Legendre Spectral Collocation Method Research Articles

Numerical approximation of space-fractional diffusion equation using Laguerre spectral collocation method

The space-fractional diffusion equation is extensively used to model various issues in engineering and mathematics. This paper addresses the numerical approximation of the space-fractional-order diffusion equation, using a fractional operator in the Caputo sense. The proposed equation is computed numerically through the Laguerre spectral collocation method combined with the finite difference scheme. The results were visualized using MATLAB R2016a, and the accuracy of the numerical scheme was validated by comparing the approximate results with those derived from exact values and the Legendre spectral collocation technique. The originality of this study lies in the application of the Laguerre spectral collocation method for fractional-order diffusion equations in the Caputo sense. Two numerical problems were included to demonstrate the validity and applicability of the proposed technique. The obtained results confirm that this method aligns well with the exact values and the Legendre spectral collocation method, as shown in tabular and graphical simulations.

Computational investigation of stochastic Zika virus optimal control model using Legendre spectral method

This study presents a computational investigation of a stochastic Zika virus along with optimal control model using the Legendre spectral collocation method (LSCM). By accumulation of stochasticity into the model through the proposed stochastic differential equations, we appropriating the random fluctuations essential in the progression and disease transmission. The stability, convergence and accuracy properties of the LSCM are conscientiously analyzed and also demonstrating its strength for solving the complex epidemiological models. Moreover, the study evaluates the various control strategies, such as treatment, prevention and treatment pesticide control, and identifies optimal combinations that the intervention costs and also minimize the proposed infection rates. The basic properties of the given model, such as the reproduction number, were determined with and without the presence of the control strategies. For R0<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_0<0$$\\end{document}, the model satisfies the disease-free equilibrium, in this case the disease die out after some time, while for R0>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_0>1$$\\end{document}, then endemic equilibrium is satisfied, in this case the disease spread in the population at higher scale. The fundamental findings acknowledge the significant impact of stochastic phonemes on the robustness and effectiveness of control strategies that accelerating the need for cost-effective and multi-faceted approaches. In last the results provide the valuable insights for public health department to enabling more impressive mitigation of Zika virus outbreaks and management in real-world scenarios.

An [formula omitted]-version Legendre collocation method for the third-kind VIEs with nonlinear vanishing delay

This paper proposes an hp-version Legendre spectral collocation method for solving third-kind VIEs with nonlinear vanishing delay, which effectively addresses the singularity of the equation and the intricate impact of the delay term. The solvability of the algebraic system is established, and the convergence of the hp-version spectral collocation method under the L∞-norm is proved. Numerical experiments are performed to validate our theoretical predictions.

Fast finite difference/Legendre spectral collocation approximations for a tempered time-fractional diffusion equation

&lt;p&gt;The present work is concerned with the efficient numerical schemes for a time-fractional diffusion equation with tempered memory kernel. The numerical schemes are established by using a $ L1 $ difference scheme for generalized Caputo fractional derivative in the temporal variable, and applying the Legendre spectral collocation method for the spatial variable. The sum-of-exponential technique developed in [Jiang et al., Commun. Comput. Phys., 21 (2017), 650-678] is used to discrete generalized fractional derivative with exponential kernel. The stability and convergence of the semi-discrete and fully discrete schemes are strictly proved. Some numerical examples are shown to illustrate the theoretical results and the efficiency of the present methods for two-dimensional problems.&lt;/p&gt;

Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives

&lt;abstract&gt;&lt;p&gt;In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders $ \alpha_i\in(0, 1) $, $ i = 1, 2, \cdots, n $). The proposed method employs a fast finite difference scheme to approximate multi-term fractional derivatives in time, requiring only $ O(1) $ storage and $ O(N_T) $ computational complexity, where $ N_T $ denotes the total number of time steps. Then we use a Legendre spectral collocation method for spatial discretization. The stability and convergence of the scheme have been thoroughly discussed and rigorously established. We demonstrate that the proposed scheme is unconditionally stable and convergent with an order of $ O\left(\left(\Delta t\right)^{2}+N^{-m}\right) $, where $ \Delta t $, $ N $, and $ m $ represent the timestep size, polynomial degree, and regularity in the spatial variable of the exact solution, respectively. Numerical results are presented to validate the theoretical predictions.&lt;/p&gt;&lt;/abstract&gt;

Legendre spectral collocation method for solving nonlinear fractional Fredholm integro-differential equations with convergence analysis

