Fractional Evolution Equations Research Articles

Global existence and attractivity for Riemann-Liouville fractional semilinear evolution equations involving weakly singular integral inequalities

In this paper, we obtain several results on the global existence, uniqueness and attractivity for fractional evolution equations involving the Riemann-Liouville type by exploiting some results on weakly singular integral inequalities in Banach spaces. Some boundedness conditions of the nonlinear term are considered to obtain the main results that generalize and improve some well-known works.

Recent advances in employing the Laplace homotopy analysis method to nonlinear fractional models for evolution equations and heat-typed problems

This study stems from the growing need for effective mathematical tools to tackle nonlinear fractional evolution equations, which find wide applications in physics and engineering. Traditional methods often struggle to provide exact closed-form solutions for such equations, leading to the development of novel approaches. In this context, the combination of the homotopy analysis method with the Laplace transform presents a promising avenue for obtaining exact solutions to these complex equations. By exploring this hybrid approach, the study aims to address the existing challenges in solving nonlinear fractional models and provide a reliable method for obtaining closed-form solutions. In this way, this study focuses on rendering the coupling between the homotopy analysis approach and the Laplace transform in the search for exact closed-form solutions for the class of fractional evolution and heat-typed equations. In fact, this class of equations features some interesting nonlinear Caputo fractional partial differential equations that model dissimilar physical processes in physics and engineering. A generalized recurrent scheme is derived, which is applicable to all the members of the governing class, and further analyzed the application of this very scheme on some test fractional initial-value problems of much concern. Eventually, the method has been found to be advantageous, with various advantages, including convergence to available exact solutions. Thus, of the advantage of the current study is innovative combination of mathematical methods, the applicability of these methods to a special class of fractional equations, the development of a generalized scheme, the successful solution of prototype problems, and the practical advantages of the method.

New Study on the Controllability of Non-Instantaneous Impulsive Hilfer Fractional Neutral Stochastic Evolution Equations with Non-Dense Domain

The purpose of this work is to investigate the controllability of non-instantaneous impulsive (NII) Hilfer fractional (HF) neutral stochastic evolution equations with a non-dense domain. We construct a new set of adequate assumptions for the existence of mild solutions using fractional calculus, semigroup theory, stochastic analysis, and the fixed point theorem. Then, the discussion is driven by some suitable assumptions, including the Hille–Yosida condition without the compactness of the semigroup of the linear part. Finally, we provide examples to illustrate our main result.

Lie symmetry analysis for fractional evolution equation with $$\zeta $$-Riemann–Liouville derivative

Lie symmetry analysis for fractional evolution equation with $$\zeta $$-Riemann–Liouville derivative

Utmost travelling wave phenomena to the fractional type nonlinear evolution equation in mathematical physics

In applied physics and engineering, non-linear fractional types partial differential equations become increasingly prominent as an estimation technique to explain a wide variety of non-linear phenomena. Throughout this study, we choose to use an adaptable extended tanh-function scheme over a conformable derivative in order to construct a comprehensive closed form traveling wave solution of the space-time fractional Cahn–Hilliard and modified Kawahara equations. The mentioned equations, which have been used as frameworks for various phenomena such as quantum theory, geosciences, water wave mechanics, and solitary waves in shallow water in the porous medium. The different values of the fractional order of the model can be employed successfully to organize the wave velocity. We recognized numerous forms of solutions by using the computational software, namely solitons type, bell types, kink types, periodic types, multiple periodic, single soliton, singular kink, and other types of solutions that are illustrated using 3D, contour, and spherical. The attained results have effectively handled the previously stated phenomena in various fields of engineering and mathematical physics. The results originating in this study were confirmed by the computational software, which was used to rewrite them as non-linear fractional partial differential equations. We ensure that the technique is revised to make it more effective, pragmatic, and credible, and we pursue more comprehensive exact results for traveling waves, and then for solitary waves.

On new computations of the time-fractional nonlinear KdV-Burgers equation with exponential memory

This paper examines the Korteweg–de Vries-Burgers (KdV-Burgers) equation with nonlocal operators using the exponential decay and Mittag-Leffler kernels. The Caputo-Fabrizio and Atangana-Baleanu operators are used in the natural transform decomposition method (NTDM). By coupling a decomposition technique with the natural transform methodology, the method provides an effective analytical solution. When the fractional order is equal to unity, the proposed approach computes a series form solution that converges to the exact values. By comparing the approximate solution to the precise values, the efficacy and trustworthiness of the proposed method are confirmed. Graphs are also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both operators outcome is examined using diagrams and numerical data. These graphs show how the approximated solution’s graph and the precise solution’s graph eventually converge as the non-integer order gets closer to 1. The outcomes demonstrate the method’s high degree of accuracy and its wide applicability to fractional nonlinear evolution equations. In order to further explain these concepts, simulations are run using a computationally packed software that helps interpret the implications of solutions. NTDM is considered the best analytical method for solving fractional-order phenomena, especially KdV-Burgers equations.

