Time-fractional Heat Equation Research Articles

Analytical study of time-fractional heat, diffusion, and Burger's equations using Aboodh residual power series and transform iterative methodologies

Analytical study of time-fractional heat, diffusion, and Burger's equations using Aboodh residual power series and transform iterative methodologies

Chebyshev Petrov-Galerkin procedure for the time-fractional heat equation with nonlocal conditions

In this research paper, we address the time-fractional heat conduction equation in one spatial dimension, subject to nonlocal conditions in the temporal domain. To tackle this challenging problem, we propose a novel numerical approach, the ‘Rectified Chebyshev Petrov-Galerkin Procedure,’ which extends the classical Petrov-Galerkin method to efficiently handle the fractional temporal derivatives involved. Our method is characterized by several key contributions; We introduce a set of basis functions that inherently satisfy the homogeneous boundary conditions of the problem, simplifying the numerical treatment. Through careful mathematical derivations, we provide explicit expressions for the matrices involved in the Petrov-Galerkin method. These matrices are shown to be efficiently invertible, leading to a computationally tractable scheme. A comprehensive convergence analysis is presented, ensuring the reliability and accuracy of our approach. We demonstrate that our method converges to the true solution as the spatial and temporal discretization parameters are refined. The proposed Rectified Chebyshev Petrov-Galerkin Procedure is found to be robust, and capable of handling a wide range of problems with nonlocal temporal conditions. To illustrate the effectiveness of our method, we provide a series of numerical examples, including comparisons with existing techniques. These examples showcase the superiority of our approach in terms of accuracy and computational efficiency.

A study of the time-fractional heat equation under the generalized Hukuhara conformable fractional derivative

In this article, we address the dual challenges posed by the complexities of fractional calculus and fuzzy uncertainty in mathematical modeling, with a specific focus on the time-fractional heat equation. We introduce a novel concept, the partial generalized Hukuhara conformable fractional derivative, as a bridge between these domains. Complementing this concept, we develop a unique fuzzy tϑϑ-Laplace transform, which enables efficient problem-solving. This transformative approach, coupled with the fuzzy Fourier transform, provides a novel method for deriving fundamental fuzzy solutions for the time-fractional heat equation. We substantiate our approach through successful examples, highlighting its efficacy in complex scenarios. This study pioneers an innovative methodology that unites fractional calculus and fuzzy logic, opening new avenues for addressing real-world challenges.

Time Fractional Heat Equation of n + 1-Dimension in Type-1 and Type-2 Fuzzy Environment

Time Fractional Heat Equation of n + 1-Dimension in Type-1 and Type-2 Fuzzy Environment

A Tau Approach for Solving Time-Fractional Heat Equation Based on the Shifted Sixth-Kind Chebyshev Polynomials

The time-fractional heat equation governed by nonlocal conditions is solved using a novel method developed in this study, which is based on the spectral tau method. There are two sets of basis functions used. The first set is the set of non-symmetric polynomials, namely, the shifted Chebyshev polynomials of the sixth-kind (CPs6), and the second set is a set of modified shifted CPs6. The approximation of the solution is written as a product of the two chosen basis function sets. For this method, the key concept is to transform the problem governed by the underlying conditions into a set of linear algebraic equations that can be solved by means of an appropriate numerical scheme. The error analysis of the proposed extension is also thoroughly investigated. Finally, a number of examples are shown to illustrate the reliability and accuracy of the suggested tau method.

Lie algebra and conservation laws for the time-fractional heat equation

The Lie symmetry method is applied to derive the point symmetries for the N-dimensional fractional heat equation. We find that the numbers of symmetries and Lie brackets are reduced significantly as compared to the integer order for all dimensions. In fact for integer order linear heat equation the number of solution symmetries is equal to the product of the order and space dimension, whereas for the fractional case, it is half of the product on the order and space dimension. The Lie algebras for the integer and fractional order equations are mentioned using the subsequent computations of Lie brackets and by inspection. Interestingly, it is observed that for the one-dimensional fractional heat equation, the Lie algebra obtained by inspection of symmetries is similar to the result obtained by computation of Lie brackets, which is . The Lie algebra using the symmetries of the two-dimensional heat equation is observed to be , whereas using the Lie brackets the algebra is deduced to be . Hence, it can be concluded that the Lie algebra obtained from the nonzero Lie brackets can be conflated to the algebra which is obtained by inspection. Further, the subsequent Lie algebras are mentioned for the three and four-dimensional integer and fractional equations and the conservation laws are explicitly stated.

