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Amit K. Verma

Publications and source records attributed to Amit K. Verma.

At least 19 recordsLinked to original sources
arXiv2026

Dynamics of Unemployment with Discouraged Workers: A Nonlinear Mathematical Model

In this article, we formulate and analyze a new non-linear mathematical model to describe the dynamics of unemployment with a discouraged working population. We consider five dynamic variables, namely, unskilled unemployed individuals, skilled unemployed individuals, discouraged individuals, employed persons, and job vacancies. Furthermore, we determine the equilibrium points of the dynamic system and investigate their local stability. To demonstrate the results, we conduct numerical simulations by presenting solution trajectories and analyzing how variations in key parameters influence the states of the dynamical variables.

physics.soc-ph↗
arXiv2025

A Novel Two-Dimensional Wigner Distribution Framework via the Quadratic Phase Fourier Transform with a Non-Separable Kernel

This paper introduces a novel time-frequency distribution, referred to as the Two-Dimensional Non-Separable Quadratic Phase Wigner Distribution (2D-NSQPWD), formulated within the framework of the Two-Dimensional Non-Separable Quadratic Phase Fourier Transform (2D-NSQPFT). By replacing the classical Fourier kernel with the NSQPFT kernel, the proposed distribution generalizes the classical Wigner distribution and effectively captures complex, non-separable signal structures. We rigorously establish several key properties of the 2D-NSQPWD, including time and frequency shift invariance, marginal behavior, conjugate symmetry, convolution relations, and Moyal's identity. Furthermore, the connection between the 2D-NSQPWD and the two-dimensional short-time Fourier transform (2D-STFT) is explored. The distribution's effectiveness is demonstrated through its application to single-, bi-, and tri-component two-dimensional linear frequency modulated (2D-LFM) signals, where it shows superior performance in cross-term suppression and signal localization.

eess.SP↗
arXiv2025

Special-Affine Wavelets: Multi-Resolution Analysis and Function Approximation in L^2(R)

The multiresolution analysis (MRA) associated with the Special affine Fourier transform (SAFT) provides a structured approach for generating orthonormal bases in \( L^2(\mathbb R) \), making it a powerful tool for advanced signal analysis. This work introduces a robust sampling theory and constructs multiresolution structures within the SAFT domain to support the formation of orthonormal bases. Motivated by the need for a sampling theorem applicable to band-limited signals in the SAFT framework, we establish a corresponding theoretical foundation. Furthermore, a method for constructing orthogonal bases in $L^2(\mathbb R)$ is proposed, and the theoretical results are demonstrated through illustrative examples.

math.FA↗
arXiv2025

Generalized Non-Standard Finite Difference Method for Fractional PDEs on Non-Uniform Grids

This paper proposes a novel Generalized Non-Standard Finite Difference (GNSFD) scheme for the numerical solution of a class of fractional partial differential equations (FrPDEs). The formulation of the method is grounded in optimization and leverages the fractional Taylor series (FrTS) expansion associated with Caputo fractional derivatives (FrDs). To discretize the time derivatives, the non-trivial denominator functions are utilized. The theoretical analysis establishes the consistency, stability, and convergence of the proposed scheme. Results are compared against existing methods to substantiate the accuracy and computational efficiency of the approach.

math.NA↗
arXiv2025

On a class of coupled fractional nonlinear singular boundary value problems arising in dusty fluid models

In this article, we introduce a new class of coupled fractional Lane-Emden boundary value problems. We employ a novel approach, the fractional Haar wavelet collocation method with the Newton-Raphson method. We analyze the conditions in two cases to present numerical experiments related to the defined system of fractional differential equations. To validate the accuracy of the proposed method we present the convergence of the method, and we demonstrate the method's effectiveness through five numerical experiments, highlighting real-world applications of fractional differential equations. Using figures and tables, we show that the residual error decreases as we increase the value of the maximum level of resolution $J$ while keeping the order of derivatives fixed, and similar trends also observe when $J$ is fixed and vary the order of fractional derivatives. We demonstrate that Mathematica software can be used effectively to solve such nonlinear singular fractional boundary value problems.

math.GM↗
arXiv2024

Uniform Haar Wavelet Solutions for Fractional Regular $β$-Singular BVPs Modeling Human Head Heat Conduction under Febrifuge Effects

