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25 pages, 375 KiB

Open AccessArticle

Spatial Discretization for Stochastic Semilinear Superdiffusion Driven by Fractionally Integrated Multiplicative Space–Time White Noise

by James A. Hoult and Yubin Yan

Foundations 2023, 3(4), 763-787; https://doi.org/10.3390/foundations3040043 - 6 Dec 2023

Viewed by 938

Abstract

We investigate the spatial discretization of a stochastic semilinear superdiffusion problem driven by fractionally integrated multiplicative space–time white noise. The white noise is characterized by its properties of being white in both space and time, and the time fractional derivative is considered in [...] Read more.

We investigate the spatial discretization of a stochastic semilinear superdiffusion problem driven by fractionally integrated multiplicative space–time white noise. The white noise is characterized by its properties of being white in both space and time, and the time fractional derivative is considered in the Caputo sense with an order

α

∈ (1, 2). A spatial discretization scheme is introduced by approximating the space–time white noise with the Euler method in the spatial direction and approximating the second-order space derivative with the central difference scheme. By using the Green functions, we obtain both exact and approximate solutions for the proposed problem. The regularities of both the exact and approximate solutions are studied, and the optimal error estimates that depend on the smoothness of the initial values are established. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions II)

33 pages, 481 KiB

Open AccessArticle

Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise

by Bernard A. Egwu and Yubin Yan

Foundations 2023, 3(2), 290-322; https://doi.org/10.3390/foundations3020023 - 29 May 2023

Viewed by 4921

Abstract

We investigate the application of the Galerkin finite element method to approximate a stochastic semilinear space–time fractional wave equation. The equation is driven by integrated additive noise, and the time fractional order

α \in (1, 2)

. The existence of [...] Read more.

We investigate the application of the Galerkin finite element method to approximate a stochastic semilinear space–time fractional wave equation. The equation is driven by integrated additive noise, and the time fractional order

α \in (1, 2)

. The existence of a unique solution of the problem is proved by using the Banach fixed point theorem, and the spatial and temporal regularities of the solution are established. The noise is approximated with the piecewise constant function in time in order to obtain a stochastic regularized semilinear space–time wave equation which is then approximated using the Galerkin finite element method. The optimal error estimates are proved based on the various smoothing properties of the Mittag–Leffler functions. Numerical examples are provided to demonstrate the consistency between the theoretical findings and the obtained numerical results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions II)

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15 pages, 309 KiB

Open AccessArticle

Random Solutions for Generalized Caputo Periodic and Non-Local Boundary Value Problems

by Bashir Ahmad, Mokhtar Boumaaza, Abdelkrim Salim and Mouffak Benchohra

Foundations 2023, 3(2), 275-289; https://doi.org/10.3390/foundations3020022 - 29 May 2023

Cited by 1 | Viewed by 1176

Abstract

In this article, we present some results on the existence and uniqueness of random solutions to a non-linear implicit fractional differential equation involving the generalized Caputo fractional derivative operator and supplemented with non-local and periodic boundary conditions. We make use of the fixed [...] Read more.

In this article, we present some results on the existence and uniqueness of random solutions to a non-linear implicit fractional differential equation involving the generalized Caputo fractional derivative operator and supplemented with non-local and periodic boundary conditions. We make use of the fixed point theorems due to Banach and Krasnoselskii to derive the desired results. Examples illustrating the obtained results are also presented. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions II)

15 pages, 321 KiB

Open AccessArticle

Existence in the Large for Caputo Fractional Multi-Order Systems with Initial Conditions

by Zachary Denton and Aghalaya S. Vatsala

Foundations 2023, 3(2), 260-274; https://doi.org/10.3390/foundations3020021 - 26 May 2023

Cited by 2 | Viewed by 1106

Abstract

One of the key applications of the Caputo fractional derivative is that the fractional order of the derivative can be utilized as a parameter to improve the mathematical model by comparing it to real data. To do so, we must first establish that [...] Read more.

One of the key applications of the Caputo fractional derivative is that the fractional order of the derivative can be utilized as a parameter to improve the mathematical model by comparing it to real data. To do so, we must first establish that the solution to the fractional dynamic equations exists and is unique on its interval of existence. The vast majority of existence and uniqueness results available in the literature, including Picard’s method, for ordinary and/or fractional dynamic equations will result in only local existence results. In this work, we generalize Picard’s method to obtain the existence and uniqueness of the solution of the nonlinear multi-order Caputo derivative system with initial conditions, on the interval where the solution is bounded. The challenge presented to establish our main result is in developing a generalized form of the Mittag–Leffler function that will cooperate with all the different fractional derivative orders involved in the multi-order nonlinear Caputo fractional differential system. In our work, we have developed the generalized Mittag–Leffler function that suffices to establish the generalized Picard’s method for the nonlinear multi-order system. As a result, we have obtained the existence and uniqueness of the nonlinear multi-order Caputo derivative system with initial conditions in the large. In short, the solution exists and is unique on the interval where the norm of the solution is bounded. The generalized Picard’s method we have developed is both a theoretical and a computational method of computing the unique solution on the interval of its existence. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions II)

