## Recursos de colección

1. #### Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation

Su, Cheng-Min; Sun, Jian-Ping; Zhao, Ya-Hong
In this paper, we study the existence and uniqueness of solutions for the following boundary value problem of nonlinear fractional differential equation: $({}^{C}{D}_{\mathrm{0}+}^{q}u)(t)=f(t,u(t))$ ,   $t\in (\mathrm{0,1})$ , $u(\mathrm{0})={u}^{\mathrm{\prime }\mathrm{\prime }}(\mathrm{0})=\mathrm{0}, ({}^{C}{D}_{\mathrm{0}+}^{{\sigma }_{\mathrm{1}}}u)(\mathrm{1})=\lambda ({I}_{\mathrm{0}+}^{{\sigma }_{\mathrm{2}}}u)(\mathrm{1})$ , where $\mathrm{2}\mathrm{0}$ , and $\lambda \ne \mathrm{\Gamma }(\mathrm{2}+{\sigma }_{\mathrm{2}})/\mathrm{\Gamma }(\mathrm{2}-{\sigma }_{\mathrm{1}})$ . The main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle.

2. #### Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation

Su, Cheng-Min; Sun, Jian-Ping; Zhao, Ya-Hong
In this paper, we study the existence and uniqueness of solutions for the following boundary value problem of nonlinear fractional differential equation: $({}^{C}{D}_{\mathrm{0}+}^{q}u)(t)=f(t,u(t))$ ,   $t\in (\mathrm{0,1})$ , $u(\mathrm{0})={u}^{\mathrm{\prime }\mathrm{\prime }}(\mathrm{0})=\mathrm{0}, ({}^{C}{D}_{\mathrm{0}+}^{{\sigma }_{\mathrm{1}}}u)(\mathrm{1})=\lambda ({I}_{\mathrm{0}+}^{{\sigma }_{\mathrm{2}}}u)(\mathrm{1})$ , where $\mathrm{2}\mathrm{0}$ , and $\lambda \ne \mathrm{\Gamma }(\mathrm{2}+{\sigma }_{\mathrm{2}})/\mathrm{\Gamma }(\mathrm{2}-{\sigma }_{\mathrm{1}})$ . The main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle.

3. #### Asymptotics for the Ostrovsky-Hunter Equation in the Critical Case

Bernal-Vílchis, Fernando; Hayashi, Nakao; Naumkin, Pavel I.
We consider the Cauchy problem for the Ostrovsky-Hunter equation ${\partial }_{x}({\partial }_{t}u-(b/\mathrm{3}){\partial }_{x}^{\mathrm{3}}u-{\partial }_{x}\mathcal{K}{u}^{\mathrm{3}})=au$ , $(t,x)\in {\mathbb{R}}^{\mathrm{2}}$ ,   $u(\mathrm{0},x)={u}_{\mathrm{0}}(x)$ , $x\in \mathbb{R}$ , where $ab>\mathrm{0}$ . Define ${\xi }_{\mathrm{0}}={(\mathrm{27}a/b)}^{\mathrm{1}/\mathrm{4}}$ . Suppose that $\mathcal{K}$ is a pseudodifferential operator with a symbol $\stackrel{^}{K}(\xi )$ such that $\stackrel{^}{K}(\pm{\xi }_{\mathrm{0}})=\mathrm{0}$ , $\mathrm{I}\mathrm{m} \stackrel{^}{K}(\xi )=\mathrm{0}$ , and $|\stackrel{^}{K}(\xi )|\le C$ . For example, we can take $\stackrel{^}{K}(\xi )=({\xi }^{\mathrm{2}}-{\xi }_{\mathrm{0}}^{\mathrm{2}})/({\xi }^{\mathrm{2}}+\mathrm{1})$ . We prove the global in time existence and the large time asymptotic behavior of solutions.

