Jacobi Spectral Collocation Technique for Time-Fractional Inverse Heat Equations

In this paper, we introduce a numerical solution for the time-fractional inverse heat equations. We focus on obtaining the unknown source term along with the unknown temperature function based on an additional condition given in an integral form. The proposed scheme is based on a spectral collocation approach to obtain the two independent variables. Our approach is accurate, efficient, and feasible for the model problem under consideration. The proposed Jacobi spectral collocation method yields an exponential rate of convergence with a relatively small number of degrees of freedom. Finally, a series of numerical examples are provided to demonstrate the efficiency and flexibility of the numerical scheme.

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Legendre Spectral Collocation Technique for Advection Dispersion Equations Included Riesz Fractional

Fractal and Fractional ◽

10.3390/fractalfract6010009 ◽

2021 ◽

Vol 6 (1) ◽

pp. 9

Author(s):

Mohamed M. Al-Shomrani ◽

Mohamed A. Abdelkawy

Keyword(s):

Numerical Methods ◽

Numerical Algorithm ◽

Spectral Collocation ◽

Theoretical Formulation ◽

Numerical Examples ◽

Dispersion Equations ◽

Advection Dispersion ◽

Collocation Technique ◽

Collocation Approach

The advection–dispersion equations have gotten a lot of theoretical attention. The difficulty in dealing with these problems stems from the fact that there is no perfect answer and that tackling them using local numerical methods is tough. The Riesz fractional advection–dispersion equations are quantitatively studied in this research. The numerical methodology is based on the collocation approach and a simple numerical algorithm. To show the technique’s performance and competency, a comprehensive theoretical formulation is provided, along with numerical examples.

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SPACE-TIME SPECTRAL COLLOCATION ALGORITHM FOR THE VARIABLE-ORDER GALILEI INVARIANT ADVECTION DIFFUSION EQUATIONS WITH A NONLINEAR SOURCE TERM

Mathematical Modelling and Analysis ◽

10.3846/13926292.2017.1258014 ◽

2017 ◽

Vol 22 (1) ◽

pp. 1-20 ◽

Cited By ~ 5

Author(s):

Mohamed A. Abd-Elkawy ◽

Rubayyi T. Alqahtani

Keyword(s):

Source Term ◽

Dimensional Space ◽

Space Time ◽

Variable Order ◽

Spectral Collocation ◽

Nonlinear Source ◽

Advection Diffusion Equation ◽

Advection Diffusion ◽

Collocation Technique ◽

Collocation Scheme

This paper presents a space-time spectral collocation technique for solving the variable-order Galilei invariant advection diffusion equation with a nonlinear source term (VO-NGIADE). We develop a collocation scheme to approximate VONGIADE by means of the shifted Jacobi-Gauss-Lobatto collocation (SJ-GL-C) and shifted Jacobi-Gauss-Radau collocation (SJ-GR-C) methods. We successfully extend the proposed technique to solve the two-dimensional space VO-NGIADE. The discussed numerical tests illustrate the capability and high accuracy of the proposed methodologies.

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A Novel Method for Solving Nonlinear Volterra Integro-Differential Equation Systems

Abstract and Applied Analysis ◽

10.1155/2018/3569139 ◽

2018 ◽

Vol 2018 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Mohammad Hossein Daliri Birjandi ◽

Jafar Saberi-Nadjafi ◽

Asghar Ghorbani

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Exact Solutions ◽

Iteration Method ◽

Spectral Collocation ◽

Numerical Examples ◽

Integro Differential Equation ◽

Collocation Technique ◽

Computational Work ◽

Novel Method

An efficient iteration method is introduced and used for solving a type of system of nonlinear Volterra integro-differential equations. The scheme is based on a combination of the spectral collocation technique and the parametric iteration method. This method is easy to implement and requires no tedious computational work. Some numerical examples are presented to show the validity and efficiency of the proposed method in comparison with the corresponding exact solutions.

