Authors:Maria C. Quintero; Juan M. Cordovez Pages: 137 - 150 Abstract: In the present study, we formulate a boundary value problem that characterizes the light and heat transfer processes in a tissue during a hypothetical superficial hyperthermia therapy. The heating process is represented with a partial differential equations system (PDEs) consisting in the diffusion and bio-heat equations whereas the tissue damage process is modeled with the Arrhenius integral equation. The PDEs system is discretized using the method of lines to obtain an ordinary differential equations system that approximates the original PDEs. The resulting model is then numerically solved to obtain (1) the spatio-temporal distribution of both absorbed light and heat, (2) the volume of tissue effectively treated, and (3) the degree of thermal damage reached. Finally, we define an optimization problem to find a set of relevant parameter values that maximize the therapeutic effect while minimizing damage to healthy tissue. The results of this work could be useful for preclinical level research, in particular for the study of strategies that modify the physical properties of the target tissue for a safe and efficient heat therapy. PubDate: 2017-04-01 DOI: 10.1007/s12591-015-0256-8 Issue No:Vol. 25, No. 2 (2017)

Authors:Jyoti Sharma; Urvashi Gupta; R. K. Wanchoo Pages: 239 - 249 Abstract: The present paper investigates the convection in a binary nanofluid layer in porous medium under the influence of rotation using Darcy–Brinkman model. A set of partial differential equations based on conservation laws for binary nanofluid convection are solved using Normal mode technique and one term weighted residual method. The problem is analyzed for both stationary as well as oscillatory convection for free-free boundaries of the layer. The oscillatory motions come into existence for bottom heavy configuration of nanoparticles in the fluid layer. As far as thermal Rayleigh number is concerned, it does not show much variation with respect to different nanoparticles (alumina, copper, titanium oxide, silver) for bottom heavy configuration. Rotation parameter is found to stabilize the system significantly. PubDate: 2017-04-01 DOI: 10.1007/s12591-015-0268-4 Issue No:Vol. 25, No. 2 (2017)

Authors:S. Chandra Sekhara Rao; Varsha Srivastava Pages: 301 - 325 Abstract: We present a parameter-robust numerical method for a time-dependent weakly coupled linear system of singularly perturbed convection-diffusion equations. A small perturbation parameter multiplies the second order spatial derivative in all the equations. The proposed numerical method uses backward Euler method in time direction on an uniform mesh together with a suitable combination of HODIE scheme and the central difference scheme in spatial direction on a Shishkin mesh. It is proved that the numerical method is parameter-robust of first order in time and almost second order in space. Numerical results are given in support of theoretical findings. PubDate: 2017-04-01 DOI: 10.1007/s12591-016-0282-1 Issue No:Vol. 25, No. 2 (2017)

Authors:Komal Bansal; Pratima Rai; Kapil K. Sharma Pages: 327 - 346 Abstract: In this paper we design two numerical schemes for solving a class of time dependent singularly perturbed parabolic convection–diffusion problems with general shift arguments in the reaction term. The discretization in both the directions is based on finite difference scheme. Special type of mesh and interpolation is used to tackle the terms containing shifts. The earlier numerical schemes for the considered problem are restricted to the case of small delay and advance arguments while in practical situations these shift arguments can be of arbitrary size (i.e., may be big or small enough in size). In this paper we propose two numerical schemes which work in both the situations i.e., when shifts are big or small enough in size. An extensive amount of analysis is presented to show the linear convergence in space and time of both the schemes. Some numerical results are given to confirm the predicted theory and to show the effect of shifts on the solution. PubDate: 2017-04-01 DOI: 10.1007/s12591-015-0265-7 Issue No:Vol. 25, No. 2 (2017)

Authors:Samir Kumar Bhowmik; Abdullah Aljouiee Pages: 361 - 371 Abstract: Wave operators play an important role modeling various physical, engineering and biological problems. We consider an initial boundary value problem modeling an elastic wave propagation. We approximate the model using a Legendre spectral Galerkin method for the spatial approximation. We analyze the accuracy of such a spatial approximation. We perform some numerical experiments to demonstrate the scheme. PubDate: 2017-04-01 DOI: 10.1007/s12591-015-0254-x Issue No:Vol. 25, No. 2 (2017)

