System Of Fractional Differential Equations Research Articles

A New Iterative Predictor-Corrector Algorithm for Solving a System of Nuclear Magnetic Resonance Flow Equations of Fractional Order

Nuclear magnetic resonance flow equations, also known as the Bloch system, are said to be at the heart of both magnetic resonance imaging (MRI) and nuclear magnetic resonance (NMR) spectroscopy. The main aim of this research was to solve fractional nuclear magnetic resonance flow equations (FNMRFEs) through a numerical approach that is very easy to handle. We present a New Iterative Predictor-Corrector Algorithm (NIPCA) based on the New Iterative Algorithm and Predictor-Corrector Algorithm to solve nonlinear nuclear magnetic resonance flow equations of fractional order involving Caputo derivatives. Graphical representation of the solutions with detailed error analysis shows the higher accuracy of the new technique. This New Iterative Predictor-Corrector Algorithm requires less computational time than previously published numerical methods. The results achieved in this article indicate that the algorithm is fit to use for other chaotic systems of fractional differential equations.

Qualitative Analysis of Coupled Fractional Differential Equations involving Hilfer Derivative

Abstract In this manuscript, we have studied the coupled system of Hilfer fractional differential equations with non-local conditions. We have used the Leray-alternative Schauder’s and the Contraction principle to obtain the results on the existence and uniqueness of the solution of the proposed problem in the weighted space of continuous functions. For the defined problem, sufficient conditions have also been developed to determine the Ulam stability of the solution. The key conclusions are well-illustrated with examples.

Existence and stability results on multidimensional fractional-order systems

The novelties of this paper is the study of existence and uniqueness results for a class of incommensurate fractional differential equation systems with general multiorders using the weighted infinity norm with respect to the classical Mittag-Leffler function via the contraction mapping principle. The lack of stability results in the Ulam–Hyers sense on fractional multidimensional differential equations motivates us to generalize our theory to stability analysis based on fixed point approach.

Extended eigenvalue–eigenvector method

In this paper, we show that the eigenvalue–eigenvector method (EEM) and its extension can be utilized to obtain the state probabilities of Poisson-type counting processes. An extension of the EEM, namely, the extended eigenvalue–eigenvector method (EEEM) is introduced which can be used to solve the linear system of fractional differential equations involving Caputo fractional derivative. As an illustration, we obtain the known state probabilities of the time fractional Poisson process and the fractional birth process using the EEEM.

Krasnoselskii type theorems in product Banach spaces and applications to systems of nonlinear transport equations and mixed fractional differential equations

Krasnoselskii type theorems in product Banach spaces and applications to systems of nonlinear transport equations and mixed fractional differential equations

Existence results for weakly coupled system of ψ-Caputo fractional differential equations with nonlinear boundary conditions

In this paper, our aim is to develop monotone iterative technique by introducing the notion of coupled upper, lower and quasi solutions and apply it to prove existence of solution of nonlinear boundary value problems for weakly coupled system of ψ-Caputo fractional differential equations. We choose suitable initial iterations and construct two monotone sequences which converge monotonically from above and below to quasi solutions of nonlinear weakly coupled system of ψ-Caputo fractional differential equations. Further we show that these sequences lead to existence of solution of nonlinear boundary value problem for weakly coupled system of ψ-Caputo fractional differential equations.

Existence and Stability for a Coupled Hybrid System of Fractional Differential Equations with Atangana–Baleanu–Caputo Derivative

The aim of this article is to investigate a coupled hybrid system of fractional differential equations with the Atangana–Baleanu–Caputo derivative which contains a Mittag–Leffler kernel function in its kernel. We firstly apply the Dhage fixed point principle to obtain the existence of mild solutions. Then, we study the Ulam–Hyers stability of the introduced fractional coupled hybrid system. Finally, an example is presented to exhibit the validity of our results.