&lt;abstract&gt;&lt;p&gt;The main purpose of this work was to develop a spectrally accurate collocation method for solving nonlinear fractional Fredholm integro-differential equations (non-FFIDEs). A proposed spectral collocation method is based on the Legendre-Gauss-Lobatto collocation (L-G-LC) method in which the main idea is to use Caputo derivatives and Legendre-Gauss interpolation for nonlinear FFIDEs. A rigorous convergence analysis is provided and confirmed by numerical tests. In addition, we provide some numerical test cases to demonstrate that the approach can preserve the non-smooth solution of the underlying problem.&lt;/p&gt;&lt;/abstract&gt;

Unraveling pine wilt disease: Comparative study of stochastic and deterministic model using spectral method

The pinewood nematode, which causes a disease known as pine wilt, is a serious threat to pine forests all over the world. In this paper, we analyze the dynamics of pine wilt disease and its effects on forest ecosystems using mathematical modeling of pine wilt disease, its numerical simulations, and analytical techniques. Understanding the nematode’s spread, evaluating the efficiency of management efforts, and foreseeing the long-term effects on pine populations are some of our research’s goals. To gain a comprehensive understanding of the dynamics of the disease, we categorize pine trees and vectors systematically into susceptible, exposed, and infected groups. We first presented disease dynamics through a flow diagram which is then transformed to system of ordinary differential equations along with non-negative initial conditions. The deterministic model is then transformed to a stochastic model. Using both deterministic and stochastic modeling approaches, this study explores the complex transmission dynamics of Pine Wilt disease. Basic reproduction number has been calculated via next generation technique and some cases of stability analysis on the basis of the values of reproduction number has been discussed. For the solution of stochastic model Legendre Spectral collocation method has been used as it is known for its high accuracy in approximating solutions to complex stochastic models. It can achieve exponential convergence rates, making it suitable for problems where high precision is required. The results of both deterministic and stochastic models are compared via different figures. We find crucial thresholds for disease transmission and highlight significant factors affecting the spread of diseases through simulations. Our research shows that preventing the severe impacts of pine wilt disease requires early detection and intervention. Additionally, our mathematical model is a useful tool for enhancing disease management plans. This research advances our knowledge of the dynamics of the pine wilt disease and provides useful advice for managing forests and promoting conservation. Our research highlights the need for quick action to protect pine forests from this harmful virus.

Spectral Quasilinearization Method for the Numerical Solution of the Non-standard Volterra Integral Equations

In this paper, an iterative scheme is proposed through quasilinearization and the Legendre spectral collocation method to approximate the solution of the non-standard Volterra integral equations (NVIEs). At first, the process of quasilinearization serves to turn a NVIE into a sequence of linear Volterra integral equations. Then, in each iteration, the linear problem is solved by the Legendre spectral collocation method. After that, a convergence analysis is performed. The efficiency and accuracy of the proposed scheme are also verified through several illustrative examples.

Dynamics and simulations of stochastic COVID-19 epidemic model using Legendre spectral collocation method

&lt;abstract&gt;&lt;p&gt;The aim of this study is to investigate the dynamics of epidemic transmission of COVID-19 SEIR stochastic model with generalized saturated incidence rate. We assume that the random perturbations depends on white noises, which implies that it is directly proportional to the steady states. The existence and uniqueness of the positive solution along with the stability analysis is provided under disease-free and endemic equilibrium conditions for asymptotically stable transmission dynamics of the model. An epidemiological metric based on the ratio of basic reproduction is used to describe the transmission of an infectious disease using different parameters values involve in the proposed model. A higher order scheme based on Legendre spectral collocation method is used for the numerical simulations. For the better understanding of the proposed scheme, a comparison is made with the deterministic counterpart. In order to confirm the theoretical analysis, we provide a number of numerical examples.&lt;/p&gt;&lt;/abstract&gt;

Numerical inversion of reaction parameter for a time-fractional diffusion equation by Legendre spectral collocation and mollification method

In this paper, we consider an inverse problem of identifying a pair of function { u ( x , t ) , r ( t ) } in the time fractional diffusion equation from two kinds of observations. By virtue of the high-precision Legendre spectral collocation and mollification method in the spatial and time direction severally, we present an efficient algorithm to solve the inverse problem. Finally, two numerical examples explicate the effectiveness and stability of the proposed method.

