Legendre Wavelet Collocation Method Research Articles

Temperature distribution in stretching/shrinking fin with variable parameters

AbstractIn this paper, we consider a mathematical model, which has a unique mechanism of heat transfer in the stretching/shrinking straight fin with an exponential profile. The thermal conductivity, internal heat generation, and heat transfer coefficient are considered temperature‐dependent. Heat is exposed to the surroundings by convection and radiation. The governing differential equation and boundary conditions are presented in a dimensionless form. In our study, we considered variable surface emissivity, that is, a constant, and the linear function of a temperature. The convective heat transfer parameter is considered a power‐low type. The novelty of this work is the application of temperature‐dependent surface emissivity, and the problem is solved by the Legendre wavelet collocation method. A comparative analysis of the present results in the context of previous findings is presented in the form of a table for validation and found exactly the same. The impacts of distinct variables are presented in the form of figures and discussed in detail. The present analysis is focused on real‐world applications and offers valuable insights for improving the design of fins.

A Legendre wavelet collocation method for solving neutral delay differential equations

The Legendre wavelet based method has been employed in this paper to investigate neutral delay differential equations. The highest order derivative is approximated by Legendre wavelet using the integral operator technique. Then integrations of Legendre wavelet are used to approximate the lower order derivatives and unknown function. To get an algebraic system of linear or nonlinear equations, approximated values of unknown function and its derivatives are substituted in neutral delay differential equations. On solving the developed system, we get unknown wavelet coefficients and subsequently the approximate solution. To analyze the theoretical usability of the approach, the upper bound of error norm is established. Moreover, the theoretical results are confirmed through few numerical experiments. A comparison of the results of presented method with method available in literature is given to conclude the superiority of the proposed method.

Legendre wavelet collocation method for investigating thermo-mechanical responses on biological tissue during laser irradiation

Objective of the present work is to propose a numerical approach based on the Legendre wavelet to address a problem of biological tissue in the presence of superficial cancer and analyze the thermo-mechanical behavior during laser irradiation. The model of thermoelasticity with variable thermal conductivity is considered to investigate transient thermoelastic coupling response in the framework of Moore–Gibson–Thompson (MGT) bioheat transfer. The Kirchhoff mapping is first performed to linearize the nonlinear governing equations due to variable thermal properties, then finite difference discretization is applied for the time domain and subsequently space domain is approximated using Legendre wavelets. The collocation points are taken to transform the system of partial differential equations into a set of algebraic equations, from which the solution can be determined. A thorough parametric analysis is carried out to examine the effects of some material and geometrical parameters on the behavior of various fields such as the displacement, temperature, stress and the tumor-normal tissue interface. The outcomes of the present work are graphically shown and compared to the predictions of other existing models. The advantages of the current numerical procedure include its simplicity, effectiveness and low computational cost even with the few collocation points.

Three-dimensional flow of hybrid nanofluid through Darcy-Forchheimer porous surface: A Legendre wavelet collocation approach

The main objective of the present study is to obtain the solution of 3D Darcy-Forchheimer flow of hybrid nanofluid (MoS2-GO/H2O) over stretchable surface using a novel Legendre Wavelet Collocation Method (LWCM). The transformed model is a semi-infinite boundary value problem of nonlinear coupled system of Ordinary Differential Equation (ODEs). Using special transformation in a space coordinate, the model is converted into finite domain of boundary value problem. Then, LWCM is implemented and problem is converted into system of algebraic equations. Now, the solution of algebraic equations is obtained by Newton-Raphson method. The ratio of volume fraction of GO and MoS2 are taken as 6% and 8% respectively. The outcomes disclose that the heat transfer rate and both drag forces regularly rises with increase in volume fraction of MoS2 and GO. Moreover, the heat transfer outlines of MoS2-GO/H2O constantly decreased when the fluid motion is in suction region, while the skin friction coefficients are increased. To validate the results obtained by the LWCM, a comparison is made with exact solution as well as with earlier published numerical results.

The Müntz–Legendre Wavelet Collocation Method for Solving Weakly Singular Integro-Differential Equations with Fractional Derivatives

We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional integration operational matrix is constructed for it. The obtained integral equation is reduced to a system of nonlinear algebraic equations using the collocation method and the operational matrix of fractional integration. The presented method’s error bound is investigated, and some numerical simulations demonstrate the efficiency and accuracy of the method. According to the obtained results, the presented method solves this type of equation well and gives significant results.