An analysis on asymptotic stability of Hilfer fractional stochastic evolution equations with infinite delay

This article deals with the existence and asymptotic stability in the p-th moment of a mild solution for Hilfer fractional stochastic delay differential equations in Hilbert spaces. Our main results are obtained by fractional calculus, semigroup theory, stochastic analysis, and the Banach fixed point theorem. Further, a new set of sufficient conditions for existence and asymptotic stability in the p-th moment is derived with the help of a fixed point technique. Additionally, we provide an example to demonstrate the reliability of our results.

Oblique Closed Form Solution for Some Type Fractional Evolution Equations in Physical Problem by Using the Homogeneous Balance Method

Oblique Closed Form Solution for Some Type Fractional Evolution Equations in Physical Problem by Using the Homogeneous Balance Method

Decay estimates in time for fractional evolution equations

Decay estimates in time for fractional evolution equations

Fractional Evolution Equations with Nonlocal Initial Conditions and Superlinear Growth Nonlinear Terms

Fractional Evolution Equations with Nonlocal Initial Conditions and Superlinear Growth Nonlinear Terms

The infinite-dimensional Pontryagin maximum principle for optimal control problems of fractional evolution equations with endpoint state constraints

<abstract><p>In this paper, we study the infinite-dimensional endpoint state-constrained optimal control problem for fractional evolution equations. The state equation is modeled by the $ \mathsf{X} $-valued left Caputo fractional evolution equation with the analytic semigroup, where $ \mathsf{X} $ is a Banach space. The objective functional is formulated by the Bolza form, expressed in terms of the left Riemann-Liouville (RL) fractional integral running and initial/terminal costs. The endpoint state constraint is described by initial and terminal state values within convex subsets of $ \mathsf{X} $. Under this setting, we prove the Pontryagin maximum principle. Unlike the existing literature, we do not assume the strict convexity of $ \mathsf{X}^* $, the dual space of $ \mathsf{X} $. This assumption is particularly important, as it guarantees the differentiability of the distance function of the endpoint state constraint. In the proof, we relax this assumption via a separation argument and constructing a family of spike variations for the Ekeland variational principle. Subsequently, we prove the maximum principle, including nontriviality, adjoint equation, transversality, and Hamiltonian maximization conditions, by establishing variational and duality analysis under the finite codimensionality of initial- and end-point variational sets. Our variational and duality analysis requires new representation results on left Caputo and right RL linear fractional evolution equations in terms of (left and right RL) fractional state transition operators. Indeed, due to the inherent complex nature of the problem of this paper, our maximum principle and its proof technique are new in the optimal control context. As an illustrative example, we consider the state-constrained fractional diffusion PDE control problem, for which the optimality condition is derived by the maximum principle of this paper.</p></abstract>

Existence and asymptotic behavior of square-mean $ S $-asymptotically periodic solutions of fractional stochastic evolution equations

This paper investigates the abstract fractional stochastic evolution equations. A new existence result of the square-mean $ S $-asymptotically periodic mild solutions are obtained under the assumption that the nonlinear terms only satisfy some growth conditions. Moreover, the uniqueness and asymptotic stability results of the square-mean $ S $-asymptotically periodic solution are presented when the nonlinear functions satisfy the general Lipschitz condition. Finally, two examples are given to illustrate our main results.

On the Existence and Uniqueness of Fuzzy Mild Solution of Fractional Evolution Equations

In this paper, the nonlocal Cauchy problem is discussed for the fuzzy fractional evolution equations in an arbitrary Banach space for order q ∈ (1, 2) and the criteria on the existence and uniqueness of mild fuzzy solutions are obtained by using Schauder’s fixed point theorem. An example to illustrate the applications of main results is also given.

Finite-Approximate Controllability of ν-Caputo Fractional Systems

This paper introduces a methodology for examining finite-approximate controllability in Hilbert spaces for linear/semilinear ν-Caputo fractional evolution equations. A novel criterion for achieving finite-approximate controllability in linear ν-Caputo fractional evolution equations is established, utilizing resolvent-like operators. Additionally, we identify a control strategy that not only satisfies the approximative controllability property but also ensures exact finite-dimensional controllability. Leveraging the approximative controllability of the corresponding linear ν-Caputo fractional evolution system, we establish sufficient conditions for achieving finite-approximative controllability in the semilinear ν-Caputo fractional evolution equation. These findings extend and build upon recent advancements in this field. The paper also explores applications to ν-Caputo fractional heat equations.

