Legendre Wavelet Galerkin Method Research Articles

A New Iterative Broyden Legendre Wavelet Galerkin FEM Applied to Unsteady State Model of Two-Dimensional Elliptic Fin

Abstract The novelty of this paper is the investigation of numerical study of a mathematical model which deals with time dependent heat flow in elliptic fin ( dry, wet and partially wet). In this paper, we developed a nonlinear model of second-order heat equations in unsteady state condition. A new iterative Broyden Legendre Wavelet Galerkin Finite Element Method is used for the solution. The central difference approximation used for discretization of second order derivatives and then utilization of Hadamard, Khatri Rao and Face splitting matrices product with Legendre Wavelet Galerkin Method transfers our main problem in to system of non-linear algebraic equations. The iterative Broyden Method provides the solution for this system. In a particular case, present solution is compared with the exact solution and is approximately the same. Effect of different parameters such as Biot number, Latent heat, Kirpichev number, Fin thickness, μ, η and ξ on the temperature distribution are discussed in detail.

LANDAU LEGENDRE WAVELET GALERKIN METHOD APPLIED TO STUDY TWO-PHASE MOVING BOUNDARY PROBLEM OF HEAT TRANSFER IN FINITE REGION

In this paper, we developed a mathematical model of solidification where specific heat and thermal conductivity are temperature dependent. This model is a two-phase moving boundary problem (MBP) of heat transfer in finite region and represents as MBP of system of parabolic nonlinear second-order partial differential equations (PDEs). We developed a Landau Legendre wavelet Galerkin method for finding the solution of the problem. The MBP of a system of PDEs is transformed into a variable boundary value problem of nonlinear ordinary defferential equations (ODEs) by the use of dimensionless variables and the Landau transform. The problem is converted into a system of algebraic equations with the application of Legendre wavelet Galerkin method. In particular case, we compared present solution with Laplace transform solution and found approximately the same. The whole investigation has been done in dimensionless form. When the specific heat and thermal conductivity exponentially vary in temperatures, it is the effect of dimensionless parameters: thermal diffusivity (α<sub>12</sub>), ratio of thermal conductivity (<i>k</i><sub>12</sub>), dimensionless temperature (θ<sub><i> f</i></sub>), Fourier number (F<sub>0</sub>), Stefan number (Ste), and ratio of densities (ρ<sub>1</sub> / ρ<sub>2</sub>) are discussed in detail.

Mathematical modelling and simulation of three phase lag bio-heat transfer model during cancer treatment

We have developed a two-dimensional mathematical model that described the study of bio-heat transfer. Our model is an initial boundary value problem of partial differential equation. The solution consists of the three step procedure- (i) transformation of problem in dimensionless form (ii) by using finite differences, the problem converted into ordinary matrix differential equation (iii) applying Legendre wavelet Galerkin method, the problem is transferred into the generalised system of Sylvester equations which are solved by applying Bartels-Stewart Algorithm of generalised inverse. We have used this method to determine the temperature profile in three different boundary conditions. The consequence of boundary conditions on temperature profile are discussed in detail. The effect of phase lag due to heat flux, phase lag due to temperature gradient and phase lag due to thermal displacement have been observed. And, we have seen that temperature profile increases when the phase lags decreases. We have also observed the effect of blood perfusion rate and metabolic heat generation in specific. Results are validated with exact results in particular case.

A numerical study of moving boundary problem involving dual phase lag model of heat mass transfer during immersion frying

In this paper, we developed a dual phase lag model of heat mass transfer during immersion frying of foods. This model is a moving boundary problem of a couple system of second order hyperbolic partial differential equations. Modified finite element Legendre wavelet Galerkin method developed for its solution. The discretization in space and then application of Legendre wavelet Galerkin method converts our problem into a coupled system of generalized Sylvester equations. In a particular case, the solution of the present model is compared with exact solution and are approximately the same. L∞ error decreases as Legendre wavelet basis functions increase and strip size decreases. Whole analysis is done in dimensionless form. The effect of variability of moving interface on Fourier number, Luikov number, Posnov number, Kossovitch number, Biot number and relaxation times is discussed in detail.

