Legendre Wavelets Research Articles

A numerical method based on Legendre wavelet and quasilinearization technique for fractional Lane-Emden type equations

A numerical method based on Legendre wavelet and quasilinearization technique for fractional Lane-Emden type equations

Legendre wavelets based approach for the solution of type-2 fuzzy uncertain smoking model of fractional order

PurposeInvestigation of the smoking model is important as it has a direct effect on human health. This paper focuses on the numerical analysis of the fractional order giving up smoking model. Nonetheless, due to observational or experimental errors, or any other circumstance, it may contain some incomplete information. Fuzzy sets can be used to deal with uncertainty. Yet, there may be some inconsistency in the membership as well. As a result, the primary goal of this proposed work is to numerically solve the model in a type-2 fuzzy environment.Design/methodology/approachTriangular perfect quasi type-2 fuzzy numbers (TPQT2FNs) are used to deal with the uncertainty in the model. In this work, concepts of r2-cut at r1-plane are used to model the problem's uncertain parameter. The Legendre wavelet method (LWM) is then utilised to solve the giving up smoking model in a type-2 fuzzy environment.FindingsLWM has been effectively employed in conjunction with the r2-cut at r1-plane notion of type-2 fuzzy sets to solve the model. The LWM has the advantage of converting the non-linear fractional order model into a set of non-linear algebraic equations. LWM scheme solutions are found to be well agreed with RK4 scheme solutions. The existence and uniqueness of the model's solution have also been demonstrated.Originality/valueTo deal with the uncertainty, type-2 fuzzy numbers are used. The use of LWM in a type-2 fuzzy uncertain environment to achieve the model's required solutions is quite fascinating, and this is the key focus of this work.

Transient Thermal Cooling of Electronics Systems using Functional Graded Fins: Hybrid Computational Analysis

Passive cooling of electronics systems requires the use of fins and spines. Improved heat transfer enhancements in the systems have been achieved through porosity, magnetic field, functionally graded materials, etc. However, the space and time-dependent analysis of passive device with inherent nonlinearities and explorations of conditions at the tip have not been fully presented. Therefore, this work presents an application of hybrid computation analysis using combined methods of Laplace transformation and Legendre wavelet collocation to the space and time-dependent nonlinear thermal models of a longitudinal conductive-radiative functionally graded extended surface. The spatial-dependent thermal conductivity is considered for three different cases. The present analysis results demonstrated good agreement with numerical results. The significances of the model's pertinent parameters on the fin’s performance are thoroughly considered and scrutinized with the aid of graphical illustrations. The fin tip thermal response decreases with an increase in conductive-convective parameters, but it increases as time progresses. The in-homogeneity index is directly proportional to the FGM fin temperature. However, the convective term is inversely proportional to the fin’s thermal distribution. The fin temperature under quadratic-law heat conductivity of the passive device shows an enhanced performance of lower thermal distribution than the linear- and exponential-law heat conductivity. Ultimately, the study provides a very useful in the design of the fin with FGM.

Identification of fractional order non– homogeneous Hammerstein-Wiener MISO continuous systems

Aiming at the problem of difficult identification of parameters in fractional non-homogeneous Hammerstein-Wiener nonlinear MISO time-delay continuous systems, an algorithm that can simultaneously identify coupled coefficients, orders, and time-delays in nonlinear fractional-order systems is designed. In this algorithm, a fractional order fitting model is constructed by using the Legendre wavelet operation matrix, and a full-parameter partial derivative model of the time-delay nonlinear system is derived. Second, the sequential quadratic programming (SQP) method is used to simultaneously estimate the non-homogeneous order, delay parameters, and coefficient parameters of the fractional order system. Finally, the effectiveness of the proposed algorithm is demonstrated through an academic example and a two-wall heat conduction system with a nonlinear fractional order system of MISO.

Extended Legendre Wavelet Method for Solving Fractional Order Time Hyperbolic Partial Differential Equation

Extended Legendre Wavelet Method for Solving Fractional Order Time Hyperbolic Partial Differential Equation

Generalized fractional-order Legendre wavelet method for two dimensional distributed order fractional optimal control problem

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 L2-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.