A study of the wave dynamics of the space–time fractional nonlinear evolution equations of beta derivative using the improved Bernoulli sub-equation function approach

The space–time fractional nonlinear Klein-Gordon and modified regularized long-wave equations explain the dynamics of spinless ions and relativistic electrons in atom theory, long-wave dynamics in the ocean, like tsunamis and tidal waves, shallow water waves in coastal sea areas, and also modeling several nonlinear optical phenomena. In this study, the improved Bernoulli sub-equation function method has been used to generate some new and more universal closed-form traveling wave solutions of those equations in the sense of beta-derivative. Using the fractional complex wave transformation, the equations are converted into nonlinear differential equations. The achieved outcomes are further inclusive of successfully dealing with the aforementioned models. Some projecting solitons waveforms, including, kink, singular soliton, bell shape, anti-bell shape, and other types of solutions are displayed through a three-dimensional plotline, a plot of contour, and a 2D plot for definite parametric values. It is significant to note that all obtained solutions are verified as accurate by substituting the original equation in each case using the computational software, Maple. Additionally, the results have been compared with other existing results in the literature to show their uniqueness. The proposed technique is effective, computationally attractive, and trustworthy to establish more generalized wave solutions.

Existence of mild solutions for a coupled system of fractional evolution equations via Mönch's fixed point theorem in Banach spaces

Existence of mild solutions for a coupled system of fractional evolution equations via Mönch's fixed point theorem in Banach spaces

Explore dynamical soliton propagation to the fractional order nonlinear evolution equation in optical fiber systems

Explore dynamical soliton propagation to the fractional order nonlinear evolution equation in optical fiber systems

Performance of affine-splitting pseudo-spectral methods for fractional complex Ginzburg-Landau equations

We evaluate the performance of novel numerical methods for solving one-dimensional nonlinear fractional dispersive and dissipative evolution equations. The methods are based on affine combinations of time-splitting integrators and pseudo-spectral discretizations using Hermite and Fourier expansions. We show the effectiveness of the proposed methods by numerically computing the dynamics of soliton solutions of the standard and fractional variants of the nonlinear Schrödinger equation (NLSE) and the complex Ginzburg-Landau equation (CGLE), and by comparing the results with those obtained by standard splitting integrators. An exhaustive numerical investigation shows that the new technique is competitive when compared to traditional composition-splitting schemes for the case of Hamiltonian problems both in terms accuracy and computational cost. Moreover, it is applicable straightforwardly to irreversible models, outperforming high-order symplectic integrators which could become unstable due to their need of negative time steps. Finally, we discuss potential improvements of the numerical methods aimed to increase their efficiency, and possible applications to the investigation of dissipative solitons that arise in nonlinear optical systems of contemporary interest. Overall, the method offers a promising alternative for solving a wide range of evolutionary partial differential equations.

Regularity theory for a new class of fractional parabolic stochastic evolution equations

AbstractA new class of fractional-order parabolic stochastic evolution equations of the form $$(\partial _t + A)^\gamma X(t) = {\dot{W}}^Q(t)$$ ( ∂ t + A ) γ X ( t ) = W ˙ Q ( t ) , $$t\in [0,T]$$ t ∈ [ 0 , T ] , $$\gamma \in (0,\infty )$$ γ ∈ ( 0 , ∞ ) , is introduced, where $$-A$$ - A generates a $$C_0$$ C 0 -semigroup on a separable Hilbert space H and the spatiotemporal driving noise $${\dot{W}}^Q$$ W ˙ Q is the formal time derivative of an H-valued cylindrical Q-Wiener process. Mild and weak solutions are defined; these concepts are shown to be equivalent and to lead to well-posed problems. Temporal and spatial regularity of the solution process X are investigated, the former being measured by mean-square or pathwise smoothness and the latter by using domains of fractional powers of A. In addition, the covariance of X and its long-time behavior are analyzed. These abstract results are applied to the cases when $$A:= L^\beta $$ A : = L β and $$Q:={\widetilde{L}}^{-\alpha }$$ Q : = L ~ - α are fractional powers of symmetric, strongly elliptic second-order differential operators defined on (i) bounded Euclidean domains or (ii) smooth, compact surfaces. In these cases, the Gaussian solution processes can be seen as generalizations of merely spatial (Whittle–)Matérn fields to space–time.

Diverse optical soliton solutions of two space-time fractional nonlinear evolution equations by the extended kudryashov method

This study investigates the inclusive optical soliton solutions to the (2+1)-dimensional nonlinear time-fractional Zoomeron equation and the space-time fractional nonlinear Chen-Lee-Liu equation using the extended Kudryashov technique. The beta derivative is used to conduct the fractional terms and investigate wide-spectral soliton solutions to the considered models. The obtained solutions yield a variety of typical soliton shapes, including ant-peakon soliton, V-shaped soliton, anti-bell-shaped soliton, kink soliton, periodic soliton, singular periodic soliton for the specific value of the parameters. The three-dimensional, contour, and two-dimensional graphs of the derived solitons have been plotted to illustrate the structure, propagation, and influence of the fractional parameter. It is observed that the fractional parameter affects the amplitudes and periods of certain solitons. The precision of the acquired solutions is confirmed by reintroducing them into the original equation using Mathematica. The findings of this study indicate that the employed method has the capability of yielding compatible, creative, and useful solutions for diverse nonlinear evolution equations with fractional derivatives. This approach could introduce novel ways for unraveling other nonlinear equations and have implications in diverse sectors of nonlinear science and engineering.