4. #### Asymptotics for the Ostrovsky-Hunter Equation in the Critical Case

Bernal-Vílchis, Fernando; Hayashi, Nakao; Naumkin, Pavel I.
We consider the Cauchy problem for the Ostrovsky-Hunter equation ${\partial }_{x}({\partial }_{t}u-(b/\mathrm{3}){\partial }_{x}^{\mathrm{3}}u-{\partial }_{x}\mathcal{K}{u}^{\mathrm{3}})=au$ , $(t,x)\in {\mathbb{R}}^{\mathrm{2}}$ ,   $u(\mathrm{0},x)={u}_{\mathrm{0}}(x)$ , $x\in \mathbb{R}$ , where $ab>\mathrm{0}$ . Define ${\xi }_{\mathrm{0}}={(\mathrm{27}a/b)}^{\mathrm{1}/\mathrm{4}}$ . Suppose that $\mathcal{K}$ is a pseudodifferential operator with a symbol $\stackrel{^}{K}(\xi )$ such that $\stackrel{^}{K}(\pm{\xi }_{\mathrm{0}})=\mathrm{0}$ , $\mathrm{I}\mathrm{m} \stackrel{^}{K}(\xi )=\mathrm{0}$ , and $|\stackrel{^}{K}(\xi )|\le C$ . For example, we can take $\stackrel{^}{K}(\xi )=({\xi }^{\mathrm{2}}-{\xi }_{\mathrm{0}}^{\mathrm{2}})/({\xi }^{\mathrm{2}}+\mathrm{1})$ . We prove the global in time existence and the large time asymptotic behavior of solutions.

5. #### A Trigonometrically Fitted Block Method for Solving Oscillatory Second-Order Initial Value Problems and Hamiltonian Systems

Ngwane, F. F.; Jator, S. N.
In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nyström method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which...

6. #### A Trigonometrically Fitted Block Method for Solving Oscillatory Second-Order Initial Value Problems and Hamiltonian Systems

Ngwane, F. F.; Jator, S. N.
In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nyström method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which...

7. #### A Family of Boundary Value Methods for Systems of Second-Order Boundary Value Problems

Biala, T. A.; Jator, S. N.
A family of boundary value methods (BVMs) with continuous coefficients is derived and used to obtain methods which are applied via the block unification approach. The methods obtained from these continuous BVMs are weighted the same and are used to simultaneously generate approximations to the exact solution of systems of second-order boundary value problems (BVPs) on the entire interval of integration. The convergence of the methods is analyzed. Numerical experiments were performed to show efficiency and accuracy advantages.

8. #### A Family of Boundary Value Methods for Systems of Second-Order Boundary Value Problems

Biala, T. A.; Jator, S. N.
A family of boundary value methods (BVMs) with continuous coefficients is derived and used to obtain methods which are applied via the block unification approach. The methods obtained from these continuous BVMs are weighted the same and are used to simultaneously generate approximations to the exact solution of systems of second-order boundary value problems (BVPs) on the entire interval of integration. The convergence of the methods is analyzed. Numerical experiments were performed to show efficiency and accuracy advantages.

9. #### Modelling the Potential Role of Media Campaigns in Ebola Transmission Dynamics

Djiomba Njankou, Sylvie Diane; Nyabadza, Farai
A six-compartment mathematical model is formulated to investigate the role of media campaigns in Ebola transmission dynamics. The model includes tweets or messages sent by individuals in different compartments. The media campaigns reproduction number is computed and used to discuss the stability of the disease states. The presence of a backward bifurcation as well as a forward bifurcation is shown together with the existence and local stability of the endemic equilibrium. Results show that messages sent through media have a more significant beneficial effect on the reduction of Ebola cases if they are more effective and spaced out.

10. #### Modelling the Potential Role of Media Campaigns in Ebola Transmission Dynamics

Djiomba Njankou, Sylvie Diane; Nyabadza, Farai
A six-compartment mathematical model is formulated to investigate the role of media campaigns in Ebola transmission dynamics. The model includes tweets or messages sent by individuals in different compartments. The media campaigns reproduction number is computed and used to discuss the stability of the disease states. The presence of a backward bifurcation as well as a forward bifurcation is shown together with the existence and local stability of the endemic equilibrium. Results show that messages sent through media have a more significant beneficial effect on the reduction of Ebola cases if they are more effective and spaced out.

11. #### Antisynchronization of Nonidentical Fractional-Order Chaotic Systems Using Active Control

Bhalekar, Sachin; Daftardar-Gejji, Varsha
Antisynchronization phenomena are studied in nonidentical fractional-order differential systems. The characteristic feature of antisynchronization is that the sum of relevant state-variables vanishes for sufficiently large value of time variable. Active control method is used first time in the literature to achieve antisynchronization between fractional-order Lorenz and Financial systems, Financial and Chen systems, and Lü and Financial systems. The stability analysis is carried out using classical results. We also provide numerical results to verify the effectiveness of the proposed theory.

