Fabrizio Fractional Derivative Research Articles

New Existence and Stability Results for ψ , w -Caputo−Fabrizio Fractional Nonlocal Implicit Problems

In the context of ψ -weighted Caputo−Fabrizio fractional derivatives, we develop and extend the existence and Ulam−Hyers stability results for nonlocal implicit differential equations. The fixed-point theorems due to Banach and Krasnoselskii are the foundation for the proof of existence and uniqueness results. Additionally, the Ulam−Hyers stability demonstrates the assurance of the existence of solutions via Gronwal inequality. Also, we offer an example as an application to explain and validate the acquired results. Finally, in terms of our outcome, we designate a more general problem for the ψ , w -Caputo−Fabrizio fractional system that includes analogous problems to the problem at hand.

Solutions of Nonlinear Fractional Differential Equations with Nondifferentiable Terms

In this research, we employ a newly developed strategy based on a modified version of the Adomian decomposition method (ADM) to solve nonlinear fractional differential equations (FDE) with both differential and nondifferential variables. FDE have disturbed the interest of many researchers. This is due to the development of both the theory and applications of fractional calculus. This track from various areas of fractional differential equations can be used to model various fields of science and engineering such as fluid flows, viscoelasticity, electrochemistry, control, electromagnetic, and many others. Several fractional derivative definitions have been presented, including Riemann–Liouville, Caputo,and Caputo– Fabrizio fractional derivative. We just need to calculate the first Adomain polynomial in this technique avoiding the hurdles in the nondifferentiable nonlinear terms' remaining polynomials. Furthermore, the proposed technique is easy to programme and produces the desired output with minimal work and time on the same processor. When compared to the exact solution, this method has the advantage of reducing calculation steps, while producing accurate results. The supporting evidence proves that modified Adomian decomposition has an advantage over traditional Adomian decomposition method which can be explained very clear with nonlinear fractional differential equations. Our computational examples with difficult issues are used to prove the new algorithm's efficiency. The results show that the modified ADM is powerful, which has a faster convergence solution than the original one. Convergence analysis is discussed, also the uniqueness is explained.

A New Numerical Approach for Solving Fractional Optimal Control Problems with the Caputo–Fabrizio Fractional Operator

In this study, the necessary optimality conditions are achieved for a class of fractional optimal control problems (FOCPs) involving the Caputo–Fabrizio (CF) fractional derivative. We offer a new numerical method based on the Chebyshev cardinal functions for solving the problem. Our goal was to reduce the original problem into a nonlinear system of algebraic equations. For doing this, we approximate the state and control variables in terms of the Chebyshev cardinal functions. The operational matrices (OMs) of left and right CF fractional integrals are also derived for the Chebyshev cardinal functions. Error estimation and convergence analysis of the method are also introduced. The numerical results illustrate the effectiveness and the rapid convergence of the proposed technique. Due to the high accuracy of the solutions, the suggested method is very useful for numerical techniques in control theory.

Existence theory and numerical solution of leptospirosis disease model via exponential decay law

<abstract><p>We investigated the leptospirosis epidemic model by using Caputo and Fabrizio fractional derivatives. Picard's successive iterative method and Sumudu transform are taken into consideration for developing the iterative solutions for the leptospirosis disease. Employing nonlinear functional analysis, the stability and uniqueness of the proposed model are established. Sensitivity analysis is taken into account to highlight the most sensitive parameters corresponding to the basic reproductive number. Various solutions to the proposed system have been interpolated by graphs with the application of Matlab software.</p></abstract>

Computational schemes between the exact, analytical and numerical solution in present of time—fractional ecological model

This paper investigates the accuracy of three recent computational schemes (the extended simplest method (ESEM), sech—tanh expansion method (STEM), and modified Kudryashov method (MKM)) through calculating the absolute value of error between their solutions and numerical solutions. The computational schemes claim to obtain exact traveling wave solutions of the investigated models; therefore, it supposes the numerical study for any models that have been analytically investigated under any constructed computational solutions that will be matching, but our study shows a different fact. (Khater et al Soft Computing (Submitted)) has studied the computational solutions of the time-fractional Lotka—Volterra (LV) model through the above-mentioned computational schemes. Many solutions have been obtained in different mathematical formulas such as exponential, trigonometric, hyperbolic, etc. These solutions describe the interaction between the high -frequency Langmuir and the low-frequent ion-acoustic waves with many applications in electromagnetic waves, plasma physics, and signal processing through optical fibers, coastal engineering, and fluid dynamics. This manuscript applies the trigonometric quintic B—spline scheme to the fractional LV model along with the Caputo and Fabrizio fractional derivatives and computational obtained solutions for investigating the numerical solutions under each employed analytical scheme. The numerical solutions are simulated in two-dimensional sketches to explain the relation between exact and numerical solutions. This study proves the computational fact hypotheses for obtaining exact solutions, and they all obtain computational solutions.

Study of mathematical model of Hepatitis <i>B</i> under Caputo-Fabrizo derivative

The current work is devoted to bring out a detail analysis including qualitative and semi-analytical study of Hepatitis B model under the Caputo- Fabrizio fractional derivative (CFFD). For the required results, fixed point theory is used to establish the conditions for existence and uniqueness of solution to the considered model. On the other hand, for semi analytical solutions, we use decomposition method of Adomian coupled with integral transform of Laplace. Moreover, the concerned solutions are presented via graphs to analyze the dynamics of different compartments of the model.

A fast compact finite difference method for quasilinear time fractional parabolic equation without singular kernel

ABSTRACTIn this paper, a fast compact finite difference method for quasilinear time fractional parabolic equation without singular kernel is developed and analysed. Compact difference scheme is used as a high order approximation for spatial derivative in the fractional parabolic equation, and the Caputo–Fabrizio (C–F) fractional derivative is discretized by a second-order approximation. We have proved that the proposed scheme has fourth-order spatial accuracy and second-order temporal accuracy.However, due to the nonlocal nature of fractional operator, numerically solving the time fractional parabolic equation with traditional direct solvers generally require memories and computational complexity, where N and M represent the number of time steps and grid points in space, respectively. We developed a fast evaluation scheme for the new C–F fractional derivative, which significantly reduced the computational complexity to , and the memory requirement to . Numerical experiments are given to verify the effectiveness and high order convergence of the proposed scheme.