Application of Legendre wavelet collocation method to the analysis of poro-thermoelastic coupling with variable thermal conductivity

Application of Legendre wavelet collocation method to the analysis of poro-thermoelastic coupling with variable thermal conductivity

Numerical Study of a Non-Linear Porous Sublimation Problem With Temperature-Dependent Thermal Conductivity and Concentration-Dependent Mass Diffusivity

Abstract Sublimation heat transfer occurs in a wide range of engineering processes, such as accelerated freeze drying (AFD), energy storage, and food technology. Particularly in the microwave AFD process, preservation of material with the least possible energy consumption is desirable. In connection with this, it is of interest to analyze the effect of temperature/concentration dependent heat/mass transfer properties. Given the limited literature available on sublimation, there is a general lack of physical understanding of this particular problem. The present work analyzes the nonlinear sublimation process driven by convective heat/mass transfer and evaporation of water vapor using the Legendre wavelet collocation method (LWCM). Results from the present work are shown to be in excellent agreement with the exact solution of the special case of a linear problem. Further, the present numerical technique shows good agreement with finite difference method in case of a completely nonlinear model. The model is used for a comprehensive investigation of the impact of the problem parameters, on the rate of sublimation. It is found that the sublimation rate increases with increasing values of β1 and decreasing values of β2. The impact of other dimensionless problem parameters such as Péclet numbers Pe1 and Pem, convection due to mass transfer of water vapor β, latent heat of sublimation l0 and Luikov number Lu on sublimation process is also discussed in detail. These observations offer a comprehensive theoretical and mathematical understanding of sublimation heat/mass transfer for improving the performance and efficiency of freeze-drying and related engineering processes.

Convective-radiative moving porous fin with temperature-dependent thermal conductivity, heat transfer coefficient and wavelength-dependent surface emissivity

PurposeIn this paper, temperature distribution and fin efficiency in a moving porous fin have been discussed. The heat transfer equation is formulated by using Darcy's model. Heat transfer coefficient and thermal conductivity vary with temperature. The surface emissivity of the fin varies with temperature as well as with wavelength. Thermal conductivity is taken as a linear and quadratic form of temperature. The entire analysis of the paper is presented in non-dimensional form.Design/methodology/approachIn this study, a new mathematical model is investigated. The novelty of this model is surface emissivity which is considered temperature and wavelength dependent. Another interesting point is the addition of porous material. The Legendre wavelet collocation method has been used to solve the nonlinear heat transfer equation. Numerical simulations are carried out in MATLAB software.FindingsAn attempt has been made to discuss temperature distribution in the presence of porosity and wavelength-temperature-dependent surface emissivity. The effect of various parameters on temperature has been discussed, including thermal conductivity, emissivity, convection-radiation, Peclet number, sink temperature, exponent “n” and porosity. Fin efficiency is also calculated for some parameters. According to the study, heat transfer rate increases with higher radiation-convection, emissivity, wavelength and porosity parameters.Originality/valueThe numerical results are carried out by using the Legendre wavelet collocation method, which has been compared with exact results in a particular case and found to be in good agreement. The percent error is calculated to find the error between the current method and the exact result. A comparison of the obtained results with the previous data is presented to validate the numerical results.

A Legendre wavelet collocation method for 1D and 2D coupled time-fractional nonlinear diffusion system

A Legendre wavelet collocation method is proposed for solving a nonlinear coupled time fractional diffusion system. We have formulated a Riemann-Liouville fractional integral operator for Legendre wavelet (RLFIO-L) adopting the definition of Riemann-Liouville fractional integral operator combined with the Laplace transformation. Both the time and space variables are discretized in terms of the Legendre wavelet and RLFIO-L. The nonlinear coupled diffusion system is quasi-linearized by making use of the Newton's method. For theoretical concerns, the upper bound of error norm of the proposed method is estimated. Some numerical experiments are presented to authenticate the computational efficacy of the method.

A hybrid approach based on Legendre wavelet for numerical simulation of Helmholtz equation with complex solution

In this article, with the help of Legendre wavelets a hybrid approach is presented for the numerical solution of the Helmholtz equation which has a complex solution. The present hybrid approach is based on the Legendre Wavelet Collocation Method (LWCM). Initially, the Helmholtz equation is converted to the coupled equation with suitable transformation, later all the derivatives in the equations including boundary conditions are approximated with the help of Legendre Wavelets. Later, with the help of the Legendre wavelet operational matrix of integration, a system of linear algebraic equations are formed for obtaining the numerical solution of Helmholtz equation. The aforementioned proposed algorithm is very simple and can easily be executed in any computer-oriented language efficiently. To demonstrate the efficiency and performance of the newly established technique, the method is tested on various well-known examples from previous literature including other Legendre wavelet-based methods. The obtained results produce better accuracy and widespread use of the newly developed technique for a range of benchmark problems which have complex solutions.