Authors:Alka Chadha; Swaroop Nandan Bora Abstract: This paper is concerned with the p-th asymptotical stability of the mild solution of an impulsive neutral stochastic integro-differential equation of Sobolev type involving Poisson jumps and non-instantaneous impulses. By utilizing Leray–Schauder fixed point theorem and compact semigroup, a set of sufficient conditions establishing the asymptotical stability of the mild solution to stochastic system in the p-moment is derived. An example is also considered for illustrating abstract results. PubDate: 2017-06-19 DOI: 10.1007/s12591-017-0371-9

Authors:Kuldip Singh Patel; Mani Mehra Abstract: In this paper, an unconditionally stable compact finite difference scheme for the solution of linear convection–diffusion equation is proposed. In the proposed scheme, second derivative approximations of the unknowns are eliminated with the unknowns itself and their first derivative approximations while retaining the fourth order accuracy and tri-diagonal nature of the scheme. Proposed compact finite difference scheme which is fourth order accurate in spatial variable and second or lower order accurate in temporal variable depending on the choice of weighted time average parameter is applied to Asian option partial differential equation. A diagonally dominant system of linear equation is obtained from the proposed scheme which can be efficiently solved. Two numerical examples are given to demonstrate the efficiency and accuracy of the proposed compact finite difference scheme. PubDate: 2017-06-15 DOI: 10.1007/s12591-017-0372-8

Authors:Maysam Maysami Sadr Abstract: Let \(\mathcal {M}\) be a manifold, \(\mathcal {V}\) be a vector field on \(\mathcal {M}\) , and \(\mathcal {B}\) be a Banach space. For any fixed function \(f:\mathcal {M}\rightarrow \mathcal {B}\) and any fixed complex number \(\lambda \) , we study Hyers–Ulam stability of the global differential equation \(\mathcal {V}y=\lambda y+f\) . PubDate: 2017-06-13 DOI: 10.1007/s12591-017-0369-3

Authors:Wensheng Yang; Xuepeng Li Abstract: A diffusive predator–prey model with ratio-dependent Holling type III functional response is considered in this work. Sufficient conditions for the global asymptotical stability of the constant positive steady-state solution are derived by constructing recurrent sequences and using an iterative method. It is shown that our result supplements one of the main results of Shi and Li’s paper (Global asymptotic stability of a diffusive predator–prey model with ratio-dependent functional response. Appl Math Comput 250:71–77, 2015). PubDate: 2017-06-12 DOI: 10.1007/s12591-017-0370-x

Authors:Hasib Khan; Hossein Jafari; Dumitru Baleanu; Rahmat Ali Khan; Aziz Khan Abstract: The study of boundary value problems (BVPs) for fractional differential–integral equations (FDIEs) is extremely popular in the scientific community. Scientists are utilizing BVPs for FDIEs in day life problems by the help of different approaches. In this paper, we apply monotone iterative technique for the existence, uniqueness and the error estimations of solutions for a coupled system of BVPs for FDIEs of orders \(\omega ,\epsilon \in \left( 3,4 \right] \) . The coupled system is given by $$\begin{aligned} D^{\omega }u\left( t \right)= & {} -G_1 \left( {t,I^{\omega }u\left( t \right) ,I^{\varepsilon }v\left( t \right) } \right) ,\\ D^{\varepsilon }v\left( t \right)= & {} -G_2 \left( {t,I^{\omega }u\left( t \right) ,I^{\varepsilon }v\left( t \right) } \right) ,\\ D^{\delta }u\left( 1 \right)= & {} 0=I^{3-\omega }u\left( 0 \right) =I^{4-\omega }u\left( 0 \right) , u(1)=\frac{{\Gamma }\left( {\omega -\delta } \right) }{{\Gamma }\left( \omega \right) }I^{\omega -\delta }\\&\quad G_1 \left( {t,I^{\omega }u\left( t \right) ,I^{\varepsilon }v\left( t \right) } \right) \left( {t=1} \right) ,\\ D^{\nu }v\left( 1 \right)= & {} 0=I^{3-\varepsilon }v\left( 0 \right) =I^{4-\nu }v\left( 0 \right) , v(1)=\frac{{\Gamma }\left( {\varepsilon -\nu } \right) }{{\Gamma }\left( \varepsilon \right) }I^{\varepsilon -\nu }\\&\quad G_2 \left( {t,I^{\omega }u\left( t \right) ,I^{\varepsilon }v\left( t \right) } \right) \left( {t=1} \right) , \end{aligned}$$ where \(t\in \left[ {0,1}\right] \) , \(\delta ,\nu \in \left[ {1,2} \right] .\) The functions \(G_1 ,G_2 :\left[ {0,1} \right] \times R\times R\rightarrow R,\) satisfy the Caratheodory conditions. The fractional derivatives \(D^{\omega },D^{\varepsilon },D^{\delta },D^{\nu }\) are in Riemann-Liouville sense and \(I^{\omega },I^{\varepsilon },I^{3-\omega },I^{4-\omega },I^{3-\varepsilon },I^{4-\varepsilon },I^{\omega -\delta },I^{\varepsilon -\nu }\) are fractional order integrals. The assumed technique is a better approach for the existence, uniqueness and error estimation. The applications of the results are examined by the help of examples. PubDate: 2017-06-08 DOI: 10.1007/s12591-017-0365-7