Combined variational iteration method with chebyshev wavelet for the solution of convection-diffusion-reaction problem

The goal of the work is to solve the nonlinear convection-diffusion-reaction problem using the variational iteration method with the combination of the Chebyshev wavelet. This work developed a hybrid iterative technique named as Variational iteration method with the Chebyshev wavelet for the solutions of nonlinear convection-diffusion-reaction problems. The aim of applying the derived algorithm is to achieve fast convergence. During the solution of the given problem, the restricted variations will be mathematically justified. The effects of the scaling and other parameters like diffusion parameter, convection parameter, and reaction parameter on the solution are also focused on by their suitable selection. The approximate results include the error profiles and the simulations. The results of variational iteration with the Chebyshev wavelet are compared with variational iteration method, the Modified variational iteration method, and the Variational iteration method with Legendre wavelet. The error profiles allow us to compare the results with well-known existing schemes.

Numerical study for temperature-dependent viscosity based unsteady flow of GP-MoS2/C2H6O2-H2O over a porous stretching sheet

The theme of this research is to attain the numerical solution of magnetized GP-MoS2/C2H6O2-H2O unsteady flow over a stretching surface, using Legendre Wavelet Collocation Technique (LWCT). The authors incorporate Legendre wavelet basis functions and its operational matrix of integration (OMI) in LWCT, which gives precise solutions for non-linear ODEs using MATLAB software. In this study, the conventional fluid is the combination of water (H2O) and ethylene glycol (C2H6O2). The ratio of volume fraction of existing nanoparticles, i.e. graphene (GP) and molybdenum disulfide (MoS2) is taken as 8% respectively. The impact of temperature dependent viscosity (TDV), magnetic field, suction and porosity and viscous dissipation on temperature, velocity, skin friction and heat transfer rate is incorporated. To validate the code, both tabular and graphical comparisons are presented with previous works. The results declare that the rate of heat transfer (HT) of GP-MoS2/C2H6O2-H2O is continuously augmented with rise in TDV and the Hartree pressure gradient. Moreover, the velocity outlines of GP-MoS2/C2H6O2-H2O hybrid nanofluid constantly reduces with increase in TDV, magnetic field and porosity parameter values.

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.

Müntz–Legendre wavelet method for solving Sturm–Liouville fractional optimal control problem with error estimates

SummaryThis paper develops a Müntz–Legendre wavelet method for solving a fractional optimal control problem with dynamic constraint as a fractional Sturm–Liouville problem. We first derive the necessary optimality conditions as a two‐point boundary value problem by using the calculus of variation and integration by part formula. The operational matrices for Müntz–Legendre scaling functions have been obtained, and by utilizing it, the two‐point boundary value problem has been converted into a system of algebraic equations. The ‐error estimates in the approximation of operations matrices and in the approximation of unknown variable by the Müntz–Legendre wavelet has been derived. In the last, illustrative examples have been taken to show the applicability of the proposed method.

Numerical Simulation for Flow and Heat Transfer of a Nanofluid Over Lubricated Stretchable Surface

The lubricants mostly used in industry have non-Newtonian properties and adhere to a variety of constitutive connections. The power law lubricant, which follows the Ostwald model and is widely used in engineering lubrication, is one of them. In this study the flow analysis of nanofluid in the vicinity of a stagnation point over a lubricated stretching surface has been investigated. The lubrication is provided by a thin layer of variable thickness of power-law fluid. Impact of thermophoresis and Brownian motion is also investigated. To acquire the dimensionless governing equations utilized an appropriate transformation. A newly developed powerful numerical procedure known as Legendre wavelet spectral collocation method (LWSCM) has been employed to acquire the similar solution. A comparison in the special cases between the published and present results substantiate the obtained solutions. Addition of the lubrication enhances the fluid velocity and reduces the temperature of the nanofluid at the stretchable surface.

Legendre wavelet operational matrix method for the analysis of thermal radiation effect on natural convection of a vertical plate embedded in a saturated porous medium

In this paper, the heat transfer characteristics of natural convection around a vertical flat surface embedded in a saturated porous medium are studied by taking into account the thermal radiation effect. With the aid of the similarity transformations, the governing equations of the problem are reduced to the system of ordinary differential equations over the infinite domain. Moreover, the Legendre wavelet operational matrix method (LWOMM) is hired to find the numerical solution to the considered problem. Initially, to ensure the efficiency of the proposed strategy, a third-order nonlinear boundary value problem having the exact solution is considered a test problem. The numerical findings elucidate that the solutions obtained by the LWOMM are in good agreement with the exact solution as compared with the solution obtained by the Haar wavelet operational matrix method, with the consumption of less computational CPU time. Also, the temperature and velocity profiles are discussed graphically.

