Hermite Wavelet Research Articles

Operational matrix method approach for fractional partial differential-equations

Abstract Fractional partial differential equations (FPDEs) have become very popular to model and analyze numerous different physical phenomena in recent years. However, it is generally complicated to find the exact solutions of those FPDEs. The objective of this study is to find the approximate numerical solution of FPDEs by introducing a wavelet-based operational matrix technique. In this study we employ Hermite wavelets (HWs) and the operational matrices of the fractional integration for Hermite wavelets. The sparsity of the obtained operational matrices provides fast and efficient computation of the proposed method. The original FPDE equations are converted to Sylvester equations, which then are solved to obtain the final solution. We provide a few numerical examples to demonstrate the versatility and efficiency of the proposed method.

MHD Mixed Convection Boundary Layer Casson Nanofluid Flow over an Exponential Stretching Sheet

This paper investigates the effects of radiation, internal heat source and magnetohydrodynamics (MHD) on the mixed convective boundary layer flow of a Casson nanofluid within a porous medium that is saturated and subject to an exponentially stretching sheet. The nanofluid model incorporates Brownian motion and thermophoresis, and the Darcy model is employed for the porous medium. By applying an appropriate similarity transformation, the nonlinear governing boundary layer equations are converted into a set of nonlinear coupled ordinary differential equations. These equations are then solved numerically using the Hermite wavelet method, with simulations conducted through the MATHEMATICA programming language. The analysis covers various aspects including temperature distribution, velocity, solute concentration and several engineering parameters such as skin friction coefficients, the Nusselt number (rate of heat transfer) and the Sherwood number (rate of mass transfer), all evaluated based on dimensionless physical parameters. The results indicate that elevated radiation intensifies temperatures and leads to thicker thermal boundary layers. As the Casson parameter increases, both the velocity and the momentum boundary layer become narrower. Additionally, a more pronounced chemical reaction rate reduces the thickness of the solutal boundary layer. The accuracy and reliability of the numerical Hermite wavelet method are validated through a comparative analysis with previous studies, demonstrating excellent concordance and confirming the robustness of the computational approach.

Modified Hermite Wavelet Discrete Matrix Approach for the Brusselator Chemical Model

AbstractThe primary goal of this study is to create a wavelet collocation technique that can be used to solve nonlinear fractional order systems of ordinary differential equations, which are equations that arise in modeling problems related to auto‐catalytic chemical reactions. Using the Hermite wavelet collocation method (HWCM), the system of nonlinear ordinary differential equations of integer and fractional order is numerically solved. The nonlinear Brusselator system is transformed into an algebraic equation system using the collocation method and the fractional derivative operational matrices. The Newton‐Raphson method is used to solve these algebraic equations, and the approximate values of the derived unknown coefficients are substituted. Through the numerical examples, the method's computational effectiveness and correctness are illustrated with various model constraints. A numerical comparison is made between the current approach ND solver, RK method, and Haar wavelet method (HWM). The efficiency and reliability of the developed strategy's performance are shown in graphs and tables. The created Hermite wavelet collocation method is resilient and has good accuracy compared to current methods found in the literature. Numerical computations are performed through Mathematica, a mathematical software.

Thermohaline convection in MHD Casson fluid over an exponentially stretching sheet

Abstract This study investigates the thermohaline convection in MHD Casson fluid over an exponentially stretching sheet. This study has practical significance in industrial processes, materials processing, energy systems, and environmental applications. The governing equations describing the conservation for an electrically conducting fluid flow, thermal and concentration transports are considered based on the principles of mass, momentum, energy and concentration equations. Our first step involves transforming the governing nonlinear partial differential equations into a coupled nonlinear ordinary differential equations with the help of suitable similarity transformations. Second step, infinite domain [0, ∞) of the problem to a finite domain [0, 1] through a coordinate transformations. This specific choice is motivated by the wavelet's significance in the finite domain of [0, 1]. Third step, we effectively solve the resulting coupled nonlinear ordinary differential equations using the numerical Hermite wavelet method (HWM). This approach proves to be a valuable technique for obtaining significant results and insights in our study. Finally, the effect of known physical parameters on velocity, temperature and concentration are analysed through tables and graphs.

Simultaneous space–time Hermite wavelet method for time-fractional nonlinear weakly singular integro-partial differential equations

An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at t=0. In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the L2 norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.

