Abstract: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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

Abstract: Abstract In this article, we consider a nonlinear Sobolev type fractional functional integrodifferential equations in a Banach space along with a nonlocal condition. Sufficient conditions for existence, uniqueness and dependence on initial data of local solutions of considered problem are derived by employing fixed point techniques and theory of classical semigroup. Further, we also render the criteria for existence of global solution. At the end, we provide an application to elaborate the obtained results. PubDate: 2019-11-13

Abstract: Abstract In this paper, following Holly (Ann Polon Math LIV 2:93–109, 1991; Univ Iagell Acta Math XXXI:154–174, 1994), Holly and Motyl (Inversion of the \(divdiv^{\star }\)-operator and three numerical methods in hydrodynamics, selected Problems in Mathematics. Cracow University of Technology, pp 35–94, 1995) and Motyl (Univ Iagell Acta Math XXXVIII:227–277, 2000; Ann Fac Sci Toulouse XXI(4):651–743, 2012), we study stability of solutions from the perspective of Hausdorff metric. To be precise, we construct a sequence of sets of approximate solutions for stationary MHD equations by using Galerkin approximation method and prove that this sequence of sets converges to the set of actual solutions of stationary MHD equations. The convergence is with respect to Hausdorff metric which is a distance defined on a family of sets. PubDate: 2019-11-11

Abstract: Abstract We consider the population dynamics of prey under the effect of the two types of predators. One of the predator types is harvested, modelled with a term with a Michaelis–Menten type functional form. Besides local stability analysis, we are interested that how harvesting could directly affect the dynamics of the ecosystem, such as existence and dynamics of coexistence equilibria and periodic solutions. Theoretical and numerical methods are used to study the role played by several bifurcations in the mathematical models. PubDate: 2019-11-06

Abstract: Abstract The paper is concerned with the extension of a monotone iterative technique to impulsive finite delay differential equations of fractional order with a nonlocal initial condition in an ordered Banach space. We study the existence of extremal mild solutions with or without assuming the compactness of a semigroup and also prove the uniqueness of the mild solution of the system. The results are obtained with the help of fractional calculus, a measure of non-compactness, the semigroup theory and monotone iterative technique. Finally, an example is provided to show the application of our main. PubDate: 2019-10-29

Abstract: Abstract In this paper, the \(\psi \) -Riemann–Liouville fractional partial integral and the \(\psi \) -Hilfer fractional partial derivative are introduced and some of its particular cases are recovered. Using the Gronwall inequality and these results, we investigate the Ulam–Hyers and Ulam–Hyers–Rassias stabilities of the solutions of a fractional partial differential equation of hyperbolic type in a Banach space \(({\mathbb {B}}, \left \cdot \right )\) , real or complex. Finally, we present an example in order to elucidate the results obtained. PubDate: 2019-10-28