Authors:Maria C. Quintero; Juan M. Cordovez Pages: 137 - 150 Abstract: 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:Ritesh Kumar Dubey; Biswarup Biswas Pages: 151 - 168 Abstract: Abstract Induced numerical oscillations in the computed solution by monotone schemes for hyperbolic conservation laws has been a focus of recent studies. In this work using a local maximum principle, the monotone stable Lax-Friedrichs (LxF) scheme is investigated to explore the cause of induced local oscillations in the computed solution. It expounds upon that LxF scheme is locally unstable and therefore exhibits induced such oscillations. The carried out analysis gives a deeper insight to characterize the type of data extrema and the solution region which cause local oscillations. Numerical results for benchmark problems are also given to support the theoretical claims. PubDate: 2017-04-01 DOI: 10.1007/s12591-016-0311-0 Issue No:Vol. 25, No. 2 (2017)

Authors:Vijitha Mukundan; Ashish Awasthi Pages: 169 - 186 Abstract: Abstract This paper proposes a higher order implicit numerical scheme to approximate the solution of the nonlinear partial differential equation (PDE). This equation is a simplified form of Navier–Stoke’s equation also known as Burgers’ equation. It is an important nonlinear PDE which arises frequently in mathematical modeling of turbulence in fluid dynamics. In order to handle nonlinearity a nonlinear transformation is used which converts the nonlinear PDE into a linear PDE. The linear PDE is semi-discretized in space by method of lines to yield a system of ordinary differential equations in time. The resulting system of differential equations is investigated and found to be a stiff system. A system of stiff differential equations is further discretized by a low-dispersion and low-dissipation implicit Runge–Kutta method and solved by using MATLAB 8.0. The proposed scheme is unconditionally stable. Moreover it is simple, easy to implement and requires less computational time. Finally, the adaptability of the scheme is demonstrated by means of numerical computations by taking three test problems. The present implicit scheme have been compared with existing schemes in literature which shows that the proposed scheme offers more accuracy with less computational time than the numerical schemes given in Jiwari (Comput Phys Comm 183:2413–2423, 2012), Kutluay et al. (J Comput Appl Math 103:251–261, 1998), Kutluay et al. (J Comput Appl Math 167:21–33, 2004). PubDate: 2017-04-01 DOI: 10.1007/s12591-016-0318-6 Issue No:Vol. 25, No. 2 (2017)

Authors:Md. Shafiqul Islam; Humaira Farzana; Samir Kumar Bhowmik Pages: 187 - 205 Abstract: Abstract In this research article, we present Galerkin weighted residual (WRM) technique to find the numerically approximated eigenvalues of the sixth order linear Sturm–Liouville problems (SLP) and Bénard layer problems. In the current method, Bernstein polynomials are being employed as the basis functions and precise matrix formulation is derived for solving eigenvalue problems. Numerical examples with homogeneous boundary conditions are considered to verify the efficiency and implementation of the proposed method. The numerical results offered in this paper are also compared with those investigated by other numerical/analytical methods and the computed eigenvalues are in good agreement. PubDate: 2017-04-01 DOI: 10.1007/s12591-016-0323-9 Issue No:Vol. 25, No. 2 (2017)

Authors:Ranjan Kumar Mohanty; Ravindra Kumar Pages: 207 - 222 Abstract: Abstract In this paper, we derive a new three level implicit method of order two in time and three in space, based on spline in tension approximation for the numerical solution of one space dimensional quasi-linear second order hyperbolic partial differential equation on a variable mesh. We also study application of the proposed method to wave equation in polar coordinates. High order approximation at first time level is briefly discussed which is applicable to solve problems both on uniform and non-uniform mesh. Numerical results are given to illustrate the usefulness of the proposed method. PubDate: 2017-04-01 DOI: 10.1007/s12591-015-0261-y Issue No:Vol. 25, No. 2 (2017)

Authors:Navnit Jha; Neelesh Kumar; Kapil K. Sharma Pages: 223 - 237 Abstract: Abstract A new non-uniform grid third order of magnitude finite differencing formula for the numerical solution of mildly nonlinear elliptic equations in two space variables is presented. Families of compact schemes of order three and four are obtained on nine stencils. The resulting formula yields a tri-block-diagonal matrix and easily computed using Gauss–Seidel method or Newton method depending on linear or nonlinear behaviour of the elliptic equation. We describe error analysis and obtain necessary criterion for the convergence to the proposed compact scheme. It is easy to prove monotone and irreducibility of the Jacobian matrix by the help of graph theory. The numerical accuracy and order of convergence for the solution variables to Chaplygin equation, two dimensional steady state heat/mass transfer equation and Dupuit–Forchheimer groundwater model have been presented to justify the theoretical analysis. PubDate: 2017-04-01 DOI: 10.1007/s12591-015-0263-9 Issue No:Vol. 25, No. 2 (2017)