Computational Scheme for the First-Order Linear Integro-Differential Equations Based on the Shifted Legendre Spectral Collocation Method

First-order linear Integro-Differential Equations (IDEs) has a major importance in modeling of some phenomena in sciences and engineering. The numerical solution for the first-order linear IDEs is usually obtained by the finite-differences methods. However, the convergence rate of the finite-differences method is limited by the order of the differences in L1 space. Therefore, how to design a computational scheme for the first-order linear IDEs with computational efficiency becomes an urgent problem to be solved. To this end, a polynomial approximation scheme based on the shifted Legendre spectral collocation method is proposed in this paper. First, we transform the first-order linear IDEs into an Cauchy problem for consideration. Second, by decomposing the system operator, we rewrite the Cauchy problem into a more general form for approximating. Then, by using the shifted Legendre spectral collocation method, we construct a computational scheme and write it into an abstract version. The convergence of the scheme is proven in the sense of L1-norm by employing Trotter-Kato theorem. At the end of this paper, we summarize the usage of the scheme into an algorithm and present some numerical examples to show the applications of the algorithm.

Quasilinearization-based Legendre collocation method for solving a class of functional Volterra integral equations

In this paper, a combination of the quasilinearization and the Legendre spectral collocation methods is introduced to approximate the solution of the nonlinear functional Volterra integral equations. Throughout this process, the quasilinearization method converts the nonlinear functional Volterra integral equation to a sequence of linear integral equations. Then, in each iteration, the obtained linear integral equation is solved using the Legendre spectral collocation method. After that, a convergence analysis is discussed in detail. Finally, several numerical examples are included to demonstrate the capability and validity of the proposed method.

Asymptotic Behavior of Three Connected Stochastic Delay Neoclassical Growth Systems Using Spectral Technique

In this study, we consider a nonlinear system of three connected delay differential neoclassical growth models along with stochastic effect and additive white noise, which is influenced by stochastic perturbation. We derived the conditions for positive equilibria, stability and positive solutions of the stochastic system. It is observed that when a constant delay reaches a certain threshold for the steady state, the asymptotic stability is lost, and the Hopf bifurcation occurs. In the case of the finite domain, the three connected, delayed systems will not collapse to infinity but will be bounded ultimately. A Legendre spectral collocation method is used for the numerical simulations. Moreover, a comparison of a stochastic delayed system with a deterministic delayed system is also provided. Some numerical test problems are presented to illustrate the effectiveness of the theoretical results. Numerical results further illustrate the obtained stability regions and behavior of stable and unstable solutions of the proposed system.

Legendre spectral collocation method for distributed and Riesz fractional convection\u2013diffusion and Schr\xf6dinger-type equation

The numerical analysis of the temporal distributed and spatial Riesz fractional problem (TDSRFP) is presented in this work. To address the two independent variables, the suggested technique employs a completely spectral Legendre collocation approach. For the current model, our technique is proven to be more accurate, efficient, and practical. The results confirmed that the spectral scheme is exponentially convergent.

On dynamics of stochastic avian influenza model with asymptomatic carrier using spectral method

In ecosystem, the most critical issue is the extinction and persistence of population. For this reason, various deterministic and stochastic models have been studied in the past decades. However, the effect of transitions which is caused by either internal or external environmental noise has given less attention. In this work, we introduce a robust and an efficient numerical scheme based on Legendre spectral collocation method (LSCM) to explore the asymptomatic behavior of stochastic influenza avian model. We consider the model in which both birds and human population are exposed class, while for the human population, the asymptomatic class is also considered, taking into account that the asymptomatic human beings may get re‐infected and migrate to symptomatic class. A fragmented treatment gives more consideration as they relocate symptomatic individuals towards the asymptomatic class. A generation approach is utilized to figure out the essential propagation number (called reproduction number) denoted by R0. For R0 &lt; 1, the system is asymptotically stable locally and has an ailment disease free equilibrium (DFE) and may have up to five endemic equilibria. Our simulations results recommend that the pace of complete predominance of influenza will be high, if all the individuals show symptoms upon infection and experience a deficient treatment. Further, the predominance pace of influenza is low, if all the individuals first move to symptomatic class and then treated effectively. An average rate of total prevalence is secured when more individuals upon disease move towards asymptomatic class. Our numerical simulations justified the theoretical results.

