We consider the initial-boundary value problem for the complex Ginzburg-Landau equation with fractional Laplacian on a hyperoctant {[ u_t(t,x)-∇ ^βu(t,x)=| u(t,x)| ^σu(t,x), 𝐱∈ 𝐃^n, t>0,; u(0,x)=u_0(x), 𝐱∈ 𝐃^n,; u| _x_j=0 =h_j(t,x_j), j=1,2,...,n, t>0, ]. where D^n=( 𝐱={ x_j} _j=1^n; x_j>0) , x_j=(x_1,x_2,...,x_j-1,x_j+1,...,x_n), n∈ N , β∈( 3/2,2) , σ >0 and ∇ ^β is a fractional Laplacian defined as ∇ ^βu=∑ _j=1^n1/ (2-β )∫ _0^x_ju_y_jy_j/( x_j-y_j) ^β -1dy_j. We study the main questions of the theory of IBV problems for nonlocal equations: (1) the existence and uniqueness of solutions, (2) the asymptotic behavior of the solution for large time, (3) the influence of initial and boundary data on the basic properties of the solution. We generalize the concept of the well-posedness of IBV problems in the L^2 based Sobolev spaces to the case of a Dirichlet problem. We give optimal relations between the orders of the Sobolev spaces to which the initial and boundary data belong in order to minimize the compatibility conditions for initial and boundary data. Our approach is based on the Riemann-Hilbert theory and the multidimensional theory of Laplace transforms.
Variable-order time-fractional derivatives provide a flexible framework for modeling systems with time-dependent memory. Most existing variable-order formulations are obtained by formally replacing the constant order with a time-dependent function, a procedure that typically leads to the loss of fundamental structural properties such as the existence of a left inverse and the validity of a fundamental theorem of calculus. In contrast, the Scarpi derivative, defined through the Laplace transform and based on Sonine kernels, preserves these essential properties in the variable-order setting. Since explicit time-domain expressions of kernels are difficult to obtain, in this work we develop analytical time-domain approximations for order-modulated Scarpi-Sonine kernels by means of asymptotic expansions of their Laplace transforms. The accuracy of the proposed approximations is assessed through systematic comparisons with kernels obtained via numerical inversion of the Laplace transform. Furthermore, we apply inverse Laplace transform techniques to compute numerical solutions of some scalar and coupled linear initial-value problems governed by variable-order Scarpi derivatives to illustrate the influence of variable orders on the solution behavior.
The paper is concerned with the three-dimensional micropolar fluid equations with Caputo time-fractional derivatives. By exploiting the smoothing effects of the Laplacian dissipation involving the time-fractional evolution mechanism, we establish global existence and uniqueness of mild solutions of the three-dimensional time-fractional micropolar fluid equations for small initial data in Besov spaces.
This paper deals with the finite-approximate controllability for the Hilfer fractional evolution equations using resolvent families of operators generated by a closed linear operator, which allows us to avoid the density assumption on D(A). We first give a new definition of mild solutions using strongly continuous operators consisting of the Wright function and resolvent operators, which is applicable to the Hilfer fractional cases of order α∈ (0,1) . Then the existence of mild solutions and the finite-approximate controllability of the Hilfer fractional evolution equations are obtained without the uniqueness.
The activation of neuronal interaction in the Caputo-Amari neuronal model involves Heaviside functions and is formulated mathematically as a Caputo inclusion system. Sigmoidal functions resulting in a system of Caputo FDEs provide an analytically and computationally convenient approximation to such systems. They are used here to obtain the existence of solutions to the fractional inclusion system.
In this paper, we investigate the asymptotic behavior of solutions to a coupled system of structurally damped wave equations with a critical nonlinearity. Our main contributions include proving the existence of a global attractor, establishing its regularity, and obtaining an upper bound for its fractal dimension. Additionally, we analyze the continuity of attractors with respect to the coupling parameter ε . To overcome the challenges posed by the critical nonlinearity, we employ the Galerkin method to establish well-posedness. Asymptotic compactness is ensured by combining the quasi-stability method with a refined decomposition of the critical nonlinear term.