Authors:Yuji Liu Abstract: Results on the existence of solutions to a new class of impulsive singular fractional differential systems with multiple base points are established. The assumptions imposed on the nonlinearities, see (D1) and (D2) in Theorem 3.1, are weaker than known ones, (i.e., (A) in Introduction section). The analysis relies on a well known fixed point theorem. An example is given to illustrate the efficiency of the main theorems. PubDate: 2017-06-08 DOI: 10.1007/s12591-017-0368-4

Authors:Stefan C. Mancas Abstract: Traveling wave solutions to Kawahara equation (KE), transmission line (TL), and Korteweg–de Vries (KdV) equation are found by using an elliptic function method which is more general than the \(\mathrm {tanh}\) -method. The method works by assuming that a polynomial ansatz satisfies a Weierstrass equation, and has two advantages: first, it reduces the number of terms in the ansatz by an order of two, and second, it uses Weierstrass functions which satisfy an elliptic equation for the dependent variable instead of the hyperbolic tangent functions which only satisfy the Riccati equation with constant coefficients. When the polynomial ansatz in the traveling wave variable is of first order, the equation reduces to the KdV equation with only a cubic dispersion term, while for the KE which includes a fifth order dispersion term the polynomial ansatz must necessary be of quadratic type. By solving the elliptic equation with coefficients that depend on the boundary conditions, velocity of the traveling waves, nonlinear strength, and dispersion coefficients, in the case of KdV equation we find the well-known solitary waves (solitons) for zero boundary conditions, as well as wave-trains of cnoidal waves for nonzero boundary conditions. Both solutions are either compressive (bright) or rarefactive (dark), and either propagate to the left or right with arbitrary velocity. In the case of KE with nonzero boundary conditions and zero cubic dispersion, we obtain cnoidal wave-trains which represent solutions to the TL equation. For KE with zero boundary conditions and all the dispersion terms present, we obtain again solitary waves, while for KE with all coefficients present and nonzero boundary condition, the solutions are written in terms of Weierstrass elliptic functions. For all cases of the KE we only find bright waves that are propagating to the right with velocity that is a function of both dispersion coefficients. PubDate: 2017-06-06 DOI: 10.1007/s12591-017-0367-5

Authors:B. Ambrosio; M. A. Aziz-Alaoui; A. Balti Abstract: In this paper, we consider networks of reaction–diffusion systems of Hodgkin–Huxley type. We give a general mathematical framework, in which we prove existence and unicity of solutions as well as the existence of invariant regions and of the attractor. Then, we illustrate some relevant numerical examples and exhibit bifurcation phenomena and propagation of bursting oscillations through one and two coupled non-homogeneous systems. PubDate: 2017-06-01 DOI: 10.1007/s12591-017-0366-6

Authors:Sanaa L. Khalaf; Ayad R. Khudair Abstract: This paper adopts the inverse fractional differential operator method for obtaining the explicit particular solution to a linear sequential fractional differential equation, involving Jumarie’s modification of Riemann–Liouville derivative, with constant coefficient s. This method depends on the classical inverse differential operator method and it is independent of the integral transforms. Several examples are then given to demonstrate the validity of our main results. PubDate: 2017-05-29 DOI: 10.1007/s12591-017-0364-8