The numerical solution of fractional integral equations via orthogonal polynomials in fractional powers

We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders, non-trivial variable coefficients, and initial-boundary conditions. The method uses an orthogonal basis that we refer to as Jacobi fractional polynomials, which are obtained from an appropriate change of variable in weighted classical Jacobi polynomials. New algorithms for building the matrices used to represent fractional integration operators are presented and compared. Even though these algorithms are unstable and require the use of high-precision computations, the spectral method nonetheless yields well-conditioned linear systems and is therefore stable and efficient. For time-fractional heat and wave equations, we show that our method (which is not sparse but uses an orthogonal basis) outperforms a sparse spectral method (which uses a basis that is not orthogonal) due to its superior stability.

Determination of time-dependent coefficient in time fractional heat equation

The aim of this work is to determine the time-dependent heat coefficient in a type of inverse problem for one dimensional time-fractional heat equations defined by the Caputo operator for (0<α<1) by applying a method based on the finite difference scheme and Tikhonov’s regularization. First, a stable implicit finite difference scheme is applied to find a numerical solution to the (forward) direct problem. While the inverse problem was reformulated as a nonlinear least-square minimization problem with a simple physical bound on the unknown coefficient and solved efficiently by MATLAB routine lsqnonlin from the optimization toolbox. But the latter problem will remain ill-posed because the presence of any error in the input data will lead to a large error in the output data. So, Tikhonov’s technique is applied to obtain stable results. In the end, two numerical test examples show that the proposed method is stable, accurate, and works well with different noise levels

Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of Chebyshev polynomials

&lt;abstract&gt;&lt;p&gt;In this article, we propose two numerical schemes for solving the time-fractional heat equation (TFHE). The proposed methods are based on applying the collocation and tau spectral methods. We introduce and employ a new set of basis functions: The unified Chebyshev polynomials (UCPs) of the first and second kinds. We establish some new theoretical results regarding the new UCPs. We employ these results to derive the proposed algorithms and analyze the convergence of the proposed double expansion. Furthermore, we compute specific integer and fractional derivatives of the UCPs in terms of their original UCPs. The derivation of these derivatives will be the fundamental key to deriving the proposed algorithms. We present some examples to verify the efficiency and applicability of the proposed algorithms.&lt;/p&gt;&lt;/abstract&gt;

Recovering Source Term and Temperature Distribution for Nonlocal Heat Equation

We consider two problems of recovering the source terms along with heat concentration for a time fractional heat equation involving the so-called mth level fractional derivative (LFD) (proposed in a paper by Luchko [1]) in time variable of order between 0 and 1. The solutions of both problems are obtained by using eigenfunction expansion method. The series solutions of the inverse problems are proved to be unique and regular. The ill-posedness of inverse problems is proved in the sense of Hadamard and some numerical examples are presented.

MULTI-TERM TIME-FRACTIONAL DERIVATIVE HEAT EQUATION FOR ONE-DIMENSIONAL DUNKL OPERATOR

In this paper, we investigate the well-posedness for Cauchy problem for multi-term time-fractional heat equation associated with Dunkl operator. The equation under consideration includes a linear combination of Caputo derivatives in time with decreasing orders in (0, 1) and positive constant coefficients and one-dimensional Dunkl operator.&nbsp; To show solvability of this problem&nbsp; we use several important properties of multinomial Mittag-Leffler functions and Dunkl transforms, since various estimates follow from the explicit solutions in form of these special functions and transforms. Then we prove the uniqueness and existence results. To achieve our goals, we use methods corresponding to the different areas of mathematics such as the theory of partial differential equations, mathematical physics, hypoelliptic operators theory and functional analysis. In particular, we use the direct and inverse Dunkl transform to establish the existence and uniqueness of solutions to this problem on the Sobolev space. The generalized solutions of this problem are studied.

Study of Convective Heating in a Thermosensitive Functionally Graded Plate with Time-Fractional Heat Equation

The proposed study emphasis on to examine the temperature and stress distribution on functionally graded thermosensitive rectangular plate. Caputo derivative of order 0 to 2 is applied to analyses the rectangular plate when subjected to the convection heat. Present article illustrated the methodology and the mathematical model opt for framing the expression to address the issues regarding the properties of ceramic based material affected by the temperature. The present article studied the Goodier’s displacement function and Boussinesq Harmonic functions for developing the mathematical model for the rectangular plate and other engineering applications.