Adaptive Neural Network Q-Learning-Based Full Recurrent Adaptive NeuroFuzzy Nonlinear Control Paradigms for Bidirectional-Interlinking Converter in a Grid-Connected Hybrid AC-DC Microgrid

The stability of a hybrid AC-DC microgrid depends mainly upon the bidirectional interlinking converter (BIC), which is responsible for power transfer, power balance, voltage solidity, frequency and transients sanity. The varying generation from renewable resources, fluctuating loads, and bidirectional power flow from the utility grid, charging station, super-capacitor, and batteries produce various stability issues on hybrid microgrids, like net active-reactive power flow on the AC-bus, frequency oscillations, total harmonic distortion (THD), and voltage variations. Therefore, the control of BIC between AC and DC buses in grid-connected hybrid microgrid power systems is of great importance for the quality/smooth operation of power flow, power sharing and stability of the whole power system. In literature, various control schemes are suggested, like conventional droop control, communication-based control, model predictive control, etc., each addressing different stability issues of hybrid AC-DC microgrids. However, model dependence, single-point-failure (SPF), communication vulnerability, complex computations, and complicated multilayer structures motivated the authors to develop online adaptive neural network (NN) Q-learning-based full recurrent adaptive neurofuzzy nonlinear control paradigms for BIC in a grid-connected hybrid AC-DC microgrid. The proposed strategies successfully ensure the following: (i) frequency stabilization, (ii) THD reduction, (iii) voltage normalization and (iv) negligible net active-reactive power flow on the AC-bus. Three novel adaptive NN Q-learning-based full recurrent adaptive neurofuzzy nonlinear control paradigms are proposed for PQ-control of BIC in a grid-connected hybrid AC-DC microgrid. The control schemes are based on NN Q-learning and full recurrent adaptive neurofuzzy identifiers. Hybrid adaptive full recurrent Legendre wavelet-based Neural Network Q-learning-based full recurrent adaptive NeuroFuzzy control, Hybrid adaptive full recurrent Mexican hat wavelet-based Neural Network Q-learning-based full recurrent adaptive NeuroFuzzy control, and Hybrid adaptive full recurrent Morlet wavelet-based Neural Network Q-learning-based full recurrent adaptive NeuroFuzzy control are modeled and tested for the control of BIC. The controllers differ from each other, based on variants used in the antecedent part (Gaussian membership function and B-Spline membership function), and consequent part (Legendre wavelet, Mexican hat wavelet, and Morlet wavelet) of the full recurrent adaptive neurofuzzy identifiers. The performance of the proposed control schemes was validated for various quality and stability parameters, using a simulation testbench in MATLAB/Simulink. The simulation results were bench-marked against an aPID controller, and each proposed control scheme, for a simulation time of a complete solar day.

Wavelets based computational algorithms for multidimensional distributed order fractional differential equations with nonlinear source term

In this manuscript we provide effective computational algorithms based on Legendre wavelet (LW), Bernstein wavelet (BW), and standard tau approach to approximate the solution of viscoelasticity damping motion nonlinear distributed order fractional differential equation (VDMNDOFDE) and multi-dimensional distributed order time-fractional nonlinear partial differential equation (DOT-FNPDE). To the best of our understanding, the proposed computational algorithm is new and has not been previously applied for solving VDMNDOFDE and DOT-FNPDE. The matrix representation of distributed order fractional derivatives, integer order derivatives and product term associated with the integral based on LW and BW are established to find the numerical solutions of the proposed VDMNDOFDE and DOT-FNPDE. Moreover, the association of standard tau rule and Legendre-Gauss quadrature (LGQ) techniques along with constructed matrix representation of differential and integral operators diminish VDMNDOFDE and DOT-FNPDE into system of nonlinear algebraic equations. Here, these systems of nonlinear algebraic equations are solved by the Newton's method to get the value of unknown vectors. Error bounds, convergence analysis, numerical algorithms of the VDMNDOFDE and DOT-FNPDE are rigorously investigated. For the reliability of the proposed computational algorithm, numerous test examples have been incorporated in the manuscript to ensure the robustness and theoretical results of proposed technique. Also, the proposed technique found to be more accurate with the existing scheme.

Identification of tissue and therapy parameters from surface temperature.