Authors:Jyoti Sharma; Urvashi Gupta; R. K. Wanchoo Pages: 239 - 249 Abstract: 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:Santosh Kumar; Paramjeet Singh Pages: 251 - 265 Abstract: Abstract The objective of this paper is to present and analyze numerical approximation for the size structured neuron model. We propose finite volume approximations based on upwind and Lax–Wendroff for the partial differential equation originating in size structured neuron model (Perthame and Ryzhik in J Differ Equ 210:155–177, 2005) which is a conservation laws with source term. The developed numerical methods are analyzed for consistency, stability, and convergence. We perform some numerical experiments to verify the predicted theory of the numerical approximations constructed in this paper. PubDate: 2017-04-01 DOI: 10.1007/s12591-016-0276-z Issue No:Vol. 25, No. 2 (2017)

Authors:Aditya Kaushik; Vijayant Kumar; Anil K. Vashishth Pages: 267 - 285 Abstract: Abstract In this paper, an extended mixed asymptotic-numerical method is presented for a one dimensional two point singularly perturbed boundary value problem. The original problem is solved separately on outer and inner region. The outer region problem called the reduced problem is solved using the q-stage Runge-Kutta method and the inner region problem is solved asymptotically. A uniform error estimate of \(\mathcal {O}(h^p),\) \((p\le q)\) is obtained. Several numerical examples are taken to illustrate the comparative study. The method is also extended to delay and advance problem. PubDate: 2017-04-01 DOI: 10.1007/s12591-016-0279-9 Issue No:Vol. 25, No. 2 (2017)

Authors:Manju Sharma Pages: 287 - 300 Abstract: Abstract A numerical study for a class of singularly perturbed partial functional differential equation has been initiated. The solution of the problem, being contaminated by a small perturbation parameter, exhibits layer behavior. There exist narrow regions, in the neighborhood of outflow boundary, where the solution has steep gradient. The presence of parasitic parameters, perturbation parameter and time delay, is often the source for the increased order and stiffness of these systems. The stiffness, attributed to the simultaneous occurrence of slow and fast phenomena and on their dependence on the past history of the physical systems. A numerical method based on standard finite difference operator is presented. The first step involves a discretization of time variable using backward Euler method. This results into a set of stationary singularly perturbed semi-discrete problems which are further discretized in space using standard finite difference operators. A priori explicit bounds on the solution of problem are established. Extensive amount of analysis is carried out in order to establish the convergence and stability of the method proposed. PubDate: 2017-04-01 DOI: 10.1007/s12591-016-0280-3 Issue No:Vol. 25, No. 2 (2017)

Authors:S. Chandra Sekhara Rao; Varsha Srivastava Pages: 301 - 325 Abstract: 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: 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:Imran Aziz; Siraj-ul-Islam Pages: 347 - 360 Abstract: Abstract In this work, we present a new method for solving elliptic partial differential equations using Haar wavelet. This work improves the earlier work (Aziz et al. in Appl Math Model 37:676–694, 2013) in terms of efficiency and contains an extension to nonlinear elliptic partial differential equations as well. In this paper the earlier algorithm (Aziz et al. in Appl Math Model 37:676–694, 2013) has been modified by starting the approximation with a fourth order mixed derivative rather than approximation of the second order derivatives with respect to x and y separately which results in a more efficient algorithm than the earlier algorithm. The use of Kronecker tensor products makes the new algorithm robust and easier to implement in a programming language. A distinguishing feature of the new method is that it can be applied to a variety of boundary conditions with a little modification of the program. The method is tested on several benchmark linear as well as nonlinear models. The numerical results show convergence, simple applicability and efficiency of the method. PubDate: 2017-04-01 DOI: 10.1007/s12591-015-0262-x Issue No:Vol. 25, No. 2 (2017)

Authors:Samir Kumar Bhowmik; Abdullah Aljouiee Pages: 361 - 371 Abstract: 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:Adel Ouannas; Ahmad Taher Azar; Toufik Ziar Abstract: 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: 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: 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: 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

Authors:Sonia Ben Makhlouf; Majda Chaieb; Zagharide Zine El Abidine Abstract: Abstract In this paper, we take up the existence and the asymptotic behavior of positive and continuous solutions to the following coupled fractional differential system $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle D^{\alpha } u=a(x)\displaystyle u^{p }\displaystyle v^{r}\quad \text { in }(0,1) , \\ \displaystyle D^{\beta } v=b(x)\displaystyle u^{s }\displaystyle v^{q}\quad \text { in }(0,1) , \\ u(0)= u(1)= D^{\alpha -3}u(0)= u^{\prime }(1)=0,\\ v(0)= v(1)= D^{\beta -3}v(0)= v^{\prime }(1)=0, \end{array} \right. \end{aligned}$$ where \( \alpha , \beta \in (3,4]\) , \(p, q\in (-1,1)\) , \(r, s\in \mathbb {R}\) such that \((1- p )(1- q )- rs > 0\) , D is the standard Riemann–Liouville differentiation and a, b are nonnegative and continuous functions in (0, 1) allowed to be singular at \(x=0\) and \(x=1\) and they are required to satisfy some appropriate conditions related to Karamata regular variation theory. PubDate: 2017-03-25 DOI: 10.1007/s12591-017-0358-6