Fractional Equations Research Articles

Enhancing Learning to Solve Multicomponent Fractional Viscoelastic Equations with U-net Fourier Neural Operators

Abstract The research of viscoelastic media is currently a hot topic in the interpretation and processing of seismic data. To accurately simulate the propagation of seismic waves in viscoelastic media, the fractional viscoelastic equation has emerged as an indispensable method. However, solving this equation numerically has proven to be challenging due to the complexity introduced by its fractional Laplacian operators. Recently, deep learning, especially Fourier neural operators (FNO), has shown excellent performance in learning to fast solve partial differential equations. Traditional FNO methods may face crosstalk problem and make it difficult to achieve satisfactory accuracy, when solving the multi-component fractional order viscoelastic equation. To solve this problem, we introduce a novel approach based on U-net Fourier neural operator (U-FNO). As an enhanced learning method to the traditional FNO-based method, the U-FNO-based method integrates a U-Fourier layer following the standard Fourier layer as a form of regularization, thereby achieving superior prediction accuracy for multi-component equations. Specifically, both the Fourier layers and U-Fourier layers in U-FNO are trained with the solutions of the equation from previous time steps as inputs. This training process enables the U-FNO to efficiently produce more accurate solutions for subsequent wavefield. Numerical simulations reveal that the U-FNO-based method efficiently learns to solve the fractional viscoelastic wave equation independent of fractional Laplacian operators. Additionally, U-FNO-based method offers superior prediction accuracy in comparison with the traditional FNO-based method.

Legendre Polynomials and Techniques for Collocation in the Computation of Variable-Order Fractional Advection-Dispersion Equations

The paper discusses a numerical approach to solving complicated partial differential equations, with a particular emphasis on fractional advection-dispersion equations of space-time variable order. With the use of fractional derivative matrices, Legendre polynomials, and numerical examples and comparisons, it surpasses current methods by utilizing spectral collocation techniques. It resolves equations involving spatial and time variables that are variable-order fractional advection–dispersion (VOFADE). Legendre polynomials serve as basis functions in this method, whereas Legendre operational matrices are employed for fractional derivatives. The technique is more computationally efficient since it reduces fractional advection–dispersion equations to systems of algebraic equations. Numerical examples and a comparison with current approaches illustrate the method’s superior performance in solving complicated partial differential equations, especially in the context of transport processes.

Global Mild Solutions For Hilfer Fractional Neutral Evolution Equation

Abstract In this paper, we present the existence of global mild solutions for the Hilfer fractional neutral evolution equations (HFNEEs), regardless of whether the semigroups are compact or noncompact. We achieve our main results by utilizing the generalization Ascoli-Arzelà theorem, Krasnoselskii’s fixed point theorem, Laplace transform, and measures of noncompactness. To demonstrate the feasibility of our method, we provide an example.

The discrete fractional Karamata theorem and its applications

Abstract We establish a fractional extension of the discrete Karamata integration theorem for two types of fractional sum operators. This result, along with other properties of regularly varying sequences and the tools such as comparison theorems and a fixed point theorem in FK-space, is then used to study asymptotic properties of solutions to a nonlinear fractional difference equation. We also establish and utilize a fractional extension of the Stolz–Cesáro theorem, i.e., of the discrete l’Hospital rule. Both, the discrete fractional Karamata theorem and the discrete fractional l’Hospital rule, are believed to find applications in a broader context within the discrete fractional calculus.

A study on the solvability of fractional integral equation in a Banach algebra via Petryshyn's fixed point theorem

This study focuses on the nonlinear fractional functional integral equation (FFIE) concerning the Riemann-Liouville operator. In certain weaker conditions, the authors demonstrate that the FFIE has a solution, which is defined within the Banach algebra 0 $ ]]> C [ 0 , a ] , a > 0 . Our analysis relies on the Petryshyn's fixed point theorem and the notion of measure of non-compactness (MNC). In addition, our results include numerous authors' work under less restrictive conditions. Furthermore, we provide an illustrative example of fractional functional integral equations to support our proven results.

Analytical study of the time-fractional Smoluchowski coagulation equation in light of different integrodifferential operators

AbstractThis article presents the derivation of the fractional Smoluchowski coagulation equation via the variational principles technique. We use the variational iteration method to solve the Caputo-type fractional coagulation equation. Furthermore, we analyze the time-fractional coagulation equation using the homotopy perturbation transform approach, considering three different fractional operators: Caputo, Caputo-Fabrizio, and Atangana-Baleanu. Our findings demonstrate that the solutions for the total number of particles during coagulation align well with existing literature, particularly in the short time limit. Additionally, we examine the impact of the time-fractional order on the dynamics of particle coagulation for each fractional operator.