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19 pages, 355 KiB

Open AccessArticle

Coupled Systems of Nonlinear Proportional Fractional Differential Equations of the Hilfer-Type with Multi-Point and Integro-Multi-Strip Boundary Conditions

by Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon

Foundations 2023, 3(2), 241-259; https://doi.org/10.3390/foundations3020020 - 24 May 2023

Cited by 3 | Viewed by 2790

Abstract

In this paper, we study a coupled system of nonlinear proportional fractional differential equations of the Hilfer-type with a new kind of multi-point and integro-multi-strip boundary conditions. Results on the existence and uniqueness of the solutions are achieved by using Banach’s contraction principle, [...] Read more.

In this paper, we study a coupled system of nonlinear proportional fractional differential equations of the Hilfer-type with a new kind of multi-point and integro-multi-strip boundary conditions. Results on the existence and uniqueness of the solutions are achieved by using Banach’s contraction principle, the Leray–Schauder alternative and the well-known fixed-point theorem of Krasnosel’skiĭ. Finally, the main results are illustrated by constructing numerical examples. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions II)

18 pages, 327 KiB

Open AccessArticle

A Comparison Result for the Nabla Fractional Difference Operator

by Jagan Mohan Jonnalagadda

Foundations 2023, 3(2), 181-198; https://doi.org/10.3390/foundations3020016 - 12 Apr 2023

Viewed by 1201

Abstract

This article establishes a comparison principle for the nabla fractional difference operator

\nabla_{ρ (a)}^{ν}

,

1 < ν < 2

. For this purpose, we consider a two-point nabla fractional boundary value problem with separated boundary conditions and derive [...] Read more.

This article establishes a comparison principle for the nabla fractional difference operator

\nabla_{ρ (a)}^{ν}

,

1 < ν < 2

. For this purpose, we consider a two-point nabla fractional boundary value problem with separated boundary conditions and derive the corresponding Green’s function. I prove that this Green’s function satisfies a positivity property. Then, I deduce a relatively general comparison result for the considered boundary value problem. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions II)

Review

Jump to: Research

64 pages, 648 KiB

Open AccessReview

Ostrowski-Type Fractional Integral Inequalities: A Survey

by Muhammad Tariq, Sotiris K. Ntouyas and Bashir Ahmad

Foundations 2023, 3(4), 660-723; https://doi.org/10.3390/foundations3040040 - 13 Nov 2023

Viewed by 1132

Abstract

This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions,

(ζ, m)

-convex [...] Read more.

This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions,

(ζ, m)

-convex functions, s-convex functions,

(s, r)

-convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions,

M T

-convex functions, P-convex functions, m-convex functions,

(s, m)

-convex functions, exponentially s-convex functions,

(β, m)

-convex functions, exponential-convex functions,

\bar{ζ}, β, γ, δ

-convex functions, quasi-geometrically convex functions,

s - e

-convex functions and n-polynomial exponentially s-convex functions. Riemann–Liouville fractional integral, Katugampola fractional integral, k-Riemann–Liouville, Riemann–Liouville fractional integrals with respect to another function, Hadamard fractional integral, fractional integrals with exponential kernel and Atagana-Baleanu fractional integrals are included. Results for Ostrowski-Mercer-type inequalities, Ostrowski-type inequalities for preinvex functions, Ostrowski-type inequalities for Quantum-Calculus and Ostrowski-type inequalities of tensorial type are also presented. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions II)

40 pages, 480 KiB

Open AccessReview

A Comprehensive Review of the Hermite–Hadamard Inequality Pertaining to Quantum Calculus

by Muhammad Tariq, Sotiris K. Ntouyas and Asif Ali Shaikh

Foundations 2023, 3(2), 340-379; https://doi.org/10.3390/foundations3020026 - 15 Jun 2023

Cited by 2 | Viewed by 1303

Abstract

A review of results on Hermite–Hadamard (H-H) type inequalities in quantum calculus, associated with a variety of classes of convexities, is presented. In the various classes of convexities this includes classical convex functions, quasi-convex functions, p-convex functions,

(p, s)

[...] Read more.

A review of results on Hermite–Hadamard (H-H) type inequalities in quantum calculus, associated with a variety of classes of convexities, is presented. In the various classes of convexities this includes classical convex functions, quasi-convex functions, p-convex functions,

(p, s)

-convex functions, modified

(p, s)

-convex functions,

(p, h)

-convex functions,

t g s

-convex functions,

η

-quasi-convex functions,

ϕ

-convex functions,

(α, m)

-convex functions,

ϕ

-quasi-convex functions, and coordinated convex functions. Quantum H-H type inequalities via preinvex functions and Green functions are also presented. Finally, H-H type inequalities for

(p, q)

-calculus, h-calculus, and

(q - h)

-calculus are also included. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions II)

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