Investigation of Third-Grade fluid flow in an inclined Microchannel: Utilizing the Hermite wavelet technique for second law analysis

Microfluidics serves as an interdisciplinary bridge between thermal and engineering fields, offering flexible solutions for addressing heat transport obstacles across diverse sectors. This study explores entropy production in the flow of a third-grade fluid through an inclined microchannel, considering radiation and convective heating effects. The analysis includes various factors such as viscous dissipation, natural convection, Joule heating, magnetic fields, and a uniform heat source/sink. The aim is to assess the energy efficiency of the system by minimizing entropy generation. To achieve this, governing equations are transformed into dimensionless forms using non-dimensional parameters. These equations are then solved using the Hermite wavelet method, a novel approach in this field. The impact of the Biot number on temperature exhibits a dual nature: temperature increases towards the upper plate of the channel while decreasing towards the lower plate. Similarly, entropy generation displays a dual behavior for higher values of Qt and Re, decreasing at the lower plate and increasing at the upper plate. This research offers valuable insights for optimizing energy efficiency in various applications, including pumps, heat exchangers, and energy systems, with potential applications in electronics cooling and renewable energy collection.

Wavelet Collocation Method for HIV-1/HTLV-I Co-Infection Model Using Hermite Polynomial.

In this study, the dynamic behavior of fractional order co-infection model with human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) is analyzed using operational matrix of Hermite wavelet collocation method. Also, the uniqueness and existence of solutions are calculated based on the fixed point hypothesis. For the fractional order co-infection model, its positivity and boundedness are demonstrated. Furthermore, different types of Ulam-Hyres stability are also discussed. The numerical solution of the model are obtained by using the operational matrix of the Hermite wavelet approach. This scheme is used to solve the system of nonlinear equationsthat are very fruitful and easy to implement. Additionally, the stability analysis of the numerical scheme is explained. The mathematical model taken in this work incorporates the biological characteristics of both HIV-1 and HTLV-I. After that all the equilibrium points of the fractional order co-infection model are found and their existence conditions are explored with the help of the Caputo derivative. The global stability of all equilibrium points of this model are determined with the help of Lyapunov functions and the LaSalle invariance principle. Convergence analysis is also discussed. Hermite wavelet operational matrix methods are more accurate and convergent than other numerical methods. Lastly, variations in model dynamics are found when examining different fractional order values. These findings will be valuable to biologists in the treatment of HIV-1/HTLV-I.

Mixed Convection Over an Exponentially Stretching Surface with the Effect of Radiation and Reaction

This study investigates the mixed convection on a vertically oriented exponentially stretching surface, considering the exponential distributions of velocity, concentration and temperature. The governing highly nonlinear partial differential equations (PDEs) are transmuted into highly nonlinear ordinary differential equations (ODEs) by considering appropriate similarity variables. The highly nonlinear differential equations are solved numerically using the Hermite wavelet method (HWM) in MATHEMATICA 12. The obtained results show excellent agreement with previous studies focusing on different special cases of the problem. A parametric analysis of the physical parameters is shown and illustrative numerical results are presented graphically. The analysis incorporates the radiation, chemical reaction and cross-diffusion effects. The interaction between radiation and chemical reaction is particularly relevant in high-temperature systems, such as combustion, industrial furnaces, photochemistry, polymerization and certain chemical reactions. The results indicate that an increase in radiation enhances the temperature profile in the momentum boundary layer. Higher reaction rate values lead to a decrease in the concentration profile, while an increase in the Dufour value results in increased velocity and temperature and decreased concentration.

An analysis of best wavelet approximation problem of a function using Hermite wavelet

In this paper, we introduce two novel wavelet approximations tailored for functions exhibiting a restricted second derivative and a bounded derivative. Employing the Hermite wavelet method, we derive these approximations to address the need for effective representations of such functions. Our findings reveal that these new wavelet approximations offer enhanced accuracy and efficiency in capturing the underlying structure of , making them valuable tools in various applications requiring precise function approximations with limited derivative constraints.

An adaptive algorithm for numerically solving fractional partial differential equations using Hermite wavelet artificial neural networks

This study aims to develop a new strategy for solving partial differential equations with fractional derivatives (FPDEs) using artificial neural networks (ANNs). Numerical solutions to FPDEs are obtained through the Hermite wavelet neural network (HWNN) model. The Caputo fractional derivative is consistently applied throughout the research to address fractional-order partial differential problems. To enhance computational efficiency and expand the input pattern, the hidden layer is removed. A neural network (NN) model featuring a feed-forward architecture and error-back propagation without supervision is employed to optimize network parameters and minimize errors. Numerical illustrations are presented to demonstrate the effectiveness of this approach in preserving computational efficiency while solving FPDEs.