AN ENHANCED WAVELET BASED METHOD FOR NUMERICAL SOLUTION OF HIGH ORDER BOUNDARY VALUE PROBLEMS

The Legendre wavelet collocation method (LWCM) is suggested in this study for solving high-order boundary value problems numerically. Eighth, tenth, and twelfth-order examples are used as test problems to ensure that the technique is efficient and accurate. In comparison to other approaches, the numerical results obtained using LWCM demonstrate that the method's accuracy is very good. The results indicate that the method requires less computational effort to achieve better results.

A new boundary element algorithm for a general solution of nonlinear space-time fractional dual-phase-lag bio-heat transfer problems during electromagnetic radiation

The main aim of this paper is to propose a new boundary element method (BEM) formulation for solving the nonlinear space-time fractional dual-phase-lag bio-heat transfer problems during electromagnetic radiation. Due to the advantages of BEM, such as not requiring a discretization of the interior of the treated region and providing a low RAM and CPU time. BEM is therefore a flexible and efficient tool for modeling bio-heat transfer problems. The efficiency of our proposed methodology has been improved by applying the communication-avoiding versions of the Arnoldi (CA-Arnoldi) preconditioner for solving the resulting linear systems arising from the BEM to reduce the iterations number and CPU time. Numerical results are depicted graphically to show the effects of time-fractional derivative order and space-fractional derivative order on the nonlinear temperature distributions. The numerical results also show the significant differences between the nonlinear temperature distributions of the classical Fourier, single-phase-lag, and dual-phase-lag bio-heat conduction models. To demonstrate the validity and accuracy of the proposed BEM methodology, numerical solutions for two-dimensional (2D) special case of the nonlinear space-time fractional dual phase lag bio-heat transfer problems are obtained and compared to experimental, Legendre wavelet collocation method (LWCM) and Fractional order Legendre functions and Galerkin method (FOLFs-GM).

Neuro-optimized numerical treatment of HIV infection model

In this paper, a neuro-optimized numerical method is presented for approximation of HIV virus progression model in the human body. The model is composed of coupled nonlinear system of differential equations (DEs) containing healthy and infected T-Cells and HIV free virus particles. The coupled system is transformed into feedforward artificial neural network (ANN) with Mexican hat wavelet function in the hidden layers. Two meta-heuristic algorithms based on chaotic particle swarm optimization (CPSO) and its hybrid version with local search technique are exploited to tune the parameters of ANN in an unsupervised manner of error function. A comprehensive testbed is established to observe the virus growth per day with performance metric containing fitness value, computational time complexity and convergence. The proposed solutions are compared with state of art Runge–Kutta method and Legendre Wavelet Collocation Method (LWCM). The core advantages of the proposed scheme are getting the solution on continuous grid, consistent convergence, simplicity in implementation and handling strong nonlinearity effectively.

Legendre wavelet collocation method for fractional optimal control problems with fractional Bolza cost

AbstractThis paper exhibits a numerical method for solving general fractional optimal control problems involving a dynamical system described by a nonlinear Caputo fractional differential equation, associated with a fractional Bolza cost composed as the aggregate of a standard Mayer cost and a fractional Lagrange cost given by a Riemann–Liouville fractional integral. By using the Lagrange multiplier within the calculus of variations and by applying integration by part formula, the necessary optimality conditions are derived in terms of a nonlinear two‐point fractional‐order boundary value problem. An operational matrix of fractional order right Riemann–Liouville integration is proposed and by utilizing it, the obtained two‐point fractional‐order boundary value problem is reduced into the solution of an algebraic system. AnL2‐error estimate in the approximation of unknown variable by Legendre wavelet is derived and in the last, illustrative examples are included to demonstrate the applicability of the proposed method.

A study of fractional order dual-phase-lag bioheat transfer model.