Existence and uniqueness results for coupled system of fractional differential equations with exponential kernel derivatives

<abstract><p>In the framework of Caputo-Fabrizio derivatives, we study a new coupled system of fractional differential equations of higher orders supplemented with coupled nonlocal boundary conditions. The existence and uniqueness results of the solutions are proved. We consider the classical fixed-point theories due to Banach and Krasnoselskii for the main results. An example illustrating the main results is introduced.</p></abstract>

Solvability of infinite systems of fractional differential equations in the space of tempered sequence space $m^beta(phi)$

The purpose of this article, is to establish the existence of solution of infinite systems of fractional differential equations in space of tempered sequence $m^beta(phi)$ by using techniques associated with Hausdorff measures of noncompactness. Finally, we provide an example to highlight and establish the importance of our main result.

A spectral collocation method for the coupled system of nonlinear fractional differential equations

<abstract><p>This paper analyzes the coupled system of nonlinear fractional differential equations involving the caputo fractional derivatives of order $ \alpha\in(1, 2) $ on the interval (0, T). Our method of analysis is based on the reduction of the given system to an equivalent system of integral equations, then the resulting equation is discretized by using a spectral method based on the Legendre polynomials. We have constructed a Legendre spectral collocation method for the coupled system of nonlinear fractional differential equations. The error bounds under the $ L^2- $ and $ L^{\infty}- $norms is also provided, then the theoretical result is validated by a number of numerical tests.</p></abstract>

Novel fixed point technique to coupled system of nonlinear implicit fractional differential equations in complex valued fuzzy rectangular $ b $-metric spaces

<abstract><p>The fundamental purpose of this research is to investigate the existence theory as well as the uniqueness of solutions to a coupled system of fractional order differential equations with Caputo derivatives. In this regard, we utilize the definition and properties of a newly developed conception of complex valued fuzzy rectangular $ b $-metric spaces to explore the fuzzy form of some significant fixed point and coupled fixed point results. We further present certain examples and a core lemma in the case of complex valued fuzzy rectangular $ b $-metric spaces.</p></abstract>

Positive solutions of infinite coupled system of fractional differential equations in the sequence space of weighted means

<abstract><p>We first discuss the existence of solutions of the infinite system of $ (n-1, n) $-type semipositone boundary value problems (BVPs) of nonlinear fractional differential equations</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} D^{\alpha}_{0_+}u_i(\rho)+\eta f_i(\rho,v(\rho)) = 0,& \rho\in(0,1), \\ D^{\alpha}_{0_+}v_i(\rho)+\eta g_i(\rho,u(\rho)) = 0,& \rho\in(0,1), \\u_i^{(j)}(0) = v_{i}^{(j)}(0) = 0,& 0\leq j\leq n-2, \\ u_{i}(1) = \zeta\int_0^1 u_i(\vartheta)d\vartheta, \ v_{i}(1) = \zeta\int_0^1 v_i(\vartheta)d\vartheta,& i\in\mathbb{N},\\ \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>in the sequence space of weighted means $ c_0(W_1, W_2, \Delta) $, where $ n\geq3 $, $ \alpha\in(n-1, n] $, $ \eta, \zeta $ are real numbers, $ 0 < \eta < \alpha, $ $ D^{\alpha}_{0_+} $ is the Riemann-Liouville's fractional derivative, and $ f_i, g_i, $ $ i = 1, 2, \ldots $, are semipositone and continuous. Our approach to the study of solvability is to use the technique of measure of noncompactness. Then, we find an interval of $ \eta $ such that for each $ \eta $ lying in this interval, the system of $ (n-1, n) $-type semipositone BVPs has a positive solution. Eventually, we demonstrate an example to show the effectiveness and usefulness of the obtained result.</p></abstract>

Analysis of COVID-19 epidemic model with sumudu transform.

In this paper, we develop a time-fractional order COVID-19 model with effects of disease during quarantine which consists of the system of fractional differential equations. Fractional order COVID-19 model is investigated with ABC technique using sumudu transform. Also, the deterministic mathematical model for the quarantine effect is investigated with different fractional parameters. The existence and uniqueness of the fractional-order model are derived using fixed point theory. The sumudu transform can keep the unity of the function, the parity of the function, and has many other properties that are more valuable. Solutions are derived to investigate the influence of fractional operator which shows the impact of the disease during quarantine on society.