A new local non-integer derivative and its application to optimal control problems

&lt;abstract&gt;&lt;p&gt;Here, a new local non-integer derivative is defined and is shown that it coincides to classical derivative when the order of derivative be integer. We call this derivative, adaptive derivative and present some of its important properties. Also, we gain and state Rolle's theorem and mean-value theorem in the sense of this new derivative. Moreover, we define the optimal control problems governed by differential equations including adaptive derivative and apply the Legendre spectral collocation method to solve this type of problems. Finally, some numerical test problems are presented to clarify the applicability of new defined non-integer derivative with high accuracy. Through these examples, one can see the efficiency of this new non-integer derivative as a tool for modeling real phenomena in different branches of science and engineering that described by differential equations.&lt;/p&gt;&lt;/abstract&gt;

Long time behavior of higher-order delay differential equation with vanishing proportional delay and its convergence analysis using spectral method

&lt;abstract&gt; &lt;p&gt;Delay differential equations (DDEs) are used to model some realistic systems as they provide some information about the past state of the systems in addition to the current state. These DDEs are used to analyze the long-time behavior of the system at both present and past state of such systems. Due to the oscillatory nature of DDEs their explicit solution is not possible and therefore one need to use some numerical approaches. In this article, we developed a higher-order numerical scheme for the approximate solution of higher-order functional differential equations of pantograph type with vanishing proportional delays. Some linear and functional transformations are used to change the given interval [0, T] into standard interval [-1, 1] in order to fully use the properties of orthogonal polynomials. It is assumed that the solution of the equation is smooth on the entire domain of interval of integration. The proposed scheme is employed to the equivalent integrated form of the given equation. A Legendre spectral collocation method relative to Gauss-Legendre quadrature formula is used to evaluate the integral term efficiently. A detail theoretical convergence analysis in L&lt;sub&gt;∞&lt;/sub&gt; norm is provided. Several numerical experiments were performed to confirm the theoretical results.&lt;/p&gt; &lt;/abstract&gt;

A spectral collocation method for the coupled system of nonlinear fractional differential equations

&lt;abstract&gt;&lt;p&gt;This paper analyzes the coupled system of nonlinear fractional differential equations involving the caputo fractional derivatives of order $ \alpha\in(1, 2) $ on the interval (0, T). Our method of analysis is based on the reduction of the given system to an equivalent system of integral equations, then the resulting equation is discretized by using a spectral method based on the Legendre polynomials. We have constructed a Legendre spectral collocation method for the coupled system of nonlinear fractional differential equations. The error bounds under the $ L^2- $ and $ L^{\infty}- $norms is also provided, then the theoretical result is validated by a number of numerical tests.&lt;/p&gt;&lt;/abstract&gt;

Modeling and analysis on the transmission of covid-19 Pandemic in Ethiopia

The newest infection is a novel coronavirus named COVID-19, that initially appeared in December 2019, in Wuhan, China, and is still challenging to control. The main focus of this paper is to investigate a novel fractional-order mathematical model that explains the behavior of COVID-19 in Ethiopia. Within the proposed model, the entire population is divided into nine groups, each with its own set of parameters and initial values. A nonlinear system of fractional differential equations for the model is represented using Caputo fractional derivative. Legendre spectral collocation method is used to convert this system into an algebraic system of equations. An inexact Newton iterative method is used to solve the model system. The effective reproduction number (R0) is computed by the next-generation matrix approach. Positivity and boundedness, as well as the existence and uniqueness of solution, are all investigated. Both endemic and disease-free equilibrium points, as well as their stability, are carefully studied. We calculated the parameters and starting conditions (ICs) provided for our model using data from the Ethiopian Public Health Institute (EPHI) and the Ethiopian Ministry of Health from 22 June 2020 to 28 February 2021. The model parameters are determined using least squares curve fitting and MATLAB R2020a is used to run numerical results. The basic reproduction number is R0=1.4575. For this value, disease free equilibrium point is asymptotically unstable and endemic equilibrium point is asymptotically stable, both locally and globally.

Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method

Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method based on Legendre’s polynomials for evaluating the numerical solutions of the pr. We then construct a scheme for the fractional-order models by using the spectral method involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic equations, which can be approximated by the Newton-Raphson method. For the second model, we also need to use the finite differences method to obtain the set of nonlinear algebraic equations, which are also approximated as in the first model. The accuracy of the results is verified in the first model by comparing it with our analytical solution. In the second and third models, the residual error functions are calculated. In all cases, the results are found to be in agreement. The method is a powerful hybrid technique of numerical and analytical approach that is applicable for partial differential equations with multi-order of fractional derivatives involving GMLK with three parameters.