The temperature distribution of human body tissues, when electromagnetic waves are applied, is studied with the help of the bioheat equation, which is characterised by several thermal parameters, such as thermal conductivity, perfusion frequency or metabolic heat. There are also electromagnetic parameters, such as electrical conductivity and dielectric constant. Besides, therapy parameters, such as the applied power should be considered. If the values of all these parameters are known, the time evolution of the temperature can be determined. Some of these values can be found in specialised databases, and others are unknown but approximate values can be obtained by reasonable estimations. The outcome of simulations depends heavily on the parameter values. Surface temperature helps in providing better estimates of the parameter values. Once these parameters are identified, medical analyses can be performed to assess the dosimetry of the radiation so that it does not damage tissues, for example. The surface temperature is obtained from a sequence of thermographic images. Based on these experimental data, an algorithm is applied to find the values of the needed parameters. The model is simulated iteratively, adjusting the parameters at each step, reducing the approximation error between the simulation and the data. This is an optimization problem that belongs to the realm of inverse problems. It can be solved using techniques based on the gradient concept, however, this problem can be ill-conditioned, so probabilistic or evolutionary algorithms are also used. In this paper, the simulation is made using a method based on Legendre wavelets. It is proposed that the subsequent optimization is made using an evolutionary algorithm, that has shown great robustness in the problems where it has been applied. As far as known, it has never been applied to the bioheat equation. This is the aim of this work.

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.

New 2D numerical integration formula based on the Legendre wavelets

Based on Legendre wavelets, a new efficient numerical integration method is proposed to estimate double integrals. This new method is expressed in terms of the operational integration matrices. Those which were introduced by Parsian, are calculated in an approximate way. In this work, we also propose the exact computation of these matrices. Better precision of the new method is observed during a comparative numerical study with the composite Simpson method.

Approximation of functions belonging to Hölder’s class and solution of Lane-Emden differential equation using Gegenbauer wavelets

In this paper, a very new technique based on the Gegenbauer wavelet series is introduced to solve the Lane-Emden differential equation. The Gegenbauer wavelets are derived by dilation and translation of an orthogonal Gegenbauer polynomial. The orthonormality of Gegenbauer wavelets is verified by the orthogonality of classical Gegenbauer polynomials. The convergence analysis of Gegenbauer wavelet series is studied in H?lder?s class. H?lder?s class H?[0,1) and H?[0,1) of functions are considered, H?[0,1) class consides with classical H?lder?s class H?[0, 1) if ?(t) = t?, 0 < ? ? 1. The Gegenbauer wavelet approximations of solution functions of the Lane-Emden differential equation in these classes are determined by partial sums of their wavelet series. In briefly, four approximations E(1) 2k?1,0, E(1) 2k?1,M, E(2) 2k?1,0, E(2) 2k?1,M of solution functions of classes H?[0, 1), H?[0, 1) by (2k?1, 0)th and (2k?1,M)th partial sums of their Gegenbauer wavelet expansions have been estimated. The solution of the Lane-Emden differential equation obtained by the Gegenbauer wavelets is compared to its solution derived by using Legendre wavelets and Chebyshev wavelets. It is observed that the solutions obtained by Gegenbauer wavelets are better than those obtained by using Legendre wavelets and Chebyshev wavelets, and they coincide almost exactly with their exact solutions. This is an accomplishment of this research paper in wavelet analysis.

Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data

Abstract We use the Legendre wavelet method to study nonlinear fractional differential equations. Based on the in-depth study of the characteristics of various fractional-order dynamic system models, this paper designs a system for solving fractional-order differential equations, and we apply them to the anomaly analysis of big computer data. This method can improve the efficiency of big data classification. The results of computer numerical simulation show that the designed algorithm for solving fractional differential equations has high accuracy. At the same time, the algorithm can avoid misclassification and omission in big data analysis.

Approximate solution of the nonlinear Fredholm equation of the second kind by the Bubnov-Galerkin method with bases in the form of Legendre wavelets

In this paper, a review of new works on approximate methods for solving nonlinear Fredholm integral equations of the second kind is carried out. Legendre wavelets are used for the numerical solution of nonlinear Fredholm integral equations of the second kind by the Galerkin-Bubnov projection method. The Newton-Kantorovich method for a system of nonlinear algebraic equations is considered and numerical calculations are carried out. The comparative analysis shows that the Bubnov-Galerkin method with basic functions in the form of Legendre wavelets is effective in terms of accuracy and convenient to implement.