions, the smoothing operator is employed to ensure the fourth-order convergence rate. Numerical illustrations for pricing European options under Merton and Kou jump-diffusion models are presented to validate the theoretical results.

Published
2018

15. Analytic and Numerical Solutions of Space-Time Fractional Diffusion Wave Equations with Different Fractional Order

Author

Singh, Abhishek Kumar, Mehra, Mani, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Paszynski, Maciej, editor, Kranzlmüller, Dieter, editor, Krzhizhanovskaya, Valeria V., editor, Dongarra, Jack J., editor, and Sloot, Peter M. A., editor

Published
2021
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17. Uncertainty Quantification in Fractional Stochastic Integro-Differential Equations Using Legendre Wavelet Collocation Method

Author

Singh, Abhishek Kumar, Mehra, Mani, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Krzhizhanovskaya, Valeria V., editor, Závodszky, Gábor, editor, Lees, Michael H., editor, Dongarra, Jack J., editor, Sloot, Peter M. A., editor, Brissos, Sérgio, editor, and Teixeira, João, editor

Published
2020
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18. A Compact Filter Regularization Method for Solving Sideways Heat Equation

Author

Shukla, Ankita, Mehra, Mani, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Sergeyev, Yaroslav D., editor, and Kvasov, Dmitri E., editor

Published
2020
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20. Applications of Wavelet in Inverse Problems

Author

Mehra, Mani, Chaubey, Yogendra Prasad, Editorial Board Member, Subrahmanyam, P. V., Editor-in-Chief, Matkowski, Janusz, Editorial Board Member, Parthasarathy, Thiruvenkatachari, Editorial Board Member, Sikirić, Mathieu Dutour, Editorial Board Member, Sharma, Bhu Dev, Editorial Board Member, and Mehra, Mani

Published
2018
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21. Other Useful Applications of Wavelet

Author

Mehra, Mani, Chaubey, Yogendra Prasad, Editorial Board Member, Subrahmanyam, P. V., Editor-in-Chief, Matkowski, Janusz, Editorial Board Member, Parthasarathy, Thiruvenkatachari, Editorial Board Member, Sikirić, Mathieu Dutour, Editorial Board Member, Sharma, Bhu Dev, Editorial Board Member, and Mehra, Mani

Published
2018
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22. Other Wavelet-Based Numerical Methods

Author

Mehra, Mani, Chaubey, Yogendra Prasad, Editorial Board Member, Subrahmanyam, P. V., Editor-in-Chief, Matkowski, Janusz, Editorial Board Member, Parthasarathy, Thiruvenkatachari, Editorial Board Member, Sikirić, Mathieu Dutour, Editorial Board Member, Sharma, Bhu Dev, Editorial Board Member, and Mehra, Mani

Published
2018
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23. Introduction to Numerical Methods

Author

Mehra, Mani, Chaubey, Yogendra Prasad, Editorial Board Member, Subrahmanyam, P. V., Editor-in-Chief, Matkowski, Janusz, Editorial Board Member, Parthasarathy, Thiruvenkatachari, Editorial Board Member, Sikirić, Mathieu Dutour, Editorial Board Member, Sharma, Bhu Dev, Editorial Board Member, and Mehra, Mani

Published
2018
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24. Wavelet Collocation Methods

Author

Mehra, Mani, Chaubey, Yogendra Prasad, Editorial Board Member, Subrahmanyam, P. V., Editor-in-Chief, Matkowski, Janusz, Editorial Board Member, Parthasarathy, Thiruvenkatachari, Editorial Board Member, Sikirić, Mathieu Dutour, Editorial Board Member, Sharma, Bhu Dev, Editorial Board Member, and Mehra, Mani

Published
2018
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25. Wavelets on Arbitrary Manifolds

Author

Mehra, Mani, Chaubey, Yogendra Prasad, Editorial Board Member, Subrahmanyam, P. V., Editor-in-Chief, Matkowski, Janusz, Editorial Board Member, Parthasarathy, Thiruvenkatachari, Editorial Board Member, Sikirić, Mathieu Dutour, Editorial Board Member, Sharma, Bhu Dev, Editorial Board Member, and Mehra, Mani