Existence and Uniqueness of Positive Solutions for a Coupled System of Fractional Differential Equations

In this article, we investigate a boundary value problem for a coupled differential system of fractional order that the nonlinear term depends on the unknown functions as well as their lower order fractional derivatives. Firstly, we give Green’s functions and prove their properties; secondly, the existence and uniqueness of positive solutions are obtained by using some fixed point theorems. In addition, two examples are presented to demonstrate the application of our main results.

On finding exact and approximate solutions to fractional systems of ordinary differential equations using fractional natural adomian decomposition method

In this work, we present proofs for new theorems that deal with natural transform method (NTM) with Caputo derivative. Also, we give exact and approximate solutions to systems of fractional differential equations along with fractional ordinary and partial differential equations using the fractional natural decomposition method (FNDM). The Caputo derivative is used here to minimize the amount of computational, and this is of great significance for large-scale problems. The work outlines the significant features of the FNDM. Our work can be considered as another technique to existing methods, and will have many applications in variant areas of science and engineering.

Numerical Simulations of One-Directional Fractional Pharmacokinetics燤odel

In this paper, we present a three-compartment of pharmacokinetics model with irreversible rate constants. The compartment consists of arterial blood, tissues and venous blood. Fick’s principle and the law of mass action were used to develop the model based on the diffusion process. The model is modified into a fractional pharmacokinetics model with the sense of Caputo derivative. The existence and uniqueness of the model are investigated and the positivity of the model is established. The behaviour of the model is investigated by implementing numerical algorithms for the numerical solution of the system of fractional differential equations. MATLAB software is used to plot the graphs for illustrating the variation of drug concentration concerning time. Therefore, the numerical simulations of the model are presented for different values of α which verified the theoretical analysis. Besides, we also observed the pattern of the simulations at the three-compartment of the model by using different values of initial conditions.

Nonlinear higher order fractional terminal value problems

<abstract><p>Terminal value problems for systems of fractional differential equations are studied with an especial focus on higher-order systems. Discretized piecewise polynomial collocation methods are used for approximating the exact solution. This leads to solving a system of nonlinear equations. For solving such a system an iterative method with a required tolerance is introduced and analyzed. The existence of a unique solution is guaranteed with the aid of the fixed point theorem. Order of convergence for the given numerical method is obtained. Numerical experiments are given to support theoretical results.</p></abstract>

Solvability of Some Nonlocal Fractional Boundary Value Problems at Resonance in ℝn

In this paper, the solvability of a system of nonlinear Caputo fractional differential equations at resonance is considered. The interesting point is that the state variable x∈Rn and the effect of the coefficient matrices matrices B and C of boundary value conditions on the solvability of the problem are systematically discussed. By using Mawhin coincidence degree theory, some sufficient conditions for the solvability of the problem are obtained.

A new study on the existence and stability to a system of coupled higher-order nonlinear BVP of hybrid FDEs under the $ p $-Laplacian operator

<abstract><p>In this paper, we study a general system of fractional hybrid differential equations with a nonlinear $ \phi_p $-operator, and prove the existence of solution, uniqueness of solution and Hyers-Ulam stability. We use the Caputo fractional derivative in this system so that our system is more general and complex than other nonlinear systems studied before. To establish the results, Green functions are used to transform the considered hybrid boundary problem into a system of fractional integral equations. Then, with the help of the topological degree theorem, we derive some sufficient conditions that ensure the existence and uniqueness of solutions for the proposed system. Finally, an example is presented to show the validity and correctness of the obtained results.</p></abstract>

Existence and Uniqueness for Coupled Systems of Hilfer Type Sequential Fractional Differential Equations Involving Riemann–Stieltjes Integral Multistrip Boundary Conditions

In this paper, we study a coupled system of Hilfer type sequential fractional differential equations supplemented with Riemann–Stieltjes integral multistrip boundary conditions. The standard tools of the fixed point theory are employed to prove the existence and uniqueness results for the considered problem. Examples are constructed for the illustration of the obtained results.