Problems For Fractional Differential Equations Research Articles

Optimal solution of fractional differential equations in solving the relief of college students’ mental obstacles

Abstract The thesis applies solutions to boundary value problems of a class of fractional differential equation coupled systems with a p-Laplacian operator. This article applies this part of the content to alleviate the psychological barriers of college students. This article transforms the tendency of individual consciousness in college students’ psychological counselling into a problem of solving boundary value problems of fractional differential equations. Studies have shown that using fractional differential equations to analyse the causes of college students’ psychological barriers has an essential role in eliminating college students’ psychological barriers.

Weighted Cauchy problem: fractional versus integer order

This work is devoted to the solvability of the weighted Cauchy problem for fractional differential equations of arbitrary order, considering the Riemann–Liouville derivative. We show the equivalence between the weighted Cauchy problem and the Volterra integral equation in the space of Lebesgue integrable functions. Finally, we point out some discrepancies between the solutions for fractional and integer order case.

DIFFERENTIAL EQUATIONS WITH TEMPERED Ψ-CAPUTO FRACTIONAL DERIVATIVE

In this paper we define a new type of the fractional derivative, which we call tempered Ψ−Caputo fractional derivative. It is a generalization of the tempered Caputo fractional derivative and of the Ψ−Caputo fractional derivative. The Cauchy problem for fractional differential equations with this type of derivative is discussed and some existence and uniqueness results are proved. We present a Henry-Gronwall type inequality for an integral inequality with the tempered Ψ−fractional integral. This inequality is applied in the proof of an existence theorem. A result on a representation of solutions of linear systems of Ψ−Caputo fractional differential equations is proved and in the last section an example is presented.

Monotone Iterative and Upper–Lower Solution Techniques for Solving the Nonlinear ψ−Caputo Fractional Boundary Value Problem

The objective of this paper is to study the existence of extremal solutions for nonlinear boundary value problems of fractional differential equations involving the ψ−Caputo derivative CDa+σ;ψϱ(t)=V(t,ϱ(t)) under integral boundary conditions ϱ(a)=λIν;ψϱ(η)+δ. Our main results are obtained by applying the monotone iterative technique combined with the method of upper and lower solutions. Further, we consider three cases for ψ*(t) as t, Caputo, 2t, t, and Katugampola (for ρ=0.5) derivatives and examine the validity of the acquired outcomes with the help of two different particular examples.

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 of solutions of boundary value problems for nonlinear fractional differential equations with integral conditions

In this work we investigate the existence and uniqueness of solutions of boundary value problems for fractional differential equations involving the Caputo fractional derivative with integral conditions and the nonlinear term depends on the fractional derivative of an unknown function. Our existence results are based on Banach contraction principle and Schauder fixed point theorem. Two examples are provided to illustrate our results.

Existence and stability criterion for the results of fractional order Φp-Laplacian operator boundary value problem

In this literature, we study the existence and stability of the solution of the boundary value problem of fractional differential equations with Φp-Laplacian operator. Our problem is based on Caputo fractional derivative of orders σ, ϵ, where k − 1 < σ, ϵ ≤ k, and k ≥ 3. By using the Schauder fixed point theory and properties of the Green function, some conditions are established which show the criterion of the existence and non-existence solution for the proposed problem. We also investigate some adequate conditions for the Hyers-Ulam stability of the solution. Illustrated examples are given as an application of our result.

Existence and Uniqueness Analysis for Fractional Differential Equations with Nonlocal Conditions

Fractional differential equations and systems are gradually becoming an essential approach to real world applications in science and engineering technology. The boundary value problems of the fractional differential equations and systems were investigated by using several different methods. Recently, much attention has been paid to investigate fractional differential equations with nonlocal conditions by using the fixed point method. In this paper, by using the Banach fixed point theorem and Krasnoselkii fixed point theorem, the existence and uniqueness of the solutions to the initial value problem of the nonlinear fractional differential equations with nonlocal conditions are investigated, and some sufficient conditions are obtained. We extend some results that already exist. Finally, an example is given to show the usefulness of the theoretical results.

Intermediate value problems for fractional differential equations

Intermediate value problems for fractional differential equations

Unique existence of solution to initial value problem for fractional differential equation involving with fractional derivative of variable order

We deal with unique existence of solution to initial value problem for fractional differential equation involving with fractional derivative of variable order. Some examples substantiate these theoretical results.

Existence and Uniqueness of Positive Solution for Boundary Value Problem of Fractional Differential Equations

Existence and Uniqueness of Positive Solution for Boundary Value Problem of Fractional Differential Equations

Existence results for a class of boundary value problems for fractional differential equations

By application of some fixed point theorems, that is, the Banach fixed point theorem, Schaefer's and the Leray-Schauder fixed point theorem, we establish new existence results of solutions to boundary value problems of fractional differential equations. This paper is motivated by Agarwal et al. (Georgian Math. J. 16 (2009) No.3, 401-411).