Published
2018
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26. Wavelet Transform

Author

Mehra, Mani, Chaubey, Yogendra Prasad, Editorial Board Member, Subrahmanyam, P. V., Editor-in-Chief, Matkowski, Janusz, Editorial Board Member, Parthasarathy, Thiruvenkatachari, Editorial Board Member, Sikirić, Mathieu Dutour, Editorial Board Member, Sharma, Bhu Dev, Editorial Board Member, and Mehra, Mani

Published
2018
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27. Wavelet-Galerkin Methods

Author

Mehra, Mani, Chaubey, Yogendra Prasad, Editorial Board Member, Subrahmanyam, P. V., Editor-in-Chief, Matkowski, Janusz, Editorial Board Member, Parthasarathy, Thiruvenkatachari, Editorial Board Member, Sikirić, Mathieu Dutour, Editorial Board Member, Sharma, Bhu Dev, Editorial Board Member, and Mehra, Mani

Published
2018
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28. Wavelets on Flat Geometries

Author

Mehra, Mani, Chaubey, Yogendra Prasad, Editorial Board Member, Subrahmanyam, P. V., Editor-in-Chief, Matkowski, Janusz, Editorial Board Member, Parthasarathy, Thiruvenkatachari, Editorial Board Member, Sikirić, Mathieu Dutour, Editorial Board Member, Sharma, Bhu Dev, Editorial Board Member, and Mehra, Mani

Published
2018
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29. Fourier Analysis

Author

Mehra, Mani, Chaubey, Yogendra Prasad, Editorial Board Member, Subrahmanyam, P. V., Editor-in-Chief, Matkowski, Janusz, Editorial Board Member, Parthasarathy, Thiruvenkatachari, Editorial Board Member, Sikirić, Mathieu Dutour, Editorial Board Member, Sharma, Bhu Dev, Editorial Board Member, and Mehra, Mani

Published
2018
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30. Preliminaries

Author

Mehra, Mani, Chaubey, Yogendra Prasad, Editorial Board Member, Subrahmanyam, P. V., Editor-in-Chief, Matkowski, Janusz, Editorial Board Member, Parthasarathy, Thiruvenkatachari, Editorial Board Member, Sikirić, Mathieu Dutour, Editorial Board Member, Sharma, Bhu Dev, Editorial Board Member, and Mehra, Mani

Published
2018
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32. Generalized fractional-order Legendre wavelet method for two dimensional distributed order fractional optimal control problem.

Author

kumar, Nitin and Mehra, Mani

Subjects
*FRACTIONAL differential equations, *BETA functions, *FRACTIONAL calculus, *LEGENDRE'S functions, *EQUATIONS, *DISTRIBUTED algorithms, *WAVELET transforms
Abstract

This paper is concerned with a two-dimensional fractional optimal control problem whose governing equations are distributed order fractional differential equations in the Caputo sense. A generalized fractional-order Legendre wavelet method has been used to solve the two-dimensional distributed-order fractional optimal control problem. An exact formula for the Riemann–Liouville integration of generalized fractional-order Legendre wavelet has been derived by using regularized beta functions. This formula and the two-dimensional Gauss–Legendre integration formula have been used to solve the two-dimensional distributed order fractional optimal control problem. Moreover, an L 2-error estimate in the approximation of an unknown function with a generalized fractional-order Legendre wavelet has been derived and the estimated order has been verified for a given function. Furthermore, convergence analysis for the proposed method has been presented. In the last, two test problems have been considered to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Published
2024
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36. Non-Overlapping Domain Decomposition for 1D Optimal Control Problems Governed by Time-Fractional Diffusion Equations on Coupled Domains: Optimality System and Virtual Controls.

Author

Leugering, Günter, Mehandiratta, Vaibhav, and Mehra, Mani

Subjects
HEAT equation, DOMAIN decomposition methods, FINITE differences, MATHEMATICAL analysis, CAPUTO fractional derivatives
Abstract

We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method. [ABSTRACT FROM AUTHOR]

Published
2024
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