12. #### Slip Effects on Fractional Viscoelastic Fluids

Unsteady flow of an incompressible Maxwell fluid with fractional derivative induced by a sudden moved plate has been studied, where the no-slip assumption between the wall and the fluid is no longer valid. The solutions obtained for the velocity field and shear stress, written in terms of Wright generalized hypergeometric functions ${}_{p}{\Psi }_{q}$ , by using discrete Laplace transform of the sequential fractional derivatives, satisfy all imposed initial and boundary conditions. The no-slip contributions, that appeared in the general solutions, as expected, tend to zero when slip parameter is $\theta \to 0$ . Furthermore, the solutions for ordinary Maxwell and Newtonian...

13. #### Boundary Value Problems with Integral Gluing Conditions for Fractional-Order Mixed-Type Equation

Berdyshev, A. S.; Karimov, E. T.; Akhtaeva, N.
Analogs of the Tricomi and the Gellerstedt problems with integral gluing conditions for mixed parabolic-hyperbolic equation with parameter have been considered. The considered mixed-type equation consists of fractional diffusion and telegraph equation. The Tricomi problem is equivalently reduced to the second-kind Volterra integral equation, which is uniquely solvable. The uniqueness of the Gellerstedt problem is proven by energy integrals' method and the existence by reducing it to the ordinary differential equations. The method of Green functions and properties of integral-differential operators have been used.

14. #### The Existence of Solutions for a Nonlinear Fractional Multi-Point Boundary Value Problem at Resonance

Han, Xiaoling; Wang, Ting
We discuss the existence of solution for a multipoint boundary value problem of fractional differential equation. An existence result is obtained with the use of the coincidence degree theory.

15. #### Existence of Positive Solutions for Fractional Differential Equation with Nonlocal Boundary Condition

Gao, Hongliang; Han, Xiaoling
By using the fixed point theorem, existence of positive solutions for fractional differential equation with nonlocal boundary condition ${D}_{0+}^{\alpha }u(t)+a(t)f(t,u(t))=0$ , \$0

16. #### Existence and Uniqueness Theorem of Fractional Mixed Volterra-Fredholm Integrodifferential Equation with Integral Boundary Conditions

We study the existence and uniqueness of the solutions of mixed Volterra-Fredholm type integral equations with integral boundary condition in Banach space. Our analysis is based on an application of the Krasnosel'skii fixed-point theorem.

17. #### Asymptotical Stability of Nonlinear Fractional Differential System with Caputo Derivative

Zhang, Fengrong; Li, Changpin; Chen, YangQuan
This paper deals with the stability of nonlinear fractional differential systems equipped with the Caputo derivative. At first, a sufficient condition on asymptotical stability is established by using a Lyapunov-like function. Then, the fractional differential inequalities and comparison method are applied to the analysis of the stability of fractional differential systems. In addition, some other sufficient conditions on stability are also presented.

18. #### Malliavin Calculus of Bismut Type for Fractional Powers of Laplacians in Semi-Group Theory

Léandre, Rémi
We translate into the language of semi-group theory Bismut's Calculus on boundary processes (Bismut (1983), Lèandre (1989)) which gives regularity result on the heat kernel associated with fractional powers of degenerated Laplacian. We translate into the language of semi-group theory the marriage of Bismut (1983) between the Malliavin Calculus of Bismut type on the underlying diffusion process and the Malliavin Calculus of Bismut type on the subordinator which is a jump process.

19. #### An Explicit Numerical Method for the Fractional Cable Equation

Quintana-Murillo, J.; Yuste, S. B.
An explicit numerical method to solve a fractional cable equation which involves two temporal Riemann-Liouville derivatives is studied. The numerical difference scheme is obtained by approximating the first-order derivative by a forward difference formula, the Riemann-Liouville derivatives by the Grünwald-Letnikov formula, and the spatial derivative by a three-point centered formula. The accuracy, stability, and convergence of the method are considered. The stability analysis is carried out by means of a kind of von Neumann method adapted to fractional equations. The convergence analysis is accomplished with a similar procedure. The von-Neumann stability analysis predicted very accurately the conditions under which the...

20. #### New Method for Solving Linear Fractional Differential Equations

Rida, S. Z.; Arafa, A. A. M.
We develop a new application of the Mittag-Leffler Function method that will extend the application of the method to linear differential equations with fractional order. A new solution is constructed in power series. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method, some examples are provided. The results reveal that the technique introduced here is very effective and convenient for solving linear differential equations of fractional order.

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