In this study, we have established a space-time fractional DPL bioheat transfer model in the presence of temperature-dependent metabolic and space-time dependent electromagnetic heat sources. Applying the Legendre wavelet collocation method, the fractional order partial differential equation is reduced into the system of algebraic equations, which has been solved using the Newton iteration method. The error bound as well as stability analysis and numerical scheme validation are provided. The time to achieve for the position of hyperthermia is discussed in three cases: the DPL model, the time-fractional DPL model, and the space-time-fractional DPL model. The effect of variability of time and space fractional derivative orders (α and β), transmitted power (P) and lagging times on the temperature profile in biological tissue at a different time are discussed in detail. We conclude that a suitable value of α, β, τT, τq, and P provides a desirable temperature at a particular time in thermal therapies. Such knowledge will be very useful in the clinical therapeutic application.

Mathematical modeling and numerical simulation of HIV infection model

In this work we implement two numerical schemes namely continuous Galerkin–Petrov (cGP(2)) and Legendre Wavelet Collocation Method (LWCM) for the approximate solution of the mathematical model which describes the behavior of CD4+ T-cells, infected CD4+ T-cells and free HIV virus particles after HIV infection. The present study discuss and analyze the effect of constant and different variable source terms (depending on the viral load) used for the supply of new CD4+ T-cells from thymus on the dynamics of CD4+ T-cells, infected CD4+ T-cells and free HIV virus. Furthermore, the model is also solve using fourth order Runge Kutta (RK4) method. Finally, the validity and reliability of the proposed schemes are verified by comparing the numerical and graphical results with the results of RK4-method. Comparison of the numerical and graphical results of cGP(2) and LWCM with RK4-method confirmed that cGP(2) and LWCM performs excellent accuracy. The present study highlights the accuracy and efficiency of the proposed schemes with the other traditional schemes such as the Laplace Adomian Decomposition Method (LADM), Variational Iteration Method (VIM), Homotopy Analysis Method (HAM), Homotopy Perturbation Method (HPM), Genetic Algorithm (GA), Interior Point Algorithm (IPA), Active Set Algorithm (ASA), Multistep Laplace Adomian Decomposition Method (MSLADM) etc.

An efficient wavelet‐based method for numerical solution of nonlinear integral and integro‐differential equations

In this work, Legendre wavelet collocation method is implemented for the numerical solution of nonlinear integral and integro‐differential equations. We approximate the solution with Legendre wavelet. The evaluations of integrals has been carried out with the help of Gaussian integration formula. The system of equations is obtained using collocation approach. The system of equations is then solved to obtain the desire solution. The method works in a direct way, which does not need any discretization, linearization, or perturbation. To check the efficiency of the scheme, we match the computed results with those available in literature. From comparison, it has been concluded that obtained results are better than those presented before and require less computational cost.

Modeling static microstructure of shape memory alloy via Legendre wavelets collocation method

The static microstructures of square-to-rectangular phase transformations are modeled in the current paper. Governing equations are described by the static Navier equations. The analysis of the simulation has been finished via Legendre wavelets collocation method. Microstructure evolutions of both 2D and 1D shape memory alloy (SMA) are capture well with a certain boundary condition and given mechanical loadings.

The impact of Legendre wavelet collocation method on the solutions of nonlinear system of two-dimensional integral equations

Numerical solutions of nonlinear system of two-dimensional integral equations have been rarely investigated in the literature. In this study, we suggest a numerically practical algorithm to approximate the solutions of nonlinear system of two-dimensional Volterra–Fredholm and Volterra integral equations. This scheme is based on two-dimensional Legendre wavelet to reduce these nonlinear systems of integral equations to a system of nonlinear algebraic equations. The main characteristic of this approach is high accuracy and computational efficiency of performing which are the consequences of Legendre wavelet properties. The main benefit of this basic function is their ability to detect singularities and their efficiency in dealing with non-sufficiently smooth function in comparison with Legendre polynomials and they minimize the error. The convergence analysis and error bound of the proposed Legendre wavelet method is investigated. Numerical examples confirm that the Legendre wavelet collocation method is accurate, reliable for solving nonlinear system of two-dimensional integral equations.

Modeling microstructure evolution in shape memory alloy rods via Legendre wavelets collocation method

Microstructures play an important role in the research on shape memory alloys (SMA). By using mathematical modeling tools to study microstructures, it is possible to predict the behaviors of these materials under applied fields. In the current paper, a thermo-mechanical model is proposed to simulate the microstructure of a SMA rod with both fixed and impact boundary conditions via Legendre wavelets collocation method. Because of the good performance of Legendre wavelets basis, this method shows excellent properties in both precision and stability. A detailed numerical algorithm is given, and the backward differentiation formula is employed to perform all the simulations in the current paper. Computational simulations are carried out to study the stress-induced phase transformation. The dynamics of microstructure evolution, presented at different temperatures, is well captured by our developed technique.