Fractional Consistent Riccati Expansion Method and Soliton-Cnoidal Solutions for the Time-Fractional Extended Shallow Water Wave Equation in (2 + 1)-Dimension

In this work, a fractional consistent Riccati expansion (FCRE) method is proposed to seek soliton and soliton-cnoidal solutions for fractional nonlinear evolutional equations. The method is illustrated by the time-fractional extended shallow water wave equation in the (2 + 1)-dimension, which includes a lot of KdV-type equations as particular cases, such as the KdV equation, potential KdV equation, Boiti–Leon–Manna–Pempinelli (BLMP) equation, and so on. A rich variety of exact solutions, including soliton solutions, soliton-cnoidal solutions, and three-wave interaction solutions, have been obtained. Comparing with the fractional sub-equation method, G′/G-expansion method, and exp-function method, the proposed method gives new results. The method presented here can also be applied to other fractional nonlinear evolutional equations.

Sign-changing solutions for a class of fractional Choquard equation with the Sobolev critical exponent in [formula omitted]

In this paper, we study the following fractional Choquard equation with critical exponent(−Δ)σu+u=(Iα⁎|u|p)|u|p−2u+|u|2σ⁎−2uinR3, where σ∈(34,1), α∈(2σ,3) and α+2σ3−2σ<p<2α,σ⁎:=3+α3−2σ, 2α,σ⁎ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, 2σ⁎:=63−2σ is the fractional Sobolev critical exponent and the operator (−Δ)σ stands for the fractional Laplacian of order σ. Based on the above assumptions, we establish the existence of positive ground state solutions, and if p∈(α+2σ3−2σ,2α3−2σ), we also have the corresponding regular property. Subsequently, by introducing some other additional hypotheses on σ, α, p and with the help of quantitative deformation lemma, we employ constrained minimization arguments on the sign-changing Nehari manifold to obtain the existence of ground state sign-changing solutions.

The multi-resolution Haar wavelets collocation procedure for fractional Riccati equations

The multi-resolution Haar wavelets collocation procedure for fractional Riccati equations

Solutions for non-autonomous fractional integrodifferential equations with delayed force term

Solutions for non-autonomous fractional integrodifferential equations with delayed force term

A semilinear problem associated to the space-time fractional heat equation in $${\mathbb {R}}^N$$

A semilinear problem associated to the space-time fractional heat equation in $${\mathbb {R}}^N$$

Analysis of the multi-term fractional Bateman equations in radioactive decay by means of Mikusiński algebraic calculus

Fractional Bateman equations model the concentrations of nuclear species through a radioactive decay chain, by considering non-Markovian behavior. Cruz-López and Espinosa-Paredes (Nuclear Engineering and Technology, 2022) solved the mathematical problem with a single memory coefficient, based on the Laplace-transform technique. Cruz-López, Espinosa-Paredes and François (Computer Physics Communications, 2022) extended the analysis to several fractional orders, with Laplace transforms and series expansions checked by induction. In this paper, we solve such a case of distinct memory effects, via the more general method of Mikusiński operational calculus. We do not distinguish between equal and different decay rates. Multivariate Mittag-Leffler functions appear in the closed-form expression.

A fast H3N3‐2σ$_\sigma$‐based compact ADI difference method for time fractional wave equations

AbstractIn the present paper, we derive a fast H3N3‐2‐based compact alternating direction implicit (ADI) scheme for the time fractional wave equation in two‐dimensional spatial domains. The time fractional derivative involved in the equation is discretized by the H3N3‐2 formula combined with sum‐of‐exponential technique. The former was first proposed by Du et al., while the latter has become a widely used technique to reduce the computational cost in the processing of convolution integrals. The spatial discretization adopts the standard compact ADI method, which can ensure that the space can reach the fourth‐order accuracy without increasing the amount of computation. The numerical stability and convergence of the difference scheme are analyzed rigorously. As an auxiliary illustration, a numerical example is given to verify the validity of the theoretical findings.