This paper introduces nonlinear fractional Lane-Emden equations of the form, $$ D^α y(x) + \fracλ{x^β}~ D^β y(x) + f(y) =0, ~ ~1 < α\leq 2, ~~ 0< β\leq 1, ~~ 0 < x < 1,$$ subject to boundary conditions, $$ y'(0) =\mathbf{a} , ~~~ \mathbf{c}~ y'(1) + \mathbf{d}~ y(1) = \mathbf{b},$$ where, $D^α, D^β$ represent Caputo fractional derivative, $\mathbf{a, b,c,d} \in \mathbb{R}$, $ λ= 1, 2$, and $f(y)$ is non linear function of $y.$ We have developed collocation method namely, uniform fractional Haar wavelet collocation method and used it to compute solutions. The proposed method combines the quasilinearization method with the Haar wavelet collocation method. In this approach, fractional Haar integrations is used to determine the linear system, which, upon solving, produces the required solution. Our findings suggest that as the values of $(α, β)$ approach $(2,1),$ the solutions of the fractional and classical Lane-Emden become identical.

math.NA↗
arXiv2024

A mathematical survey on Fourier type integral transform and their offshoots: windowed Fourier transform, wavelet transform and Stockwell transform

This comprehensive review paper delves into the intricacies of advanced Fourier type integral transforms and their mathematical properties, with a particular focus on fractional Fourier transform (FrFT), linear canonical transform (LCT), quadratic phase Fourier transform (QPFT), and their associated offshoots: windowed Fourier transform, wavelet transform, and Stockwell transform. In the pursuit of a deeper understanding of these transformations, we explore their convolution properties, shedding light on their capacity to define windowed, wavelet and Stockwell transforms in the realm of Fourier, fractional Fourier and quadratic phase Fourier transforms. This review also expands its purview to the realm of uncertainty principles. Several uncertainty principles, like Heisenberg, logarithmic, local, Rényi uncertainty principles, etc., within the context of fractional Fourier, linear canonical, and quadratic phase Fourier transforms, as well as their derivative offshoots are presented in the paper both for the functions of complex as well as quatenrion valued. In particular, the counterpart of several important inequalities of classical Fourier transform are also presented in details for the quaternion case. This article also reviews that multiresolution analysis that has been developed in the literature so far.

math.CA↗
arXiv2023

Uncertainty principles associated with the short time quaternion coupled fractional Fourier transform

In this paper, we extend the coupled fractional Fourier transform of a complex valued functions to that of the quaternion valued functions on $\mathbb{R}^4$ and call it the quaternion coupled fractional Fourier transform (QCFrFT). We obtain the sharp Hausdorff-Young inequality for QCFrFT and obtain the associated Rènyi uncertainty principle. We also define the short time quaternion coupled fractional Fourier transform (STQCFrFT) and explore its important properties followed by the Lieb's and entropy uncertainty principles.

math.GM↗
arXiv2022

Linear Canonical Stockwell Transform and the associated Multiresolution Analysis

In this article, we give a new definition of the linear canonical Stockwell transform (LCST) and study its basic properties along with the inner product relation, reconstruction formula and also characterize the range of the transform and show that its range is the reproducing kernel Hilbert space. We also develop a multiresolution analysis (MRA) associated with the proposed transform together with the construction of the orthonormal basis for $L^2(\mathbb{R}).$

math.FA↗
arXiv2022

Short time quaternion quadratic phase Fourier transform and its uncertainty principles

In this paper, we extend the quadratic phase Fourier transform of a complex valued functions to that of the quaternion valued functions of two variables. We call it the quaternion quadratic phase Fourier transform (QQPFT). Based on the relation between the QQPFT and the quaternion Fourier transform (QFT) we obtain the sharp Hausdorff-Young inequality for QQPFT. We define the short time quaternion quadratic phase Fourier transform (STQQPFT) and explore some of its properties including inner product relation and inversion formula. We find its relation with that of the 2D quaternion ambiguity function and the quaternion Wigner-Ville distribution associated with QQPFT and obtain the Lieb's uncertainty and entropy uncertainty principles for these three transforms.

eess.SP↗
arXiv2022

Linear canonical wavelet transform and the associated uncertainty principles

We define a novel time-frequency analyzing tool, namely linear canonical wavelet transform (LCWT) and study some of its important properties like inner product relation, reconstruction formula and also characterize its range. We obtain Donoho-Stark's and Lieb's uncertainty principle for the LCWT and give a lower bound for the measure of its essential support. We also give Shapiro's mean dispersion theorem for the proposed LCWT.

math.FA↗
arXiv2022

Multidimensional Fractional Wavelet Transforms and Uncertainty Principles

In this paper, we have given a new definition of continuous fractional wavelet transform in $\mathbb{R}^N$, namely the multidimensional fractional wavelet transform (MFrWT) and studied some of the basic properties along with the inner product relation and the reconstruction formula. We have also shown that the range of the proposed transform is a reproducing kernel Hilbert space and obtain the associated kernel. We have obtained the uncertainty principle like Heisenberg's uncertainty principle, logarithmic uncertainty principle and local uncertainty principle of the multidimensional fractional Fourier transform (MFrFT). Based on these uncertainty principles of the MFrFT we have obtained the corresponding uncertainty principles i.e., Heisenberg's, logarithmic and local uncertainty principles for the proposed MFrWT.