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A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise

Symmetry ◽

10.3390/sym12060904 ◽

2020 ◽

Vol 12 (6) ◽

pp. 904 ◽

Cited By ~ 5

Author(s):

Afshin Babaei ◽

Hossein Jafari ◽

S. Banihashemi

Keyword(s):

Order Of Convergence ◽

Additive Noise ◽

Numerical Solutions ◽

Numerical Test ◽

Test Problems ◽

Heat Equations ◽

Algebraic Equations ◽

Linear Algebraic Equations ◽

Stochastic Heat Equations ◽

Collocation Approach

A spectral collocation approach is constructed to solve a class of time-fractional stochastic heat equations (TFSHEs) driven by Brownian motion. Stochastic differential equations with additive noise have an important role in explaining some symmetry phenomena such as symmetry breaking in molecular vibrations. Finding the exact solution of such equations is difficult in many cases. Thus, a collocation method based on sixth-kind Chebyshev polynomials (SKCPs) is introduced to assess their numerical solutions. This collocation approach reduces the considered problem to a system of linear algebraic equations. The convergence and error analysis of the suggested scheme are investigated. In the end, numerical results and the order of convergence are evaluated for some numerical test problems to illustrate the efficiency and robustness of the presented method.

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On the Number of Isolating Integrals in Systems with Three Degrees of Freedom

International Astronomical Union Colloquium ◽

10.1017/s0252921100028530 ◽

1971 ◽

Vol 10 ◽

pp. 110-117

Author(s):

Claude Froeschle

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Weak Coupling ◽

Degrees Of Freedom ◽

Model Problem ◽

Energy Integral ◽

Dimensional Mapping ◽

Three Degrees Of Freedom

AbstractDynamical systems with three degrees of freedom can be reduced to the study of a four-dimensional mapping. We consider here, as a model problem, the mapping given by the following equations: We have found that as soon as b ≠ 0, i.e. even for a very weak coupling, a dynamical system with three degrees of freedom has in general either two or zero isolating integrals (besides the usual energy integral).

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Shifted Legendre spectral collocation technique for solving stochastic Volterra–Fredholm integral equations

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2020-0144 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Mohamed A. Abdelkawy

Keyword(s):

Brownian Motion ◽

Error Analysis ◽

Integral Equations ◽

Lagrange Interpolation ◽

Fredholm Integral Equations ◽

Algebraic Equations ◽

Spectral Collocation ◽

Numerical Examples ◽

Collocation Technique

Abstract This paper addresses a spectral collocation technique to treat the stochastic Volterra–Fredholm integral equations (SVF-IEs). The shifted Legendre–Gauss–Radau collocation (SL-GR-C) method is developed for approximating the FSV-IDEs. The principal target in our technique is to transform the SVF-IEs to a system of algebraic equations. For computational purposes, the Brownian motion W(x) is discretized by Lagrange interpolation. While the integral terms are interpolated by Legendre–Gauss–Lobatto quadrature. Some numerical examples are given to test the accuracy and applicability of our technique. Also, an error analysis is introduced for the proposed method.

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Method of External Potential in Solution of Cauchy Mixed Problem for the Heat Equation

ISRN Mathematical Analysis ◽

10.1155/2013/640985 ◽

2013 ◽

Vol 2013 ◽

pp. 1-5

Author(s):

T. Sh. Kalmenov ◽

N. E. Tokmagambetov

Keyword(s):

Heat Equation ◽

Mixed Problem ◽

A Priori Estimates ◽

A Priori ◽

Practical Importance ◽

Fourier Method ◽

Integral Form ◽

Heat Equations ◽

Main Research ◽

The One

Numerous research works are devoted to study Cauchy mixed problem for model heat equations because of its theoretical and practical importance. Among them we can notice monographers Vladimirov (1988), Ladyzhenskaya (1973), and Tikhonov and Samarskyi (1980) which demonstrate main research methods, such as Fourier method, integral equations method, and the method of a priori estimates. But at the same time to represent the solution of Cauchy mixed problem in integral form by given and known functions has not been achieved up to now. This paper completes this omission for the one-dimensional heat equation.