In this paper, we developed Fujita’s critical exponent to determine whether or not the following fractional reaction-diffusion system admits global solutions or only blow-up solutions: (S){[ u_t=Δ _α u+ v^p, in ℝ^N× (0,∞ ),; v_t=Δ _α v+ u^q, in ℝ^N× (0,∞ ),; u(· ,0)=u_0≥ 0, v(· ,0)=v_0≥ 0, in ℝ^N, ]. where p≥ q ≥ 1 with pq>1 and Δ _α:=-( -Δ) ^α /2 denotes the fractional Laplace operator with 0<α≤ 2 on ℝ^N, N≥ 2 . More precisely, for the number (pq)^*:= α (p+1)/N +1 , we proved the following:
In this paper, we investigate the strong and weak type boundedness of the fractional maximal operator in Musielak-Orlicz-Morrey spaces. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness in less restrictive terms.
We study a class of stochastic Caputo fractional evolution equations with additive Q -Wiener noise in a separable Hilbert space. By combining sectorial operator theory with tools from fractional calculus, we employ a Green–Caputo representation for mild solutions. Using this representation, we establish well-posedness, moment estimates, and mean-square stability properties. We also investigate Hyers–Ulam–Rassias stability in the mean-square setting and provide sufficient conditions for this property. In addition, we introduce a Green–Caputo type time-stepping scheme and analyze its strong convergence and discrete stability behavior. Numerical experiments are presented to support the theoretical results.
The paper is devoted to the study of a six-parameter generalization of the Krätzel function introduced by E. Krätzel [Integral transformations of Bessel type, in Generalized Functions and Operational Calculus, Bulgarian Academy of Sciences, Sofia, 1979, pp. 148-155 ]. The representation of the function in terms of a series and an H-function is presented. The analytic properties like log-convexity, complete monotonicity and Turán type inequality of the function are substantiated. Furthermore, the concordance of the dual pathway generalized Krätzel function with the Weyl fractional integral and differential operators is examined.
Compartmental systems permit the analysis of epidemiological and sociological phenomena, due to the possibility of capturing transitions in stages and interactions. Recently, the authors of this paper studied the role of Caputo fractional calculus in compartmental modeling, in the partial and pure versions of the formulation, where the non-Markovian property is related to variable latency or decision periods and the fractional index controls the decreasing risk of transition. Several mechanistic considerations were made that pointed out some issues with the fractional calculus. In the present work, we provide an expository analysis of fractional compartmental modeling and compare it with the traditional use of delay differential equations, using recent studies on delay models in epidemiology. In general terms, the contribution of fractional calculus remains to be fully justified.
In this paper, we study the following double critical Schrödinger-Poisson system involving the fractional p-Laplacian in ℝ^3 of the form: {[ (-Δ )^s_p u-ϕ |u|^p_s^♯ -2u=λ |u|^p-2u+μ |u|^q-2u+|u|^p_s^*-2u in ℝ^3,; (-Δ )^sϕ =|u|^p_s^♯ in ℝ^3, ]. and prescribed mass ∫ _ℝ^3 |u|^pdx=a^p, where (-Δ )^s_p is the fractional p-Laplace operator, s∈ (0,1) , sp<3 , μ , a>0, λ∈ℝ , q∈ (p, p_s^*) and p_s^*:= 3p/3-sp , p_s^♯ :=p(3+2s)/2(3-sp) is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. For the L^p -subcritical case, we show the existence of multiple normalized solutions by using the truncation technique and the genus theory. For the L^p -supercritical case, we obtain a couple of normalized solutions by using the auxiliary functional. For both cases, in order to overcome the loss of compactness of the energy functional due to the double critical growth, the concentration-compactness principle is needed to overcome this difficulty. In a sense, we generalize some of the previous results [2, 24, 33, 54]. As far as we know, this study seems to be the first contribution regarding existence of normalized solutions for double critical Schrödinger-Poisson system involving the fractional p-Laplacian.
This paper is concerned with a class of time-fractional subdiffusion-normal transport equations that describe the transition from subdiffusion to normal diffusion. Our primary contribution is the establishment of the well-posedness and Strichartz estimates for global solutions on the unbounded domain ℝ^N . Furthermore, we extend our analysis to bounded domains with smooth boundaries, proving well-posedness and Strichartz estimates for small initial data and under the condition of a polynomial growth nonlinearity.
