Abstract: This manuscript is about infinite-delayed semilinear fractional differential systems. The considered system is of weighted type so that the initial condition may be non null at time zero. The principle contribution is the approximate controllability of the considered system. The state space is not required to be reflexive and the non linear function is not supposed to be Lipschitz. More than that, the nonlinear function in this work can involve spatial derivatives which enlarges the area of application of the obtained result. PubDate: 2020-02-18

Abstract: In this article, we address the Kadomtsev–Petviashvili (KP) equation in which a small competing dispersion effect is present. We examine the nature of solutions under the influence of dispersion effect by exploiting dynamical system theory and Lyapunov function. We prove the existence of bounded traveling wave solutions when KP equation has external dispersion effect. The retrieved traveling wave solutions are in the form of solitary waves, periodic and elliptic functions. We obtain the general solution of the equation in presence and absence of the dispersion effect in terms of Weirstrass \(\wp \) functions and Jacobi elliptic functions. Furthermore, we apply a new method which is based on factorization method, use of functional transformation and the Abel’s first order nonlinear equation and obtain a new form of kink-type solutions. Finally, we analyze the stability analysis of the dispersive solutions explicitly by using dynamical system theory. We show that the traveling wave velocity behaves like a bifurcation parameter which classifies different classes of waves. Moreover, we also obtain a transcritical bifurcation which occurs at a critical velocity. PubDate: 2020-02-12

Abstract: In this paper, two generalized variable mesh finite difference schemes based on cubic spline has been developed to solve the system of nonlinear singular boundary value problems. The suggested methods are pertinent to singular boundary value problem and are of second and third order. Numerical examples are provided to prove the precision and competence of the schemes. PubDate: 2020-02-11

Abstract: In this study, we consider the singular Sturm-Liouville (S-L) problem with transmission conditions (TC). The eigenvalues of the singular S-L problem with TC are investigated. By defining a new Hilbert space which is related to TC, the self-adjointness of the S-L problem in this associated Hilbert space is proved. And we give the condition for \(\lambda \) being the eigenvalue of the singular S-L problem. Furthermore, the asymptotic behavior of eigenvalues of the singular S-L problem is described. PubDate: 2020-02-01

Abstract: In this paper we present the comparison of experiments and numerical simulations for bubble cutting by a wire. The air bubble is surrounded by water. In the experimental setup an air bubble is injected on the bottom of a water column. When the bubble rises and contacts the wire, it is separated into two daughter bubbles. The flow is modeled by the incompressible Navier–Stokes equations. A meshfree method is used to simulate the bubble cutting. We have observed that the experimental and numerical results are in very good agreement. Moreover, we have further presented simulation results for liquid with higher viscosity. In this case the numerical results are close to previously published results. PubDate: 2020-01-30

Abstract: Divergence free vector fields are called magnetic vector fields in three-dimensional semi-Riemannian manifolds. When a charged particle enters the magnetic vector field, it traces a new trajectory called magnetic curve by the influenced of magnetic field. In the present paper, we investigate the magnetic curves on the lightlike surfaces corresponding to the Killing magnetic fields in 3D semi-Riemannian manifolds. Moreover, we give some characterizations of these curves. As an application, we determine all magnetic curves on the lightlike cone. Finally, we give various examples to confirm the main results. PubDate: 2020-01-28

Abstract: The effects of stenosis and mass transfer on arterial flow have been studied in the present investigation through a mathematical model. The flowing blood is considered to be non-Newtonian (Casson model) and the arterial wall is treated as axisymmetric with an outline of the stenosis obtained from a casting of a mildly stenosed artery. The mass transfer to blood is controlled by the convection–diffusion equation. The governing equations of motion accompanied by the appropriate choice of the boundary conditions are solved numerically by Marker and Cell method and the results obtained are checked for numerical stability with the desired degree of accuracy. The effects of two different types of stenosis model, different severity of stenosis and different values of yield stress parameter are investigated and represented graphically. Particular attention is focused on the effect of stenosis on the wall shear stress and Sherwood number. The present results show quite consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration. PubDate: 2020-01-27