Authors:R. Nageshwar Rao; P. Pramod Chakravarthy Abstract: In this paper, we presented exponentially fitted finite difference methods for a class of time dependent singularly perturbed one-dimensional convection diffusion problems with small shifts. Similar boundary value problems arise in computational neuroscience in determination of the behavior of a neuron to random synaptic inputs. When the shift parameters are smaller than the perturbation parameter, the shifted terms are expanded in Taylor series and an exponentially fitted tridiagonal finite difference scheme is developed. The proposed finite difference scheme is unconditionally stable and is convergent with order \(O(\Delta t+h^{2})\) where \(\Delta t\) and h respectively the time and space step-sizes. When the shift parameters are larger than the perturbation parameter a special type of mesh is used for the space variable so that the shifts lie on the nodal points and an exponentially fitted scheme is developed. This scheme is also unconditionally stable. By means of two examples, it is shown that the proposed methods provide uniformly convergent solutions with respect to the perturbation parameter. On the basis of the numerical results, it is concluded that the present methods offer significant advantage for the linear singularly perturbed partial differential difference equations. PubDate: 2017-05-09 DOI: 10.1007/s12591-017-0363-9

Authors:Adel Ouannas; Ahmad Taher Azar; Toufik Ziar Abstract: Referring to continuous-time chaotic dynamical systems, this paper investigates the inverse full state hybrid function projective synchronization (IFSHFPS) of non-identical systems characterized by different dimensions. By taking a master system of dimension n and a slave system of dimension m, the method enables each master system state to be synchronized with a linear combination of slave system states, where the scaling factor of the linear combination can be any arbitrary differentiable function. The approach, based on the Lyapunov stability theory and stability of linear continuous-time systems, presents some useful features: (i) it enables non-identical chaotic systems with different dimension \(n<m\) or \(n>m\) to be synchronized; (ii) it can be applied to a wide class of chaotic (hyperchaotic) systems for any differentiable scaling function; (iii) it is rigorous, being based on two theorems, one for the case \(n<m\) and the other for the case \(n>m\) . Two different numerical examples are reported. The examples clearly highlight the capability of the conceived approach in effectively achieving synchronized dynamics for any differentiable scaling function. PubDate: 2017-04-27 DOI: 10.1007/s12591-017-0362-x

Authors:D. Shakti; J. Mohapatra Abstract: In this article, a class of nonlinear singularly perturbed boundary value problems depending on a parameter are considered. To solve this class of problems; first we apply the backward Euler finite difference scheme on Shishkin type meshes [standard Shishkin mesh (S-mesh), Bakhvalov–Shishkin mesh (B–S-mesh)]. The convergence analysis is carried out and the method is shown to be convergent with respect to the small parameter and is of almost first order accurate on S-mesh and first order accurate on B–S-mesh. Then, to improve the accuracy of the computed solution from almost first order to almost second order on S-mesh and from first order to second order on B–S-mesh, the post-processing method namely, the Richardson extrapolation technique is applied. The proof for the uniform convergence of the proposed method is carried out on both the meshes. Numerical experiments indicate the high accuracy of the proposed method. PubDate: 2017-04-24 DOI: 10.1007/s12591-017-0361-y

Authors:Fei Xu; Yong Li; Yixian Gao; Xu Xu Abstract: This paper is devoted to study the existence and uniqueness of solutions for a class of nonlinear fractional dynamical systems with affine-periodic boundary conditions. We can show that there exists a solution for an \(\alpha \) -fractional system via the homotopy invariance of Brouwer degree, where \(0<\alpha \le 1\) . Furthermore, using Gronwall–Bellman inequality, we can prove the uniqueness of the solution if the nonlinearity satisfies the Lipschitz continuity. We apply the main theorem to the fractional kinetic equation and fractional oscillator with constant coefficients subject to affine-periodic boundary conditions. And in appendix, we give the proof of the nonexistence of affine-periodic solution to a given \((\alpha ,Q,T)\) -affine-periodic system in the sense of Riemann–Liouville fractional integral and Caputo derivative for \(0<\alpha <1\) . PubDate: 2017-04-08 DOI: 10.1007/s12591-017-0360-z

Authors:Irene Benedetti; Luca Bisconti Abstract: We give sufficient conditions for the existence, the uniqueness and the continuous dependence on initial data of the solution to a system of integro-differential equations with superlinear growth on the nonlinear term. As possible applications of our methods we consider two epidemic models: a perturbed versions of the well-known integro-differential Kendall SIR model, and a SIRS-like model. PubDate: 2017-04-04 DOI: 10.1007/s12591-017-0359-5