The use of Hermite wavelet collocation method for fractional cancer dynamical system

Tumour is a very serious and dangerous health risk, which is defined as a mass or lump of tissue formalized by the aggregation of abnormal cells. In this paper, we have presented a comparative and chaotic investigation of tumour and effector cells using non-integer order tumour immune dynamical model. Also, we have looked at the interactions between different tumour cell populations and immunological composition using a model of a real-world medical research problem. Additionally, stability analysis of the proposed model is established with the help of fixed-point iteration. Further bifurcation diagrams, as well as phase portraits, have been used to study the proposed system numerically and to analyse its behaviour. For numerical simulations, we have used the Hermite wavelet operational matrix and Taufik-Atangana methods using Caputo fractional derivatives. models. The main objective of the paper is to convert nonlinear fractional systems into algebraic equations. Our findings will be valuable to biologists in cancer treatment.

New algorithm based on Hermite wavelets and Green’s function to solve Thomas-Fermi type differential equations

This article introduces a novel method using Hermite wavelets and Green’s function to solve Thomas-Fermi-type equations with specific boundary conditions. The equation is first transformed into an integral form and then expanded using Hermite wavelets to create an algebraic expression. The collocation technique generates a set of nonlinear equations, solved iteratively by methods like Newton-Raphson. Several examples demonstrate the method’s effectiveness and accuracy. It is useful for a range of linear and nonlinear differential equations in various fields. Comparative analysis shows it is more accurate and computationally efficient than traditional methods. All computations are performed using MAT LAB.

Effects of heat transfer on MHD suction–injection model of viscous fluid flow through Differential transformation and Hermite wavelet techniques

Effects of heat transfer on MHD suction–injection model of viscous fluid flow through Differential transformation and Hermite wavelet techniques

Cross diffusion effects on MHD double diffusive viscous flow through Hermite wavelet method

AbstractDouble-diffusive convection is a form of fluid flow that occurs when two processes of molecular diffusion are active in a fluid at the same time, causing instabilities and also complicated behaviour. One chemical or biological species concentration can cause a flux of another species, either linearly or nonlinearly, a phenomenon known as cross-diffusion. The cross-diffusion effects on double-diffusive MHD fluid flow through the Hermite wavelet method is examined. The governing coupled partial differential equations of the problem under consideration are transformed to highly nonlinear ordinary differential equations over a finite domain with the help of similarity transformations. The results are obtained for the skin friction coefficient, as well as the velocity, temperature and the concentration profiles for some values of the governing parameters, namely, the cross diffusion terms, Hartmann number, thermophoresis parameter, squeeze number, Prandtl number and suction/injection parameter. The obtained results are validated against previously published results for special case of the problems.

Numerical solution of fractional PDEs through wavelet approach

To solve fractional partial differential equations (FPDEs) under various physical conditions, this study developed a novel method known as the Hermite wavelet method employing the functional integration matrix. The method that has been suggested is based on the Hermite wavelet collocation process. To determine the solution of the fractional differential equations, the Caputo fractional derivative operator of order α∈(0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1]$$\\end{document} is used. With the use of appropriate grid points, this method converts FPDEs into a system of nonlinear algebraic equations. We achieve a solution by solving these nonlinear algebraic equations by the Newton–Raphson method. Tables and graphs show that the suggested method produces superior results. We provide various illustrative examples to establish the effectiveness of the suggested concept, and the outcomes support the applicability of the suggested strategy. Obtained results are numerically expressed in terms of absolute errors. Finally, convergence analyses are discussed as some theorem with proof.

Time-varying stretching velocity analysis for an unsteady flow of Williamson fluid by Hermite wavelet

This study investigates unsteady velocity Uw=ξx/t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${U}_{w}=\\xi x/t$$\\end{document} for a Williamson nanofluid film flowing over a moving surface. This work can be used to outline the effects of an applied angled magnetic-field on liquid film flow, which occurs in numerous real-world solicitations such as coating industries for wire or sheet, labs, painting, and several others. Analyzing williamson nanoliquid film flow over a stretching sheet is the main aim of this investigation. The leading Navier–Stokes models are reduced to third-order nonlinear ODE through similarity transformations that are then undertaken using the Hermite wavelet method (HWM). Both 2-dimensional and axisymmetric film flow circumstances have been analyzed. The moving surface parameter ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi$$\\end{document} is said to have a limited range for which the solution exists. Specifically, ξ≤-1/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi \\le -1/4$$\\end{document} for axisymmetric flow and ξ≥-1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi \\ge -1/2$$\\end{document} for two-dimensional flow. Before decreasing to the boundary condition, the velocity climbs until it reaches its maximum. By taking into account the stretching (ξ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi >0$$\\end{document}) and shrinking (ξ<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi <0$$\\end{document}) wall conditions, streamlines are also examined for axisymmetric and 2-dimensional flow patterns.

