Fractional Integral Conditions Research Articles

A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition

Abstract In this work, we present a sophisticated operating algorithm, the reproducing kernel Hilbert space method, to investigate the approximate numerical solutions for a specific class of fractional Begley-Torvik equations (FBTE) equipped with fractional integral boundary condition. Such fractional integral boundary condition allows us to understand the non-local behavior of FBTE along with the given domain. The algorithm methodology depends on creating an orthonormal basis based on reproducing kernel function that satisfies the constraint boundary conditions so that the solution is finally formulated in the form of a uniformly convergent series in ϖ 3 [ a , b ] {\varpi }_{3}\left[a,b] . From a numerical point of view, some illustrative examples are provided to determine the appropriateness of algorithm design and the effect of using non-classical boundary conditions on the behavior of solutions approach.

Analysis of coupled system of q$$ q $$‐fractional Langevin differential equations with q$$ q $$‐fractional integral conditions

In this dissertation, we study the coupled system of ‐fractional Langevin differential equations involving ‐Caputo derivative having ‐fractional integral conditions. With the help of some adequate conditions, we investigate the uniqueness and existence of mild solution of the aforementioned system. We also analyze various kinds of Ulam's stability. Banach fixed point theorem and Leray–Schauder of cone type are used to illustrate the existence and uniqueness results. We also used non‐linear functional analysis methods to explore variety of stability types. An example is provided to clearly demonstrate our theoretical outcomes.

Solvability of a ϱ-Hilfer Fractional Snap Dynamic System on Unbounded Domains

This paper is devoted to studying the ϱ-Hilfer fractional snap dynamic system under the ϱ-Riemann–Liouville fractional integral conditions on unbounded domains [a,∞),a≥0, for the first time. The results concerning the existence and uniqueness, along with the Ulam–Hyers, Ulam–Hyers–Rassias, and semi-Ulam–Hyers–Rassias stabilities, are established in an appropriate special Banach space according to fractional calculus, fixed point theory, and nonlinear analysis. At the end, a numerical example is presented for the interpretation of the main results.

Analysis of q‐fractional coupled implicit systems involving the nonlocal Riemann–Liouville and Erdélyi–Kober q‐fractional integral conditions

The solutions to fractional differential equations are a developing area of current research, given that these equations arise in various domains. In this article, we provide some necessary criteria for the existence, uniqueness, and different types of Ulam stability for a coupled implicit system requiring the conditions for nonlocal Riemann–Liouville and Erdélyi–Kober q‐fractional integral conditions. The uniqueness and existence results for the suggested coupled system are demonstrated using Banach fixed point theorem and Leray–Schauder of cone type. We also explore the various types of stability using classical methods of nonlinear functional analysis. To verify the effectiveness of our theoretical outcomes, we study an interesting example.

Observability of fractional order differential impulsive multi control problem with fractional integral nonlocal initial condition

In this paper ,the obsevability of fractional differential impulsive multi control abstract problem with fractional integral nonlocal initial condition have been studied as abstract Cauchy problem for using Banach fixed point which defined on space of presented problem which is piecewise continuous space and proprieties of the initial observable condition for their problem.

On Atangana–Baleanu-Type Nonlocal Boundary Fractional Differential Equations

This research paper is devoted to investigating two classes of boundary value problems for nonlinear Atangana–Baleanu-type fractional differential equations with Atangana–Baleanu fractional integral conditions. The applied fractional derivatives work as the nonlocal and nonsingular kernel. Upon using Krasnoselskii’s and Banach’s fixed point techniques, we establish the existence and uniqueness of solutions for proposed problems. Moreover, the Ulam–Hyers stability theory is constructed by using nonlinear analysis. Eventually, we provide two interesting examples to illustrate the effectiveness of our acquired results.

Positive solutions for fractional differential equations with non-separated type nonlocal multi-point and multi-term integral boundary conditions

In this paper, we investigate a class of nonlinear fractional differential equations that contain both the multi-term fractional integral boundary condition and the multi-point boundary condition. By the Krasnoselskii fixed point theorem we obtain the existence of at least one positive solution. Then, we obtain the existence of at least three positive solutions by the Legget-Williams fixed point theorem. Two examples are given to illustrate our main results.

Boundary value problems for Hilfer fractional differential equations with Katugampola fractional integral and anti-periodic conditions

The purpose of this paper is to investigate the existence and uniqueness of solutions for a new class of nonlinear fractional differential equations involving Hilfer fractional operator with fractional integral boundary conditions. Our analysis relies on classical fixed point theorems and the Boyd-Wong nonlinear contraction. At the end, an illustrative example is presented. The boundary conditions introduced in this work are of quite general nature and can be reduce to many special cases by fixing the parameters involved in the conditions.

New results for higher‐order Hadamard‐type fractional differential equations on the half‐line

The purpose of this paper is to analyze a new kind of Hadamard‐type fractional boundary value problem combining integral boundary condition and multipoint fractional integral boundary condition on an infinite interval. By the help of the Bai‐Ge's fixed point theorem, multiplicity results of positive solutions are derived for the Hadamard‐type fractional boundary value problem. In the end, an example is also presented to illustrative the main result.

