## Recursos de colección

1. #### Existence and Uniqueness of Solution of Stochastic Dynamic Systems with Markov Switching and Concentration Points

Lukashiv, Taras; Malyk, Igor
In this article the problem of existence and uniqueness of solutions of stochastic differential equations with jumps and concentration points are solved. The theoretical results are illustrated by one example.

2. #### An Analysis of the Replicator Dynamics for an Asymmetric Hawk-Dove Game

Kohli, Ikjyot Singh; Haslam, Michael C.
We analyze, using a dynamical systems approach, the replicator dynamics for the asymmetric Hawk-Dove game in which there is a set of four pure strategies with arbitrary payoffs. We give a full account of the equilibrium points and their stability and derive the Nash equilibria. We also give a detailed account of the local bifurcations that the system exhibits based on choices of the typical Hawk-Dove parameters $v$ and $c$ . We also give details on the connections between the results found in this work and those of the standard two-strategy Hawk-Dove game. We conclude the paper with some examples of numerical simulations that further illustrate some global...

3. #### Critical Oscillation Constant for Euler Type Half-Linear Differential Equation Having Multi-Different Periodic Coefficients

We compute explicitly the oscillation constant for Euler type half-linear second-order differential equation having multi-different periodic coefficients.

4. #### Approximate Controllability of Semilinear Control System Using Tikhonov Regularization

Katta, Ravinder; Sukavanam, N.
For an approximately controllable semilinear system, the problem of computing control for a given target state is converted into an equivalent problem of solving operator equation which is ill-posed. We exhibit a sequence of regularized controls which steers the semilinear control system from an arbitrary initial state ${x}^{\mathrm{0}}$ to an $\mathrm{ϵ}$ neighbourhood of the target state ${x}_{\tau }$ at time $\tau >\mathrm{0}$ under the assumption that the nonlinear function $f$ is Lipschitz continuous. The convergence of the sequences of regularized controls and the corresponding mild solutions are shown under some assumptions on the system operators. It is also proved that the target state corresponding to the regularized...

5. #### An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations

Cengizci, Süleyman
In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with...

6. #### Fractional Variational Iteration Method for Solving Fractional Partial Differential Equations with Proportional Delay

Singh, Brajesh Kumar; Kumar, Pramod
This paper deals with an alternative approximate analytic solution to time fractional partial differential equations (TFPDEs) with proportional delay, obtained by using fractional variational iteration method, where the fractional derivative is taken in Caputo sense. The proposed series solutions are found to converge to exact solution rapidly. To confirm the efficiency and validity of FRDTM, the computation of three test problems of TFPDEs with proportional delay was presented. The scheme seems to be very reliable, effective, and efficient powerful technique for solving various types of physical models arising in science and engineering.

7. #### 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.

8. #### 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.

9. #### 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 are inherent in predictor-corrector methods. The stability property of the...

10. #### 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.

11. #### 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.

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