A New Legendre Wavelet Galerkin Method Applied to Study the Heat Transfer in Two-Dimensional Fin

A New Legendre Wavelet Galerkin Method Applied to Study the Heat Transfer in Two-Dimensional Fin

Analysis of DPL Bioheat Transfer Model During Thermal Treatment

In this paper, we studied temperature distribution in tissue using different approximations of the DPL model under the most generalized non-Fourier boundary condition in the presence of metabolic and external heat sources. The different approximations of the DPL bioheat transfer model are solved using two different techniques: (1) Laplace transform technique where inversion evaluated using the Stehfest algorithm, (2) finite element Legendre wavelet Galerkin method (FELWGM). The solution obtained using FELWGM provides a better result in comparison to the Stehfest method. The results obtained from different approximations of the DPL model are compared with experimental observation of Stolwijk and Hardy (Pflug Arch 291(2):129–162, 1966). It has been observed that the second approximation of $$\tau _q$$ in the DPL model provide better result in comparison of the rest models. The effect on temperature distribution of the variability of different parameters such as relaxation time, metabolic heat source, spatial heat source, different type boundary conditions are analyzed and discussed in detail. An inequality of parameters for diathermia, hyperthermia, and thermal ablation for boundary conditions of the first, second, and third kinds are given at the end.

Wavelet based numerical approach of non-classical moving boundary problem with convection effect and variable latent heat under the most generalized boundary conditions

The major goal of this article is to analysis a mathematical model of a non-classical one-dimensional moving boundary problem in the presence of convection effect when one surface subjected to the most generalized boundary conditions. The control function describes the cooling and heating effect depends on the evolution of heat flux at x=0. The thermo-physical properties are assumed to be constant while latent heat depends on the position of moving interface and its velocity. The parameters affecting the thermal behaviour of the melting process in various enclosures, i.e. the effect of Peclet number, control function, variable latent heat, heat flux and convecting heat transfer coefficient on the moving interface and temperature distribution are comprehensively investigated by the Legendre wavelet Galerkin computational method. In specific cases the Legendre wavelet Galerkin method is used for the numerical solution when surface is subjected to the first kind boundary condition with constant latent heat and thus the results obtained are compared with Turkyilmazoglu (2018) results and found in good compliance. Further when latent heat is variable, the present numerical solution obtained is compared with exact solution and are found in high accuracy and good flexibility.

Discontinuous Legendre Wavelet Galerkin Method for Optimal Control of Time Delayed Systems

Time-delay systems arise in many important applications in science and engineering and optimal control of delay differential equations are of theoretical and practical importance. This paper presents discontinuous Legendre wavelet Galerkin (DLWG) approach for solving optimal control problem of time-delayed systems. This new method demonstrates that operational matrices of derivative, delay and product are lower dimensions and sparse because of calculation only on each subinterval. The advantages are implemented to solve algebraic equations transformed from the time-delayed systems with less storage space and execution time. Finally, an experiment is included to illustrate the effectiveness and applicability of the proposed method

Numerical treatment for Burgers–Fisher and generalized Burgers–Fisher equations

In this paper, the discontinuous Legendre wavelet Galerkin method is proposed for the numerical solution of the Burgers–Fisher and generalized Burgers–Fisher equations. This method combines both the discontinuous Galerkin and the Legendre wavelet Galerkin methods. Various properties of Legendre wavelets have been used to find the variational form of the governing equation. This variational form transforms it into a system of ordinary differential equations which will be solved numerically. Some illustrative examples are presented to emphasize the efficiency and reliability of the proposed method.

A computational solution of a phase-change material in the presence of convection under the most generalized boundary condition

This article presents a mathematical model describing inward melting process of a phase change material in the presence of convection under the most generalized boundary condition. It is assumed that the material within a container have different geometrical configuration like circular cylinder or sphere. All thermophysical properties of the solid and liquid regions are assumed to be homogeneous. Initially, we convert the mathematical model into an initial value problem in the form of vector matrix representation using a finite difference technique. Two numerical methods; the operational matrix of integration for Bessel functions and finite element Legendre wavelet Galerkin method, are applied to solve the initial value problem. Thus, the obtained results from both methods analyzed for constant (or time depending) temperature or constant (or time depending) heat flux. The whole study is presented in dimensionless form. The effect of Stefan number, Peclet number, Kirpichev number and Biot number on dimensionless temperature profile and dimensionless moving front are illustrated graphically.

A study of heat transfer during cryosurgery of lung cancer.