We establish a space–time duality principle for fractional powers of sectorial operators and for right-sided fractional identities satisfied by their semigroup orbits. Let 0<α <1 . If A is an injective sectorial operator of angle strictly smaller than π /2 and admits a bounded holomorphic H^∞ -calculus, then the semigroup orbit u(t)=e^-tA^αφ which solves the space–fractional problem ∂ _tu+A^α u=0 , also satisfies a right-sided Liouville identity of order 1/α involving the original spatial operator A. Thus the same orbit of the semigroup generated by -A^α admits an additional future-dependent fractional representation involving the local spatial operator A. In the selfadjoint nonnegative Hilbert-space case the abstract identity becomes a genuine right-sided Liouville identity on the decaying spectral component of the solution. This duality is then used to analyze nonlocal diffusion on graphs. For proper graph Laplacians L on infinite locally finite graphs, the fractional powers L^α are shown to be mixed nonlocal: their heat flows have a spatially nonlocal realization through L^α , while the same orbits satisfy an additional right-sided Liouville identity involving the local Laplacian L. By contrast, transformed d–path Laplacians, including Estrada-type operators on the infinite path, are purely spatially nonlocal: they cannot be represented as fractional powers of finite-range operators. On finite graphs, fractional roots always exist by spectral calculus, but they are typically dense rather than local. Finally, for Mellin–transformed d–path Laplacians on ℤ we prove stable-type scaling and long-time heat-kernel asymptotics: for 13 the classical Gaussian regime is recovered.
This work deals with certain class of Caputo-type fractional stochastic differential inclusions driven by Brownian motion. Under suitable measurability and Lipschitz conditions on the multivalued drift and diffusion terms, we establish the existence of solutions first via a fixed-point argument and then via a Filippov-type approximation procedure. The proof relies on stochastic analysis, fractional calculus, and a measurable selection argument. Our result extends the classical Filippov theorem to the fractional stochastic framework and partially generalizes prior work by Da Prato and Frankowska (1994).
We introduce a Tricomi-type generalized fractional calculus in the Sonine kernel framework. The key result is that the Tricomi branch is a Stieltjes function in the admissible parameter range, so its reciprocal is a complete Bernstein function. This fact induces a Sonine fractional calculus together with the canonical Tricomi integral and the associated Riemann–Liouville-type and Caputo-type derivatives. We also prove that, within the Kummer class, the Tricomi branch is the unique Stieltjes representative, once the natural asymptotic normalization is fixed, and we derive the corresponding Lévy–Khintchine representation, Volterra formulation, and scalar Cauchy problem. A distinctive feature of the resulting operators is the emergence of two independent asymptotic orders.
In this work, we prove that the effect of damping given by tempered Caputo derivative does not lead the porous elastic model to exponential stability. Although damping acts on both equations of the system, it is not strong enough to lead the model to exponential stabilization, which is novel for this type of model, where a single damping factor provides exponential stability depending on a relationship between the velocity coefficients. Thus, we prove that the model is well-posed and polynomially stable.
It is well known that investigations on exact solutions of nonlinear fractional partial differential equations (PDEs) are very difficult compare with those investigations on integer-order nonlinear PDEs. In this paper, a new method called the separation method of semi-fixed variables together with the dynamical system method is introduced. The connection and difference between the classical separation method of variables and the separation method of semi-fixed variables are compared in details. As example, a generalized nonlinear time-fractional reaction-diffusion equation with higher-order terms is studied under the definition of Riemann-Liouville fractional derivative. In different parametric regions, different kinds of phase portraits of the system derived from the generalized model are presented. Existence and dynamic properties of solutions of the generalized model are investigated. In some special parametric conditions, some exact solutions of the generalized equation are obtained. In the absence of the reaction term, a very strange and interesting phenomenon of the model is found.
Recent research in financial volatility modeling demonstrates that rough volatility models, driven by fractional Brownian motion (fBm) with a small Hurst parameter, play a crucial role in understanding the implied volatility surface and improving option pricing. However, the affine structure of the standard Heston model only provides analytical expressions for the characteristic function and density function of the variance process under specific forms of volatility dynamics. To overcome this limitation, this paper introduces a non-affine coefficient, enabling the model to more flexibly capture complex characteristics of asset returns such as excess kurtosis and heavy tails. At the macro level, we replace the Brownian motion in the non-affine Heston model with fBm, constructing the rough non-affine Heston model. At the micro level, we establish a theoretical foundation for the model by linking nearly unstable Hawkes processes to fractional volatility models. Finally, empowered by a robust Lifted Semi-Implicit Euler scheme and the CRN technique, we empirically calibrate the models using S P 500 index options during the March 2020 crash. The results demonstrate that the rough non-affine model outperforms its affine counterpart by capturing extreme left-tail skewness and drastically reducing downside pricing errors.