math.FA↗
arXiv2021

A generalized novel approach based on orthonormal polynomial wavelets with an application to Lane-Emden equation

Capturing solution near the singular point of any nonlinear SBVPs is challenging because coefficients involved in the differential equation blow up near singularities. In this article, we aim to construct a general method based on orthogonal polynomials as wavelets. We discuss multiresolution analysis for wavelets generated by orthogonal polynomials, e.g., Hermite, Legendre, Chebyshev, Laguerre, and Gegenbauer. Then we use these wavelets for solving nonlinear SBVPs. These wavelets can deal with singularities easily and efficiently. To deal with the nonlinearity, we use both Newton's quasilinearization and the Newton-Raphson method. To show the importance and accuracy of the proposed methods, we solve the Lane-Emden type of problems and compare the computed solutions with the known solutions. As the resolution is increased the computed solutions converge to exact solutions or known solutions. We observe that the proposed technique performs well on a class of Lane-Emden type BVPs. As the paper deals with singularity, non-linearity significantly and different wavelets are used to compare the results.

math.NA↗
arXiv2019

A note on continuous fractional wavelet transform in $\mathbb{R}^n$

In this paper, we have studied continuous fractional wavelet transform (CFrWT) in $n$-dimensional Euclidean space $\mathbb{R}^n$ with dilation parameter $\boldsymbol a=(a_{1},a_{2},\ldots,a_{n}),$ such that none of $a_{i}'s$ are zero. Necessary and sufficient condition for the admissibility of a function is established with the help of fractional convolution. Inner product relation, reconstruction formula and the reproducing kernel for the CFrWT depending on two wavelets are obtained. Heisenberg's uncertainty inequality and Local uncertainty inequality for the CFrWT are obtained. Finally, boundedness of the transform on the Morrey space $L^{1,ν}_{M}(\mathbb{R}^n)$ and the estimate of $L^{1,ν}_{M}(\mathbb{R}^n)$-distance of the CFrWT of two argument functions with respect to different wavelets are discussed.

math.FA↗
arXiv2019

An improved Haar wavelet quasilinearization technique for a class of generalized Burger's equation

Solving Burgers' equation always poses challenge to researchers as for small values of viscosity the analytical solution breaks down. Here we propose to compute numerical solution for a class of generalised Burgers' equation described as $$ \frac{\partial w}{\partial t}+ w^μ\frac{\partial w}{\partial x_{*}}=νw^δ\frac{\partial^2 w}{\partial x^{2}_{*}},\hspace{0.2in}a \leq x_{*}\leq b,~~t\geq 0,$$ based on the Haar wavelet (HW) coupled with quasilinearization approach. In the process of numerical solution, finite forward difference is applied to discretize the time derivative, Haar wavelet to spatial derivative and non-linear term is linearized by quasilinearization technique. To discuss the accuracy and efficiency of the method $L_{\infty}$ and $L_{2}$-error norm are computed and they are compared with some existing results. We have proved the convergence of the proposed method. Computer simulations show that the present method gives accurate and better result even for small number of grid points for small values of viscosity.

math.NA↗
arXiv2019

System of Lane-Emden equations as IVPs BVPs and Four Point BVPs & Computation with Haar Wavelets

In this work we present Haar wavelet collocation method and solve the following class of system of Lane-Emden equation defined as \begin{eqnarray*} -(t^{k_1} y'(t))'=t^{-ω_1} f_1(t,y(t),z(t)),\\ -(t^{k_2} z'(t))'=t^{-ω_2} f_2(t,y(t),z(t)), \end{eqnarray*} where $t>0$, subject to initial values, boundary values and four point boundary values: \begin{eqnarray*} \mbox{Initial Condition:}&&y(0)=γ_1,~y'(0)=0,~z(0)=γ_2,~z'(0)=0,\\ \mbox{Boundary Condition:}&&y'(0)=0,~y(1)=δ_1,~z'(0)=0,~z(1)=δ_2,\\ \mbox{Four~point~Boundary~Condition:}&&y(0)=0,~y(1)=n_1z(v_1),~z(0)=0,~z(1)=n_2y(v_2), \end{eqnarray*} where $n_1$, $n_2$, $v_1$, $v_2$ $\in (0,1)$ and $k_1\geq 0$, $k_2\geq0$, $ω_1<1$, $ω_2<1$ are real constants. Results are compared with exact solutions in the case of IVP and BVP. In case of four point BVP we compare the result with other methods. Convergence of these methods is also established and found to be of second order. We observe that as resolution is increased to $J=4$ we get the exact values for IVPs and BVPs. For four point BVPs also at $J=4$, we get highly accurate solutions, e.g., the $L^\infty$ error is of order $10^{-16}$ or $10^{-17}$.

math.NA↗