Fractional Euler numbers and generalized proportional fractional logistic differential equation

We solve a logistic differential equation for generalized proportional Caputo fractional derivative. The solution is found as a fractional power series. The coefficients of that power series are related to the Euler polynomials and Euler numbers as well as to the sequence of Euler’s fractional numbers recently introduced. Some numerical approximations are presented to show the good approximations obtained by truncating the fractional power series. This generalizes previous cases including the Caputo fractional logistic differential equation and Euler’s numbersOpen access funding provided by Università degli Studi di Bari Aldo Moro within the CRUI-CARE Agreement. This work has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain under Grant PID2020-113275GB-I00, cofinanced by the European Community fund FEDER, as well as Xunta de Galicia grant ED431C 2019/02 for Competitive Reference Research Groups (2019-22)S

The Monotone Iterative Technique for Three-Point Second-Order Integrodifferential Boundary Value Problems with p-Laplacian

A monotone iterative technique is applied to prove the existence of the extremal positive pseudosymmetric solutions for a three-point second-order p-Laplacian integrodifferential boundary value problem.The research of the second author was partially supported by Ministerio de Educacion´ y Ciencia and FEDER, Project MTM2004-06652-C03-01, and by Xunta de Galicia and FEDER, Project PGIDIT05PXIC20702PNS

Application of Non-singular Kernel in a Tumor Model with Strong Allee Effect

We obtain the analytical solutions in implicit form of a tumor cell population differential equation with strong Allee effect. We consider the ordinary case and then a fractional version. Some particular cases are plottedThe research of Juan J. Nieto has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain under grant PID2020-113275GB-I00, and by Xunta de Galicia, grant ED431C 2019/02. Subhas Khajanchi acknowledges the financial support from the Department of Science and Technology (DST), Govt. of India, under the Scheme “Fund for Improvement of S &T Infrastructure (FIST)” [File No. SR/FST/MS-I/2019/41]. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer NatureS

On the fractional Allee logistic equation in the Caputo sense

In the framework of population models, logistic growth and fractional logistic growth has been analyzed. In some situations the so-called Allee effect gives more accurate approximation. In this work, fractional Allee differential equation in the Caputo sense is considered. The solution is obtained by considering formal power series. Numerical computations are presented to compare the truncating series with the classical Allee differential equation.Agencia Estatal de Investigación | Ref. PID2020-113275GB-I00Xunta de Galicia | Ref. Ref. ED431C 2019/0

System of fractional boundary value problem with p-Laplacian and advanced arguments

In this paper, we discuss the existence and multiplicity of positive solutions for a system of fractional differential equations with boundary condition and advanced arguments. The existence result is proved via Leray–Schauder’s fixed point theorem type in a vector Banach space. Further, by using a new fixed point theorem in order Banach space, we study the multiplicity of positive solutions. Finally, some examples are given to illustrate our resultsThe research of J.J. Nieto has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain under Grant MTM2016-75140-P, co-financed by the European Community fund FEDER, and by Xunta de Galicia under grant ED431C 2019/02S

Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative

In this paper, we establish the existence and uniqueness of solutions for a class of initial value problem for nonlinear implicit fractional differential equations with Riemann-Liouville fractional derivative, also, the stability of this class of problem. The arguments are based upon the Banach contraction principle and Schaefer’s fixed point theorem. An example is included to show the applicability of our resultsThe research of J.J. Nieto has been partially supported by the AEI of Spain under Grant MTM2016-75140-P and co-financed by European Community fund FEDERS

A Novel Technique to Control the Accuracy of a Nonlinear Fractional Order Model of COVID-19: Application of the CESTAC Method and the CADNA Library

In this paper, a nonlinear fractional order model of COVID-19 is approximated. For this aim, at first we apply the Caputo–Fabrizio fractional derivative to model the usual form of the phenomenon. In order to show the existence of a solution, the Banach fixed point theorem and the Picard–Lindelof approach are used. Additionally, the stability analysis is discussed using the fixed point theorem. The model is approximated based on Indian data and using the homotopy analysis transform method (HATM), which is among the most famous, flexible and applicable semi-analytical methods. After that, the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library, which are based on discrete stochastic arithmetic (DSA), are applied to validate the numerical results of the HATM. Additionally, the stopping condition in the numerical algorithm is based on two successive approximations and the main theorem of the CESTAC method can aid us analytically to apply the new terminations criterion instead of the usual absolute error that we use in the floating-point arithmetic (FPA). Finding the optimal approximations and the optimal iteration of the HATM to solve the nonlinear fractional order model of COVID-19 are the main novelties of this studyThe work of J.J.N. has been partially supported by the Xunta de Galicia under grant ED431C 2019/02, as well as by Instituto de Salud Carlos III and the Ministerio de Ciencia e Innovación of Spain, research grant COV20/00617. The work of S. Noeiaghdam has been supported by a grant from the Academic Council in the direction of the scientific school of Irkutsk National Research Technical University No. 14-NSH-RAN-2020S

Multiple Positive Solutions of the Singular Boundary Value Problem for Second-Order Impulsive Differential Equations on the Half-Line

This paper uses a fixed point theorem in cones to investigate the multiple positive solutions of a boundary value problem for second-order impulsive singular differential equations on the half-line. The conditions for the existence of multiple positive solutions are established.This work is supported by the National Nature Science Foundation of P. R.China 10871063 and Scientific Research Fund of Hunan Provincial Education Department 07A038 , partially supported by Ministerio de Educacion y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, project no.PGIDIT06PXIB207023PRS

Permanence and Periodic Solution of Predator-Prey System with Holling Type Functional Response and Impulses

We considered a nonautonomous two dimensional predator-prey system with impulsive effect. Conditions for the permanence of the system and for the existence of a unique stable periodic solution are obtained.S