Abstract: This paper deals with the non-autonomous Lotka–Volterra type one prey-two competitive predator systems with pure time delays. Based on the the comparison method and construction of the multiple Lyapunov functional, some new sufficient conditions on the boundedness, permanence, extinction and global attractivity of the system are established. In addition, two examples with numerical simulations are presented to illustrate the obtained theoretical results. PubDate: 2020-01-25

Abstract: The present article is devoted to develop an adaptive scheme for the numerical solution of fractional integro-differential equations with weakly singular kernel. The adaptive scheme is based on the product integration method of Huber. An error estimate is provided for the discretisation error occurring at each step to calculate the step size of next integration step. A parameter known as error tolerance is predefined to control local discretisation error. Computations verify that by controlling local discretisation error, the true global errors match fairly well to the error tolerance parameter. The first integration step size \(h_{{ initial}}\) is introduced in order to make a controlled evaluation of local discretisation error at the first integration step also, where the error estimate is not available. Finally, the computations and results of some numerical experiments validate the accuracy and applicability of the adaptive scheme. PubDate: 2020-01-18

Abstract: In this article we consider an abstract Cauchy problem with the Hilfer fractional derivative and an almost sectorial operator. We introduce a suitable definition of a mild solution for this evolution equation and establish the existence result for a mild solution. We also give an example to highlight the applicability of theoretical results established. PubDate: 2020-01-13

Abstract: In this paper, we consider a class of fourth order elliptic equations of Kirchhoff type with variable exponents $$\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2_{p(x)}u-M\left( \int _\Omega \frac{1}{p(x)} \nabla u ^{p(x)}\,dx\right) \Delta _{p(x)}u = \lambda f(x,u) \quad \text { in }\Omega ,\\ u=\Delta u = 0 \quad \text { on } \partial \Omega , \end{array}\right. \end{aligned}$$where \(\Omega \subset {\mathbb {R}}^N\), \(N \ge 3\), is a smooth bounded domain, \(\Delta _{p(x)}^2u=\Delta ( \Delta u ^{p(x)-2} \Delta u)\) is the operator of fourth order called the p(x)-biharmonic operator, \(\Delta _{p(x)}u = {\text {div}} \left( \nabla u ^{p(x)-2}\nabla u\right) \) is the p(x)-Laplacian, \(p:{{\overline{\Omega }}} \rightarrow {\mathbb {R}}\) is a log-Hölder continuous function, \(p^- > \max \{1,\frac{N}{2}\}\), \(\lambda >0\) and \(f: {\overline{\Omega }}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function satisfying some certain conditions. Using one smooth version of Ricceri’s variational principle [6], we establish some existence results of infinitely many solutions for the problem in an appropriate space of functions. PubDate: 2020-01-09

Abstract: Numerical estimation for higher order eigenvalue problems are promising and has accomplished significant importance, mainly due to existence of higher order derivatives and boundary conditions relating to higher order derivatives of the unknown functions. In this article, we perform a numerical study of linear hydrodynamic stability of a fluid motion caused by an erratic gravity field. We employ two methods, collocation and spectral collocation based on Bernstein and Legendre polynomials to solve the linear hydrodynamic stability problems and Benard type convection problems. In order to handle boundary conditions, our techniques state all the unknown coefficients of boundary conditions derivatives in terms of known co-efficient. The schemes have been carried out to several test problems to establish the efficiency of the two methods. PubDate: 2019-12-21

Abstract: The system of differential and operator equations is considered. This system is assumed to enjoy an equilibrium point. The Cauchy problem with the initial condition with respect to one of the desired functions is formulated. The second function controls the corresponding nonlinear dynamic process. The sufficient conditions of the global classical solution’s existence and stabilisation at infinity to the equilibrium point are formulated in the main theorem. Solution can be constructed by the method of successive approximations. If the conditions of the main theorem are not satisfied, then several solutions may exist. Some solutions can blow-up in a finite time, while others stabilise to an equilibrium point. The special case of considered systems of differential-operator equations are nonlinear systems of differential-algebraic equations which model various nonlinear phenomena in power systems, chemical processes and many other processes. PubDate: 2019-12-19