In this study, a mathematical model describing two-dimensional bio-heat transfer during cryosurgery of lung cancer is developed. The lung tissue is cooled by a cryoprobe by imposing its surface at a constant temperature or a constant heat flux or a constant heat transfer coefficient. The freezing starts and the domain is distributed into three stages namely: unfrozen, mushy and frozen regions. In stage I where the only unfrozen region is formed, our problem is an initial-boundary value problem of the hyperbolic partial differential equation. In stage II where mushy and unfrozen regions are formed, our problem is a moving boundary value problem of parabolic partial differential equations and in stage III where frozen, mushy, and unfrozen regions are formed, our problem is a moving boundary value problem of parabolic partial differential equations. The solution consists of the three-step procedure: (i) transformation of problem in non-dimensional form, (ii) by using finite differences, the problem converted into ordinary matrix differential equation and moving boundary problem of ordinary matrix differential equations, (iii) applying Legendre wavelet Galerkin method the problem is transferred into the generalized system of Sylvester equations which are solved by applying Bartels-Stewart algorithm of generalized inverse. The complete analysis is presented in the non-dimensional form. The consequence of the imposition of boundary conditions on moving layer thickness and temperature distribution are studied in detail. The consequence of Stefan number, Kirchoff number and Biot number on moving layer thickness are also studied in specific.

Variable thermal conductivity approach for bioheat transfer during thermal ablation

The aim of the article is to present a mathematical model that predicts tissue temperature during thermal therapy (thermal ablation). The mathematical formulation of variable thermal conductivity of thermal wave bioheat transfer model for a generalized coordinate system due to external heat source during thermal ablation has been given. The governing bioheat transfer equation is simplified by Kirchhoff’s transformation. The numerical solution of the problem has been achieved using a finite difference scheme to discretize in space coordinate; the reduced system of second order ordinary differential equation is solved by the finite element Legendre wavelet Galerkin method (FELWGM). The obtained result from FELWGM is compared with exact analytical solution and shows good agreement. Parametric study is performed to evaluate the effect of variable thermal conductivity on tissue temperature distribution for different physical parameters and illustrated graphically. Particular cases are also deduced from present investigation. The effect of variability of thermal conductivity parameter, variability of time, generalized coordinate system, lagging time and external heat source coefficient on tissue temperature is discussed in detail. Thus, the consideration of the variable thermal conductivity is extremely beneficial for the clinical therapeutic application in the treatment of cancerous cells.

A study of cryosurgery of lung cancer using Modified Legendre wavelet Galerkin method

In this paper, we have developed a new mathematical model describing bio-heat transfer during cryosurgery of lung cancer. The lung tissue cooled by a flat probe whose temperature decreases linearly with time. The freezing process occurs in three stages and the whole region is divided into solid, mushy and liquid region. The heat released in the mushy region is considered as discontinuous heat generation. The model is an initial-boundary value problem of the hyperbolic partial differential equation in stage 1 and moving boundary value problem of parabolic partial differential equations in stage 2 and 3. The method of the solution consists of four-step procedure as transformation of problem in dimensionless form, the problem of hyperbolic partial differential equation converted into ordinary matrix differential equation and the moving boundary problem of parabolic partial differential equations converted into moving boundary problem of ordinary matrix differential equations by using finite differences in space, transferring the problem into the generalized system of Sylvester equations by using Legendre wavelet Galerkin method and the solution of the generalized system of Sylvester equation are solved by using Bartels-Stewart algorithm of generalized inverse. The whole analysis is presented in dimensionless form. The effect of cryoprobe rate on temperature distribution and the effect of Stefan number on moving layer thickness is discussed in detail.

Discontinuous Legendre wavelet Galerkin method for the generalised Burgers-Fisher equation

This paper presents discontinuous Legendre wavelet Galerkin (DLWG) technique for solving the generalised Burgers-Fisher equation. Weak formulation of this equation and numerical fluxes are addressed by utilising the advantages of the both Legendre wavelet and discontinuous Galerkin (DG) approach. Finally, numerical experiments demonstrate the validity and utility of the DLWG method.