Numerical solutions of wavelet neural networks for fractional differential equations

Neural network has good self‐learning and adaptive capabilities. In this paper, a wavelet neural network is proposed to be used to solve the value problem of fractional differential equations (FDE). We construct a wavelet neural network (WNN) with the structure 1 ×N× 1 based on the wavelet function and give the conditions for the convergence of the given algorithm. This method uses the truncated power series of the solution function to transform the original differential equation into an approximate solution, then, using WNN, update the parameters, and finally get the FDE solution. Simulation results prove the validity of WNN.

Cauchy type problems for fractional differential equations

While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann–Liouville derivatives is less understood. In this paper, we propose new type initial, inner, and inner-boundary value problems for fractional differential equations with the Riemann–Liouville derivatives. The results on the existence and uniqueness are proved, and conditions on the solvability are found. The well-posedness of the new type of initial, inner, and inner-boundary conditions is also discussed. Moreover, we give explicit formulas for the solutions. As an application fractional partial differential equations for general positive operators are studied.

On establishing qualitative theory to nonlinear boundary value problem of fractional differential equations

In this article, we study a class of nonlinear fractional differential equation for the existence and uniqueness of a positive solution and the Hyers–Ulam-type stability. To proceed this work, we utilize the tools of fixed point theory and nonlinear analysis to investigate the concern theory. We convert fractional differential equation into an integral alternative form with the help of the Greens function. Using the desired function, we studied the existence of a positive solution and uniqueness for proposed class of fractional differential equation. In next section of this work, the author presents stability analysis for considered problem and developed the conditions for Ulam’s type stabilities. Furthermore, we also provided two examples to illustrate our main work.

An Integral Boundary Value Problem of Fractional Differential Equations with a Sign-Changed Parameter in Banach Spaces

This paper is to investigate the existence and uniqueness of solutions for an integral boundary value problem of new fractional differential equations with a sign-changed parameter in Banach spaces. The main used approach is a recent fixed point theorem of increasing Ψ − h , r -concave operators defined on ordered sets. In addition, we can present a monotone iterative scheme to approximate the unique solution. In the end, two simple examples are given to illustrate our main results.

Boundary Value Problems for Fractional Differential Equation in Special Banach Space

This paper studies the existence of solutions of boundary value problem for fractional differential equations on the half-line in a special Banach space. The main result is based on Monch fixed point theorem combining with a suitable measure of non-compactness, an example is given to illustrate our approach.

Multiple solutions for a class of boundary value problems of fractional differential equations with generalized Caputo derivatives

&lt;abstract&gt;&lt;p&gt;This paper is mainly concerned with the existence of multiple solutions for the following boundary value problems of fractional differential equations with generalized Caputo derivatives:&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \hskip 3mm \left\{ \begin{array}{lll} ^{C}_{0}D^{\alpha}_{g}x(t)+f(t, x) = 0, \ 0&amp;lt;t&amp;lt;1;\\ x(0) = 0, \ ^{C}_{0}D^{1}_{g}x(0) = 0, \ ^{C}_{0}D^{\nu}_{g}x(1) = \int_{0}^{1}h(t)^{C}_{0}D^{\nu}_{g}x(t)g'(t)dt, \end{array}\right. $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where $ 2 &amp;lt; \alpha &amp;lt; 3 $, $ 1 &amp;lt; \nu &amp;lt; 2 $, $ \alpha-\nu-1 &amp;gt; 0 $, $ f\in C([0, 1]\times \mathbb{R}^{+}, \mathbb{R}^{+}) $, $ g' &amp;gt; 0 $, $ h\in C([0, 1], \mathbb{R}^{+}) $, $ \mathbb{R}^{+} = [0, +\infty) $. Applying the fixed point theorem on cone, the existence of multiple solutions for considered system is obtained. The results generalize and improve existing conclusions. Meanwhile, the Ulam stability for considered system is also considered. Finally, three examples are worked out to illustrate the main results.&lt;/p&gt;&lt;/abstract&gt;

On definition of solution of initial value problem for fractional differential equation of variable order

We propose a new definition of continuous approximate solution to initial value problem for differential equations involving variable order Caputo fractional derivative based on the classical definition of solution of integer order (or constant fractional order) differential equation. Some examples are presented to illustrate these theoretical results.

Controllability and Hyers-Ulam stability results of initial value problems for fractional differential equations via generalized proportional-Caputo fractional derivative

Controllability and Hyers-Ulam stability results of initial value problems for fractional differential equations via generalized proportional-Caputo fractional derivative