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Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

Axioms ◽

10.3390/axioms7030048 ◽

2018 ◽

Vol 7 (3) ◽

pp. 48 ◽

Cited By ~ 4

Author(s):

Konstantin Zhukovsky ◽

Dmitrii Oskolkov ◽

Nadezhda Gubina

Keyword(s):

Thin Films ◽

Differential Operators ◽

Exchange Process ◽

Integral Form ◽

Heat Propagation ◽

Heat Equations ◽

Elementary Functions ◽

Heat Exchange Process ◽

Ballistic Properties ◽

Low Dimensional

One-dimensional equations of telegrapher’s-type (TE) and Guyer–Krumhansl-type (GK-type) with substantial derivative considered and operational solutions to them are given. The role of the exponential differential operators is discussed. The examples of their action on some initial functions are explored. Proper solutions are constructed in the integral form and some examples are studied with solutions in elementary functions. A system of hyperbolic-type inhomogeneous differential equations (DE), describing non-Fourier heat transfer with substantial derivative thin films, is considered. Exact harmonic solutions to these equations are obtained for the Cauchy and the Dirichlet conditions. The application to the ballistic heat transport in thin films is studied; the ballistic properties are accounted for by the Knudsen number. Two-speed heat propagation process is demonstrated—fast evolution of the ballistic quasi-temperature component in low-dimensional systems is elucidated and compared with slow diffusive heat-exchange process. The comparative analysis of the obtained solutions is performed.

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DYNAMICAL ASPECTS OF INEXTENSIBLE CHAINS

International Journal of Modern Physics B ◽

10.1142/s0217979211102204 ◽

2012 ◽

Vol 26 (02) ◽

pp. 1250009

Author(s):

FRANCO FERRARI ◽

MACIEJ PYRKA

Keyword(s):

Path Integral ◽

Degrees Of Freedom ◽

Planck Equation ◽

Integral Form ◽

Langevin Equations ◽

Thermal Bath ◽

Fixed Temperature ◽

Generating Functional ◽

A Chain ◽

Non Gaussian

In the present work, a method to impose the inextensibility constraints on the dynamics of a chain fluctuating in a thermal bath at fixed temperature is investigated. The final goal is to construct the probability function of the chain and the generating functional of the correlation functions of the relevant degrees of freedom of the system. First, we study the dynamics of a freely hinged chain composed by massive beads connected together by massless segments of fixed length. It is shown that a system of this kind may be described by a set of Langevin equations in which the noise is characterized by a non-gaussian probability distribution. Starting from these Langevin equations, the generating functional of the freely hinged chain is derived in path integral form. A connection with a stochastic process governed by a Fokker–Planck equation is established. Next, a chain composed by one-dimensional bars with constant mass distribution is considered. A path integral expression of the generating functional for a chain of this type is derived. Finally, it is verified that in the limit in which the chain becomes continuous, both generating functionals of the freely hinged chain and of the freely jointed bar chain converge to the same result as expected.

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Inconsistent Spectral Evolution in Operational Wave Models due to Inaccurate Specification of Nonlinear Interactions

Journal of Physical Oceanography ◽

10.1175/jpo-d-17-0162.1 ◽

2019 ◽

Vol 49 (3) ◽

pp. 705-722 ◽

Cited By ~ 3

Author(s):

Dorukhan Ardag ◽

Donald T. Resio

Keyword(s):

Degrees Of Freedom ◽

Source Term ◽

Wave Spectra ◽

Nonlinear Interactions ◽

Spectral Evolution ◽

Nonlinear Source ◽

Discrete Interaction ◽

Wave Models ◽

Equilibrium Range ◽

Spectral Shapes

AbstractThe introduction of third-generation (3G) models was based on the premise that wave spectra could evolve without prior shape restrictions only if the representation for nonlinear interactions contained as many degrees of freedom as the discretized spectrum being modeled. It is shown here that a different criterion is needed to accurately represent nonlinear spectral evolution within models, a more rigorous criterion such that the number of degrees of freedom in the nonlinear source term must be equal to the intrinsic number of degrees of freedom in the theoretical form of this source term, which is larger than the degrees of freedom in the spectrum. Evolution of spectral shapes produced by the current approximation for nonlinear interactions in 3G models, the discrete interaction approximation (DIA), is compared to the full integral solution for three different time scales: 1) relaxation of the equilibrium range following a perturbation, 2) spectral evolution of the equilibrium range during an interval of constant winds, and 3) the evolution of spectral shape during transition to swell during propagation over long distances. It is shown that the operational nonlinear source term produces significant deviations in the evolution of the wave spectra at all of these scales because of its parametric reduction of the number of degrees of freedom and incorrect energy flux scaling. It is concluded that the DIA does not meet the critical criterion for allowing a spectrum to evolve to spectral shapes consistent with those observed in nature.

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