Discontinuous Legendre wavelet Galerkin method for the generalised Burgers-Fisher equation

Discontinuous Legendre wavelet Galerkin method for the generalised Burgers-Fisher equation

Numerical simulation of time fractional dual-phase-lag model of heat transfer within skin tissue during thermal therapy

In this paper, we investigated the thermal behavior in living biological tissues using time fractional dual-phase-lag bioheat transfer (DPLBHT) model subjected to Dirichelt boundary condition in presence of metabolic and electromagnetic heat sources during thermal therapy. We solved this bioheat transfer model using finite element Legendre wavelet Galerkin method (FELWGM) with help of block pulse function in sense of Caputo fractional order derivative. We compared the obtained results from FELWGM and exact method in a specific case, and found a high accuracy. Results are interpreted in the form of standard and anomalous cases for taking different order of time fractional DPLBHT model. The time to achieve hyperthermia position is discussed in both cases as standard and time fractional order derivative. The success of thermal therapy in the treatment of metastatic cancerous cell depends on time fractional order derivative to precise prediction and control of temperature. The effect of variability of parameters such as time fractional derivative, lagging times, blood perfusion coefficient, metabolic heat source and transmitted power on dimensionless temperature distribution in skin tissue is discussed in detail. The physiological parameters has been estimated, corresponding to the value of fractional order derivative for hyperthermia treatment therapy.

Galerkin and collocation methods for solution of initial value problem of generalised van der Pol equation

In this paper, we proposed a van der Pol oscillator with delay feed-back including duffing oscillators and a periodic forcing term model. For solution of proposed model, using collocation and Galerkin method with Legendre wavelet as a basis functions. Convergence and stability analysis of the method are discussed. An algorithm provided for computing numerical data. The solution obtained by both Legendre wavelet collocation method and Legendre wavelet Galerkin method is exactly same as exact solution and those obtained by method of averaging (Atay, 1998), PEM (Kimiaeifar et al., 2010). The Legendre wavelet collocation method for different M and k provides better results in lesser time than Legendre wavelet Galerkin method. It has been observed that the displacement have cyclic behaviour with respect to velocity for different feedback gain and angular frequency of the driving force. The displacement decreases as delay parameter increases in small domain. The displacement also decreases as duffing parameter increases and angular frequency of the driving force decreases.

Galerkin and collocation methods for solution of initial value problem of generalised van der Pol equation

In this paper, we proposed a van der Pol oscillator with delay feed-back including duffing oscillators and a periodic forcing term model. For solution of proposed model, using collocation and Galerkin method with Legendre wavelet as a basis functions. Convergence and stability analysis of the method are discussed. An algorithm provided for computing numerical data. The solution obtained by both Legendre wavelet collocation method and Legendre wavelet Galerkin method is exactly same as exact solution and those obtained by method of averaging (Atay, 1998), PEM (Kimiaeifar et al., 2010). The Legendre wavelet collocation method for different M and k provides better results in lesser time than Legendre wavelet Galerkin method. It has been observed that the displacement have cyclic behaviour with respect to velocity for different feedback gain and angular frequency of the driving force. The displacement decreases as delay parameter increases in small domain. The displacement also decreases as duffing parameter increases and angular frequency of the driving force decreases.

Discontinuous Legendre wavelet Galerkin method for reaction–diffusion equation

ABSTRACTThis paper proposes a novel numerical method, that is, discontinuous Legendre wavelet Galerkin technique for solving reaction–diffusion equation (RDE). Specifically, variational formulation and corresponding numerical fluxes of this type equation are devised by utilizing the advantages of both Legendre wavelet bases and discontinuous Galerkin approach. Furthermore, adaptive algorithm, stability and error analysis of this method have been discussed. Especially, the distinctive features of the presented approach are easy to cope with a variety of boundary conditions and able to effectively approximate solution of the RDE with less execution and storage space. Finally, numerical tests affirm better accuracy for a range of benchmark problems and demonstrate the validity and utility of this approach.

A study on DPL model of heat transfer in bi-layer tissues during MFH treatment

In this paper, dual-phase-lag bioheat transfer model subjected to Fourier and non-Fourier boundary conditions for bi-layer tissues has been solved using finite element Legendre wavelet Galerkin method (FELWGM) during magnetic fluid hyperthermia. FELWGM localizes small scale variation of solution and fast switching of functional bases. It has been observed that moderate hyperthermia temperature range (41–46°C) can be better achieved in spherical symmetric coordinate system and treatment method will be independent of the Fourier and non-Fourier boundary conditions used. The effect of phase-lag times has been observed only in tumor region. FCC FePt magnetic nano-particle produces more effective treatment with respect to other magnetic nano-particles. The effect of variability of magnetic heat source parameters (magnetic induction, frequency, diameter of magnetic nano-particles, volume fractional of magnetic nano-particles and ligand layer thickness) has been investigated. The physical property of these parameters has been described in detail during magnetic fluid hyperthermia (MFH) treatment and also discussed the clinical application of MFH in Oncology.