Abstract: The present paper deals with a problem of two competing fish species where each of the species release a substance toxic for the other species. It is considered that each of the species migrate due to toxic substance released by other species. Conditions for the existence and stability of all feasible equilibrium points are obtained. The dynamical behavior of the system is examined in the presence of migration. Lastly, the optimal harvesting policy is obtained by using the Pontryagin’s maximum principle. MATLAB is used to simulate the system using hypothetical set of parametric values. PubDate: 2019-12-19

Abstract: In this paper, we consider the non-separated boundary value problem for system of nonlinear Riemann–Liouville fractional differential equations $$\begin{aligned} \begin{aligned} D_{0^+}^{\alpha }x(t)+\lambda f(t, x(t), y(t))=0,~~0<t<1,\\ D_{0^+}^{\alpha }y(t)+\mu g(t, x(t), y(t))=0, ~~0<t<1, \end{aligned} \end{aligned}$$subject to the boundary conditions $$\begin{aligned} \begin{aligned} x(0)=y(0)=0,~~ u_1D_{0^+}^{\beta }x(1)=v_1D_{0^+}^{\beta }y(\xi ),\\ u_2D_{0^+}^{\beta }y(1)=v_2D_{0^+}^{\beta }x(\eta ),~~\eta ,\xi \in (0,1), \end{aligned} \end{aligned}$$where the coefficients \(u_{i},v_{i},i=1,2\) are real positive constants, we give sufficient conditions on \(\lambda , \mu , f\) and g such that the system has no positive solutions. An example is given to demonstrate the main result. PubDate: 2019-12-13

Abstract: In this paper, we construct and implement a new improvement of third order weighted essentially non-oscillatory (WENO) scheme in the finite difference framework for hyperbolic conservation laws. In our approach, a modification in the global smoothness measurement is reported by applying all three points on global stencil \((i-1,i,i+1)\) which is used for convergence of non-linear weights towards the optimal weights at critical points and achieves the desired order of accuracy for third order WENO scheme. We use the third order accurate total variation diminishing (TVD) Runge-Kutta time stepping method. The major advantage of the proposed scheme is its better numerical accuracy in smooth regions. The computational performance of the proposed WENO scheme with this global smoothness measurement is verified in several benchmark one- and two-dimensional test cases for scalar and vector hyperbolic equations. Extensive computational results confirm that the new proposed scheme achieves better performance as compared with WENO-JS3, WENO-Z3 and WENO-F3 schemes. PubDate: 2019-12-03

Abstract: In this paper, we apply the reproducing kernel Hilbert space method (RKHSM) for solving third order differential equations with multiple characteristics in a rectangular domain. The exact solution is expressed in a series form. The numerical examples are given to demonstrate the good performance of the presented method. The results obtained indicate that the method is simple and effective. PubDate: 2019-11-30

Abstract: In this article, we introduce the new type of contractions for a mapping and a relation and prove certain coincidence point theorems which generalize some known results in this area. Moreover, an application to the existence of a unique solution for the integral equation is also provided. PubDate: 2019-11-28

Abstract: This article is about finding the possible solutions of fractional kinetic equation associated with the generalized multiindex Bessel function using the method of Laplace transform. Numerical results and graphical solutions of the main theorems are also presented. PubDate: 2019-11-21

Abstract: The role of astrocytes in physiological processes is always a matter of interest for biologists, mathematicians and computer scientists. Similar to neurons, astrocytes propagate Ca2+ over long distances in response to stimulation and release gliotransmitters in a Ca2+-dependent manner to modulate various important brain functions. There are various processes and parameters that affect the cytoplasmic calcium concentration level of astrocytes like calcium buffering, influx via calcium channels, etc. Buffers bind with calcium ion (Ca2+) and makes calcium bound buffers. Thus, it decreases the calcium concentration [Ca2+] level. Ca2+ enters into the cytosol through voltage gated calcium channel (VGCC) and thus it increases the concentration level. In view of above, a three-dimensional mathematical model is developed for combined study of the effect of buffer and VGCC on cytosolic calcium concentration in astrocytes. Finite element method is applied to find the solution using hexagonal elements. A computer programme is developed for entire problem to simulate the results. The obtained results show that high affinity buffer reveals the effect of VGCC and at low buffer concentration VGCC effects more significantly. PubDate: 2019-11-13