Sequential Riemann–Liouville and Hadamard–Caputo Fractional Differential Equation with Iterated Fractional Integrals Conditions

In the present research, we initiate the study of boundary value problems for sequential Riemann–Liouville and Hadamard–Caputo fractional derivatives, supplemented with iterated fractional integral boundary conditions. Firstly, we convert the given nonlinear problem into a fixed point problem by considering a linear variant of the given problem. Once the fixed point operator is available, we use a variety of fixed point theorems to establish results regarding existence and uniqueness. Some properties of iteration that will be used in our study are also discussed. Examples illustrating our main results are also constructed. At the end, a brief conclusion is given. Our results are new in the given configuration and enrich the literature on boundary value problems for fractional differential equations.

Existence and uniqueness solutions for nonlinear fractional differential equations with fractional integral boundary conditions

The theory of Schuder fixed point and Banach contraction theorems are used to explain the behavior of solution for differential equation of fractional order β ∈ (2, 3] with fractional integral limit conditions given by .We prove the existence and uniqueness solution of the above equations.

Existence of solutions for a coupled system of Caputo-Hadamard type fractional di¤erential equations with Hadamard fractional integral conditions

In this work, we study existence and uniqueness of solutions for a coupled system of nonlinear fractional di¤erential equations involving two Caputo-Hadamard-type fractional derivatives. By applying the Banach's fixed point theorem and Shaefer's fixed point theorem, the existence of solutions is obtained.The results obtained in this work are well illustrated with the aid of examples.

Multiple solutions for a nonlocal fractional boundary value problem with fractional integral conditions on infinite interval

This paper discusses the existence and multiplicity of positive solutions for fractional differential equations with nonlocal fractional integro-differential boundary conditions set on an unbounded domain. Using properties of Green’s function and the fixed point theory, some existence results were obtained. Three examples illustrate the existence theorems.

Katugampola Fractional Differential Equation with Erdelyi-Kober Integral Boundary Conditions

In this paper, we study the existence and uniqueness of solutions for nonlinear fractional Katugampola differential equation with Erdely-Kober fractional integral conditions, new existence and uniqueness results are established using Banach's contraction principle, nonlinear contractions, Krasnoselskii's and Leray-Schauder's fixed theorems. Four examples are given in order to clarify theoretical results.

Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration

In the three-dimensional open rectangular domain, the problem of the identification of the redefinition function for a partial differential equation with Gerasimov–Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the 0<α≤1 order. A positive parameter is present in the mixed derivatives. The solution of this fractional differential equation is studied in the class of regular functions. The Fourier series method is used, and a countable system of ordinary fractional differential equations with degeneration is obtained. The presentation for the redefinition function is obtained using a given additional condition. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proven.

Hilfer fractional differential inclusions with Erd\xe9lyi\u2013Kober fractional integral boundary condition

In this article, we debate the existence of solutions for a nonlinear Hilfer fractional differential inclusion with nonlocal Erdélyi–Kober fractional integral boundary conditions (FIBC). Both cases of convex- and nonconvex-valued right-hand side are considered. Our obtained results are new in the framework of Hilfer fractional derivative and Erdélyi–Kober fractional integral with FIBC via the fixed point theorems (FPTs) for a set-valued analysis. Some pertinent examples demonstrating the effectiveness of the theoretical results are presented.

On a nonlinear parabolic equation with fractional Laplacian and integral conditions

The paper deals with non-classical initial-boundary value problems for parabolic equations with a fractional Laplacian. We study the existence and uniqueness of a mild solution to our problem. The continuous dependence of the solution on the given data is shown and the ill-posedness of the mild solution at t = 0 is also considered. In order to avoid such ill-posedness, we construct a regularized solution using the Fourier truncation method. An error estimate and the convergence rate between the regularized solution and the exact solution are obtained.

The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative

In the finance market, it is well known that the price change of the underlying fractal transmission system can be modeled with the Black-Scholes equation. This article deals with finding the approximate analytic solutions for the time-fractional Black-Scholes equation with the fractional integral boundary condition for a European option pricing problem in the Katugampola fractional derivative sense. It is well known that the Katugampola fractional derivative generalizes both the Riemann–Liouville fractional derivative and the Hadamard fractional derivative. The technique used to find the approximate analytic solutions of the time-fractional Black-Scholes equation is the generalized Laplace homotopy perturbation method, the combination of the generalized Laplace transform and homotopy perturbation method. The approximate analytic solution for the problem is in the form of the generalized Mittag-Leffler function. This shows that the generalized Laplace homotopy perturbation method is one of the most effective methods to construct approximate analytic solutions of the fractional differential equations. Finally, the approximate analytic solutions of the Riemann–Liouville and Hadamard fractional Black-Scholes equation with the European option are also shown.

Nonlocal boundary value problems for integro-differential Langevin equation via the generalized Caputo proportional fractional derivative

Results reported in this paper study the existence and stability of a class of implicit generalized proportional fractional integro-differential Langevin equations with nonlocal fractional integral conditions. The main theorems are proved with the help of Banach’s, Krasnoselskii’s, and Schaefer’s fixed point theorems and Ulam’s approach. Finally, an example is given to demonstrate the applicability of our theoretical findings.

Existence Results for Sequential Riemann–Liouville and Caputo Fractional Differential Inclusions with Generalized Fractional Integral Conditions

Under different criteria, we prove the existence of solutions for sequential fractional differential inclusions containing Riemann–Liouville and Caputo type derivatives and supplemented with generalized fractional integral boundary conditions. Our existence results rely on the endpoint theory, the Krasnosel’skiĭ’s fixed point theorem for multivalued maps and Wegrzyk’s fixed point theorem for generalized contractions. We demonstrate the application of the obtained results with the help of examples.