Infinite-time bubble towers in the fractional heat equation with critical exponent

Applications of the Laplace variational iteration method to fractional heat like equations

The Fractional Power Series Method (FPSM) is a method which provides systematic procedure to obtain exact solution of the Fractional Partial Differential Equations (FPDEs). Recently, the FPSM has been applied in science and engineering to address physical problems in heat conduction, fluid dynamics, quantum mechanics, viscoelastic and so on. Interestingly, applying the FPSM to look for the solution of the FPDE contains the Mittag–Leffler function. The solution is feasible due to the involvement of the Mittag–Leffler function with one parameter. However, the Prabhakar function which generalizes the Mittag–Leffler function has been overlooked by the researchers across the globe. This Prabhakar function contains an additional parameter, the Pochhammer symbol, which when incorporated in the FPSM yields not only exact solution but also a continuum solution of the FPDE in a functional space. In this paper, a modified version, known as the Fractional Prabhakar Power Series Method (FPPSM), is introduced to find the solutions to both the fractional heat and telegraph equations. Thus, this improved method incorporates the Prabhakar function instead of the Mittag–Leffler function. The additional parameter in the Prabhakar function provides the fast convergent solution in terms of the number of steps involved in obtaining its solution. The FPPSM, when applied to find for the solution of FPDE, yields a series which converges to the exact solution of the FPDE. The FPPSM is applied to obtain the solutions of the fractional heat equations in multidimensions: two and three dimensions and the fractional telegraph equation in one dimension. The series obtained using the FPPSM is derived to be in Sobolev spaces, ensuring the existence of a unique solution on the grounds that the Sobolev spaces are complete. Also, a unique stable solution with respect to small perturbation in the initial conditions of the fractional heat and telegraph equation is established in this paper, verifying that both the fractional heat and telegraph equations are well posed. The comparative analysis among the FPPSM and other existing methods such as the HAM, the VIM and the ADM is provided to ensure the efficacy of the FPPSM.

في هذه الورقة تمت دراسة معادلة انتشار الحرارة باستخدام التفاضل الكسري وكان التركيز على البارومترات ∝و β= 1⁄2 وحصلنا على الحل التحليلي باستخدام طرق لابلاس و فوريير وقد كان هذا الحل متقاربا للحل الكلاسيكي (حل انتشار الموجة الحرارية).

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.

Recently, a number of researchers have turned their attention to the creation of isometrically invariant shape descriptors based on the heat equation. The reason for this surge in interest is that the Laplace-Beltrami operator, associated with the heat equation, is highly dependent on the topology of the underlying manifold, which may lead to the creation of highly accurate descriptors. In this paper, we propose a generalisation based on the fractional heat equation. While the heat equation enables one to explore the shape with a Markovian Gaussian random walk, the fractional heat equation explores the manifold with a non-Markovian Levy random walk. This generalisation provides two advantages. These are, first, that the process has a memory of the previously explored geometry and, second, that it is possible to correlate points or vertices which are not part of the same neighbourhood. Consequently, a highly accurate, contextual shape descriptor may be obtained.

The Fractional Power Series Method (FPSM) is an effective and efficient method that offers an analytic method to find exact solution for Fractional Partial Differential Equations (FPDEs) in a functional space. In recent time, the FPSM has been applied in various science and engineering fields to solve physical problems in areas such as fluid dynamics, quantum mechanics, viscoelasticity, and heat conduction. This paper introduces a modification of the FPSM called the Fractional Gauss Hypergeometric Power Series Method (FGHPSM) which employs the so‐called Gauss Hypergeometric Function (GHF) to replace the Mittag–Leffler function in the FPSM on the grounds that the GHF generalizes the Mittag–Leffler function. This GHF when integrated into the FPSM provides not only exact solution but a generalized solution as compared with solution of the same equation using the FPSM. The FGHPSM is applied to solve fractional heat equation in two and three dimensions as well as fractional telegraph equation in a single dimension. The series obtained by the FGHPSM is derived to be in Sobolev spaces ensuring the existence of a unique solution. Also, a unique stable solution with respect to small perturbation in the initial conditions of the fractional heat and telegraph equations is established in this paper. MSC2020 Classification: 35J10

Boundary control strategy for three kinds of fractional heat equations with control-matched disturbances

Existence of solutions for an inhomogeneous fractional semilinear heat equation

<p>The highly successful Budyko-Sellers energy balance models are based on the classical continuum mechanics heat equation in two spatial dimensions. When extended to the third dimension using the correct conductive-radiative surface boundary conditions, we show that surface temperature anomalies obey the (nonclassical) Half-order energy balance equation (HEBE, with exponent H = ½) implying heat is stored in the subsurface with long memory. </p><p> </p><p>Empirically, we find that both internal variability and the forced response to external variability are compatible with H ≈ 0.4.  Although already close to the HEBE and classical continuum mechanics, we argue that an even more realistic “effective media” macroweather model is a generalization: the fractional heat equation (FHE) for long-time (e.g. monthly scale anomalies).  This model retains standard diffusive and advective heat transport but generalize the (temporal) storage term.  A consequence of the FHE is that the surface temperature obeys the Fractional EBE (FEBE), generalizing the HEBE to 0< H ≤1.  We show how the resulting FEBE can be been used for monthly and seasonal forecasts as well as for multidecadal climate projections.  We argue that it can also be used for understanding and modelling past climates.</p>

In this article we analyse quantitative approximation properties of a certain class of nonlocal equations: Viewing the fractional heat equation as a model problem, which involves both local and nonlocal pseudodifferential operators, we study quantitative approximation properties of solutions to it. First, relying on Runge type arguments, we give an alternative proof of certain qualitative approximation results from [ 9 ]. Using propagation of smallness arguments, we then provide bounds on the cost of approximate controllability and thus quantify the approximation properties of solutions to the fractional heat equation. Finally, we discuss generalizations of these results to a larger class of operators involving both local and nonlocal contributions.

This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is formulated in terms of the Fourier transform of its spatially and temporally homogeneous Green function. The spectral density of the resulting solution is then obtained explicitly. The result implies that the solution of the fractional heat equation may possess spatial long-range dependence asymptotically.

This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is formulated in terms of the Fourier transform of its spatially and temporally homogeneous Green function. The spectral density of the resulting solution is then obtained explicitly. The result implies that the solution of the fractional heat equation may possess spatial long-range dependence asymptotically.

In this work, the exponential stability of the nonlocal fractional heat equation is studied. The fractional Laplacian is defined via a singular integral. Using the spectral properties of the fractional Laplacian and a state decomposition, a feedback control is constructed by considering the first N modes and an observer defined via a bounded operator. Different configurations are examined, including interior controller with interior observation, and interior controller with exterior observation. Using the recent result about the simplicity of the eigenvalues (Fall et al. in Calc Var Partial Differ Equ 62(8):233, 2023), some of our stabilization results are valid for s∈(0,1), in particular for s∈(0,1/2) in which case the fractional heat equation is not null controllable.

Nonlinear fractional stochastic heat equation driven by Gaussian noise rough in space

We prove that the initial-value problem for the fractional heat equation admits a solution provided that the (possibly unbounded) initial datum has a conveniently moderate growth at infinity. Under the same growth condition we also prove that the solution is unique. Our result does not require any sign assumption, thus complementing the Widder's type theorem of Barrios et al. (Arch. Rational Mech. Anal. 213 (2014) 629-650) for positive solutions. Finally, we show that the fractional heat flow preserves convexity of the initial datum. Incidentally, several properties of stationary convex solutions are established.