Subdiffusion of heavy quarks in hot QCD matter by the fractional Langevin equation

The subdiffusion phenomena are studied for heavy quarks dynamics in the hot QCD matter. Our approach aims to provide a more realistic description of heavy quark dynamics through detailed theoretical analyses and numerical simulations, utilizing the fractional Langevin equation framework with the Caputo fractional derivative. I present numerical schemes for the fractional Langevin equation for subdiffusion and calculate the time evolution mean squared displacement and mean squared momentum of the heavy quarks. My results indicate that the mean squared displacements of the heavy quarks for the subdiffusion process deviate from a linear relationship with time. Further, I calculate the normalized momentum correlation function, kinetic energy, and momentum spread. Finally, I show the effect of subdiffusion on experimental observables, the nuclear modification factor. Published by the American Physical Society 2024

A new error estimate of a finite difference scheme for a fractional transport-advection equation with zero order term

In this work, we propose a finite difference scheme for Caputo’s fractional derivative transport equation in time and space, with a zero-order term. A new error estimation of the approximate solution has been demonstrated. By introducing an approximation of the Caputo derivative, we proved that the convergence is of order 2-α in time, and 2-β in space, for 0<α;β<1. Results of conditional stability and convergence of the numerical method are discussed. Finally a numerical implementation was presented to show the conformity between the theoretical and numerical approach of the finite difference scheme.

Existence, Uniqueness, and Stability of Solutions for Nabla Fractional Difference Equations

In this paper, we study a class of nabla fractional difference equations with multipoint summation boundary conditions. We obtain the exact expression of the corresponding Green’s function and deduce some of its properties. Then, we impose some sufficient conditions in order to ensure existence and uniqueness results. Also, we establish some conditions under which the solution to the considered problem is generalized Ulam–Hyers–Rassias stable. In the end, some examples are included in order to illustrate our main results.

An Exact Solution of the Fractional Transient Electromagnetic Field Inside an Atmospheric Duct

The objective of this paper is to find the exact solution of the fractional electromagnetic (EM) equation within an atmospheric duct (dielectric medium) of the transient EM field caused by a vertical magnetic dipole and discuss the results in both fractional D-dimensional space and integer space. The novelty of this work is the exact solution of the fractional EM equation in terms of Bessel and Mittage-Leffler functions based on Caputo fractional derivative order α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} and the Laplacian operator in D -dimensional fractional space. Using the separation of variables method, the resulting magnetic field can be controlled by D=α1+α2+α3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=\\alpha _1+\\alpha _2+\\alpha _3$$\\end{document} and alpha as spatial and time fractional parameters, respectively. The transient magnetic field behaviour inside the duct is plotted through some of figures depending on fractional parameters D and α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document}. The classical integer results in usual space are recovered from fractional solution at D=α=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=\\alpha =1$$\\end{document}. The main results highlighted are that spatial frequencies grow at a faster rate during EM wave propagation; thus, the amplitude of the propagated wave in integer space is greater than that in fractional space. Increasing the fractional parameters D and α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} simultaneously increases the amplitude of the EM wave and can be used to control the power and energy of the EM wave.

Existence of solutions for systems of conformable fractional dynamic equations on time scales

Existence of solutions for systems of conformable fractional dynamic equations on time scales

Oscillatory and Asymptotic Criteria for a Fifth-Order Fractional Difference Equation

In this paper, using the properties of the conformable fractional difference and fractional sum, we initially establish some oscillatory and asymptotic criteria for a fifth-order fractional difference equation. Several critical inequalities, the Riccati transformation technique, and the integral technique are used in the deduction process. We provide some example to test the results. The established criteria are new results in the study of oscillation, and can be extended to other types of high-order fractional difference equations as well as fractional differential equations with more complicated forms.

Алгоритм для численного решения диффузионно-реакционно-дрейфового уравнения с дробной производной по времени и координате

&lt;p&gt;The paper is devoted to the construction and program implementation of the computational algorithm for modeling a process of diffusion-drift nature based on the fractional diffusion approach. The mathematical model is formulated as an initial-boundary value problem for the time-space fractional diffusion-drift equation in a limited domain. Time and space fractional derivatives are considered in the sense of Caputo and Riemann &amp;ndash; Liouville, respectively. A modified implicit finite-difference scheme is constructed. The concept of the considered mathematical problem provides an example of a deterministic model of the charging process of dielectric materials. An application program has been developed that implements the constructed numerical algorithm. The results were verified using the example of solving a test problem.&lt;/p&gt;