On the time fractional heat equation with obstacle

We study a Caputo time fractional degenerate diffusion equation which we prove to be equivalent to the fractional parabolic obstacle problem, showing that its solution evolves for any α∈(0,1) to the same stationary state, the solution of the classic elliptic obstacle problem. The only thing which changes with α is the convergence speed. We also study the problem from the numerical point of view, comparing some finite different approaches, and showing the results of some tests. These results extend what recently proved in [1] for the case α=1.

Direct and inverse problems for time-fractional heat equation generated by Dunkl operator

Abstract In this paper, we study non–local in time evolution type equations generated by the Dunkl operator. Direct and inverse problems are investigated with the Caputo time-fractional heat equation with the parameter 0 &lt; γ ≤ 1 {0&lt;\gamma\leq 1} . In particular, well-posedness properties are established for the forward problem. To adopt techniques of the harmonic analysis, we solve the problems in the Sobolev type spaces associated with the Dunkl operator. Our special interest is an inverse source problem for the Caputo–Dunkl heat equation. As additional data, the final time measurement is taken. Since our inverse source problem is ill-posed, we also show the stability result. Moreover, as an advantage of our calculus used here, we derive explicit formulas for the solutions of the direct and inverse problems.

Tychonoff Solutions of the Time-Fractional Heat Equation

In the literature, one can find several applications of the time-fractional heat equation, particularly in the context of time-changed stochastic processes. Stochastic representation results for such an equation can be used to provide a Monte Carlo simulation method, upon proving that the solution is actually unique. In the classical case, however, this is not true if we do not consider any additional assumption, showing, thus, that the Monte Carlo simulation method identifies only a particular solution. In this paper, we consider the problem of the uniqueness of the solutions of the time-fractional heat equation with initial data. Precisely, under suitable assumptions about the regularity of the initial datum, we prove that such an equation admits an infinity of classical solutions. The proof mimics the construction of the Tychonoff solutions of the classical heat equation. As a consequence, one has to add some addtional conditions to the time-fractional Cauchy problem to ensure the uniqueness of the solution.

On generalized fractional integral with multivariate Mittag-Leffler function and its applications

The fractional calculus (FC) has been extensively studied by researchers due to its vast applications in sciences in the last few years. In fractional calculus, multivariate Mittag–Leffler functions are considered the powerful extension of the classical Mittag–Leffler functions. This paper defines the generalized fractional integral operator with multivariate Mittag–Leffler (M-L) function. We prove certain basic properties of the proposed operators, such as an expansion of an infinite series of Riemann–Liouville integrals, Laplace transform (LT), semigroup property, composition with Riemann–Liouville integrals. Also, we present the fractional differential operators and their properties. The application of the proposed operators like the fractional kinetic differential and the time-fractional heat equation are also discussed.

Stochastic solutions for time-fractional heat equations with complex spatial variables

We deal with complex spatial diffusion equations with time-fractional derivative and study their stochastic solutions. In particular, we complexify the integral operator solution to the heat-type equation where the time derivative is replaced with the convolution-type generalization of the regularized Caputo derivative. We prove that this operator is solution of a complex time-fractional heat equation with complex spatial variable. This approach leads to a wrapped Brownian motion on a circle time-changed by the inverse of the related subordinator. This time-changed Brownian motion is analyzed and, in particular, some results on its moments, as well as its construction as weak limit of continuous-time random walks, are obtained. The extension of our approach to the higher dimensional case is also provided.

Approximate analytical solution of time-fractional non-linear heat equation via fractional power series method

A time-fractional heat equation arising in a quiescent medium is established, and its approximate analytical solution is obtained by the fractional power series method. The results show that the method performs extremely well in terms of efficiency and simplicity.

Solution to unsteady fractional heat conduction in the quarter-plane via the joint Laplace-Fourier sine transforms

In this article, the author implemented the joint transform method, for solving the boundary value problems of time fractional heat equation. The results reveal that the integral transform method is reliable and efficient. Illustrative examples are also provided.

Existence and Uniqueness of Solutions in Some Boundary Conditions for Fractional Differential Systems

In this article, fractional case of periodic problems is discussed. Considering the time fractional heat equation, inverse problems with periodic and anti-periodic boundary conditions were created. For these problems, the Fourier method was used to obtain existence and uniqueness results. The fractional derivative of a periodic function was analyzed along the real axis, and the periodic behavior of linear systems in case of fractions was investigated.