A Study of Nanofluid Flow with Free Bio-Convection in 3D Nearby Stagnation Point by Hermite Wavelet Technique

A new wavelet-numerical method for solving a system of partial differential equations describing an incompressible bio-convection nanofluid flow in a three-dimensional region close to the stagnation point is the primary focus of this article. Hermite wavelets form the basis of the algorithm. An assortment of similitude factors is utilized to improve on the overseeing conditions addressing the protection of all out mass, force, nuclear power, nanoparticles, and microorganisms to a bunch of completely connected nonlinear common differential conditions. The most important physical quantities that have a practical impact on the spread of motile bacteria are presented and analyzed in this paper. During bio-convection, the Prandtl, Lewis, Peclet, Schmidt, and Rayleigh numbers can alter the distribution of moving molecules. The dispersion of microorganisms can be emphatically affected by the kinds of nanoparticles and by the varieties in the temperature as well as volumetric part of the nanoparticles between the wall and the encompassing liquid. With excellent agreement for coupled nonlinear differential equations in engineering applications, our result demonstrates how powerful and simple the HWM is for solving these coupled nonlinear ordinary differential equations.

Solution of generalized fractional Jaulent–Miodek model with uncertain initial conditions

This paper analyses a coupled system of generalized coupled system of fractional Jaulent–Miodek equations, including uncertain initial conditions with fuzzy extension. In this regard, an extension of the homotopy with a generalized integral algorithm is adopted for a class of time-fractional fuzzy Jaulent–Miodek models by mixing the fuzzy q-homotopy analysis algorithm with a generalized integral transform and Caputo fractional derivative. The triangular fuzzy numbers (TFNs)are expressed in double parametric form using κ-cut and r-cut and utilized to explain the uncertainties arising in the initial conditions of highly nonlinear differential equations with generalized Hukuhara differentiability (gH-differentiability). The TFNs are controlled by the κ-cut and r-cut, and the variability of uncertainty is examined using a “triangular membership function” (TMF). The results are analyzed by finding the solutions for different spatial coordinate values of time with κ-cut and r-cut for both lower and upper bounds and validated through numerical and graphical representations in crisp cases. Finally, it can be seen that the uncertain probability density function rapidly decreases at the left and right edges when the fractional order is increased, and it is observed that the obtained solutions are more accurate than the existing results through the Hermite wavelet method in the literature.

An optimal control model for Covid-19 spread with impacts of vaccination and facemask

A non-linear system of differential equations was used to explain the spread of the COVID-19 virus and a SEIQR model was developed and tested to provide insights into the spread of the pandemic. This article, which is related to the aforementioned work as well as other work covering variations of SIR models, Hermite Wavelets Transform, and also the Generalized Compartmental COVID-19 model, we develop a mathematical control model and apply it to represent optimal vaccination strategy against COVID-19 using Pontryagin's Maximum Principle and also factoring in the effect of facemasks on the spread of the virus. As background work, we analyze the mathematical epidemiology model with the facemask effect on both reproduction number and stability, we also analyze the difference between confirmed COVID-19 cases of the Quarantine class and anonymous cases of the Infectious class that is expected to recover. We also apply control theory to mine insights for effective virus spread prevention strategies. Our models are validated using Matlab mathematical model validation tools. Statistical tests against data from Jordan are used to validate our work including the modeling of the relation between the facemask effect and COVID-19 spread. Furthermore, the relation between control measure ξ, cost, and Infected cases is also studied.

Hermite wavelets collocation method for solving a fredholm integro-differential equation with fractional Caputo-Fabrizio derivative

In this paper, we investigate the numerical study of nonlinear Fredholm integro-differential equation with the fractional Caputo-Fabrizio derivative. We use the Hermite wavelets and collocation technique to approximate the exact solution by reducing the Fredholm integro-differential equation to a nonlinear algebraic system. Furthermore, we apply this numerical method on certain examples to check its accuracy and validity.