Solution of a fractional logistic ordinary differential equation

We solve the logistic differential equation of fractional order and non-singular kernel. The analytical solution is obtainedThis work has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain under Grant MTM2016-75140-P, cofinanced by the European Community fund FEDER, Spain, as well as by Instituto de Salud Carlos III, grant COV20/00617. JJN is beneficiary of Xunta de Galicia grant ED431C 2019/02 for Competitive Reference Research Groups, Spain (2019-22)S

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

Dissipativity of Fractional Navier–Stokes Equations with Variable Delay

We use classical Galerkin approximations, the generalized Aubin–Lions Lemma as well as the Bellman–Gronwall Lemma to study the asymptotical behavior of a two-dimensional fractional Navier–Stokes equation with variable delay. By modifying the fractional Halanay inequality and the comparison principle, we investigate the dissipativity of the corresponding system, namely, we obtain the existence of global absorbing set. Besides, some available results are improved in this work. The existence of a global attracting set is still an open problemThe work of Lin F. Liu has been partially supported by NSF of China (Nos. 11901448, 11871022 and 11671142) as well as by China Postdoctoral Science Foundation Grant (Nos. 2018M643610). The work of Juan J. Nieto has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain, co-financed by the European Fund for Regional Development (FEDER) corresponding to the 2014-2020 multiyear financial framework, project MTM2016-75140-P, Xunta de Galicia under grant ED431C 2019/02; by Instituto de Salud Carlos III (Spain), grant COV20/00617S

Basic control theory for linear fractional differential equations with constant coefficients

In this paper we present an analogous result of the famous Kalman controllability criterion for first order linear ordinary differential equations with constant coefficients that applies to the case of linear differential equations of fractional order with constant coefficients. We use the fractional Gramian matrix, the range space and the Kalman matrix as main tools to derive a sufficient and necessary condition for the controllability of the fractional system. Moreover, we provide some simple examples, including a linear fractional harmonic oscillator, to illustrate our results. Finally, several open problems arising from this topic are suggested, including another simple linear system of incommensurate fractional ordersThis research has been partially supported by the AEI of Spain under Grant MTM2016-75140-P, co-financed by European Community fund FEDER and XUNTA de Galicia under grant ED431C 2019/02. Sebastián Buedo-Fernández also acknowledges current funding from Ministerio de Educación, Cultura y Deporte of Spain (FPU16/04416) and previous funding from Xunta de Galicia (ED481A-2017/030)S

Existence of extremal solutions for quadratic fuzzy equations

Some results on the existence of solution for certain fuzzy equations are revised and extended. In this paper, we establish the existence of a solution for the fuzzy equation , where , , , and are positive fuzzy numbers satisfying certain conditions. To this purpose, we use fixed point theory, applying results such as the well-known fixed point theorem of Tarski, presenting some results regarding the existence of extremal solutions to the above equation.Research partially supported by Ministerio de Educación y Ciencia and FEDER, Projects BFM2001 – 3884 – C02 – 01 and MTM2004 – 06652 – C03 – 01, and by Xunta de Galicia and FEDER, Project PGIDIT02PXIC20703PNS

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

Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited

We consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we find an equivalent Volterra integral equation with two parameter Mittag–Leffler function in the kernel to the mentioned equation. We used the contraction mapping theorem and Weissinger’s fixed point theorem to obtain existence and uniqueness of global solution in the spaces of Lebesgue integrable functions. The new representation formula of the general solution helps us to find the fixed point problem associated with the fractional Langevin equation which its contractivity constant is independent of the friction coefficient. Two examples are discussed to illustrate the feasibility of the main theoremsThe authors are thankful to the Editor(s) and reviewers of the manuscript for their helpful comments. The work of H. Fazli and H. Sun was supported by the National Key R&D Program of China (2017YFC0405203), the National Natural Science Foundation of China under Grant No. 11972148. The reserach of J. J. Nieto was partially supported by Xunta de Galicia, ED431C 2019/02, and by project MTM2016-75140-P of AEI/FEDER (Spain)S

A Reliable Algorithm for a Local Fractional Tricomi Equation Arising in Fractal Transonic Flow

The pivotal proposal of this work is to present a reliable algorithm based on the local fractional homotopy perturbation Sumudu transform technique for solving a local fractional Tricomi equation occurring in fractal transonic flow. The proposed technique provides the results without any transformation of the equation into discrete counterparts or imposing restrictive assumptions and is completely free of round-off errors. The results of the scheme show that the approach is straightforward to apply and computationally very user-friendly and accurateS