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 Differential Equations and Dynamical SystemsJournal Prestige (SJR): 0.306 Citation Impact (citeScore): 1Number of Followers: 4      Hybrid journal (It can contain Open Access articles) ISSN (Print) 0974-6870 - ISSN (Online) 0971-3514 Published by Springer-Verlag  [2626 journals]
• Grid-Connected Photovoltaic System Stability Enhancement Using Ant Lion
Optimized Model Reference Adaptive Control Strategy
• Abstract: Abstract An inductive-capacitive-inductive (LCL) type filters are broadly utilized in grid-connected voltage source inverters (VSIs) since they can give substantially improved attenuation of switching harmonics in currents injected into the grid with lower cost, weight and power losses than their L and LC type counterparts. This paper discusses an improved model reference adaptive control (MRAC) strategy for designing the control parameters to voltage source converter and improves stability of photovoltaic (PV) generation in different grid capabilities. An improved damping strategy is commonly referred to as MRAC, which has been developed using the proportional resonant (PR) controller and its gain parameters are optimized by Ant Lion optimization (ALO) algorithm. The error elimination at steady-state and gain at particular frequency were provided by the PR current controller. The main focus is set on the mathematical modelling of grid-connected PV, LCL filter and PR current controller and its parameters specifically, resonant and proportional gains. Further, the phase margin and gain of the controller transfer function are presented by means of bode plot to evaluate the operating condition of the controller for different gain parameters. Moreover, a large number of control strategies are mandatory for optimizing the controller parameters and to stabilize the system with zero steady-state error. The obtained results of improved MRAC strategy is presented and its effectiveness is compared with some existing controllers.
PubDate: 2020-03-05

• On the Existence of Solitary Waves for an Internal System of the
Benjamin–Ono Type
• Abstract: Abstract We derive a Benjamin–Ono type system with high dispersion to describe the propagation of internal waves in the case of wave speed large enough. We also establish the existence of solitary wave solutions for the Benjamin–Ono type system, by adapting the positive operator theory in a cone on Fréchet spaces introduced originally by Krasnosel’skii and by using Tuck’s Result (everywhere-convex functions have everywhere-positive Fourier-cosine transforms) to guarantee the positiveness of the kernels involved in the fixed point setting.
PubDate: 2020-03-05

• Existence and Stability of Periodic Orbits for a Hamiltonian System with
Homogeneous Potential of Degree Five
• Abstract: Abstract In this paper we consider the autonomous Hamiltonian system with two degrees of freedom associated to the function $$H=\dfrac{1}{2} (x^2+y^2)+ \frac{1}{2}(p_x^2+ p_y^2)+ V_5(x, y)$$, where $$V_5(x,y)=\Big (\dfrac{A}{5}x^5+Bx^3y^2+\dfrac{C}{5}xy^4\Big )$$ which is related to a homogeneous potential of degree five. We prove the existence of different families of periodic orbits and the type of stability is analyzed through the averaging theory which guarantee the existence of such orbits on adequate sets defined by the parameters A, B, C.
PubDate: 2020-03-05

• Second Order Compact Difference Scheme for Time Fractional Sub-diffusion
Fourth-Order Neutral Delay Differential Equations
• Abstract: Abstract In this paper, we propose a compact difference scheme of second order temporal convergence for the analysis of sub-diffusion fourth-order neutral fractional delay differential equations. In this regard, a difference scheme combining the compact difference operator for spatial discretization along with $$L2-1_{\sigma }$$ formula for Caputo fractional derivative is constructed and analyzed. Unique solvability, stability, and convergence of the proposed scheme are proved using the discrete energy method in $$L_2$$ norm. Established scheme is of second-order convergence in time and fourth-order convergence in spatial dimension, i.e., $$O(\tau ^{3-\alpha }+h^4)$$, where $$\tau$$ and h are time and space mesh sizes respectively and $$\alpha \in (0,1)$$. Finally, some numerical experiments are given to show the authenticity, efficiency, and accuracy of our theoretical results.
PubDate: 2020-03-04

• Existence of Solutions for $${\mathrm {m}}$$m -Point Boundary Value
Problems on an Infinite Time Scale
• Abstract: Abstract In this study, we present the existence result for the the second order $${\mathrm {m}}$$-point boundary value problems on infinite time scales. Nagumo condition, lower and upper solutions play an important role in the arguments.
PubDate: 2020-03-03

• Approximate Controllability for Weighted Semilinear Riemann–Liouville
Fractional Differential Systems with Infinite Delay
• Abstract: 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

• Explicit Weierstrass $$\wp$$℘ Traveling Wave Solution and Stability of
Dispersive Solutions to the Kadomtsev–Petviashvili Equation with
Competing Dispersion Effect
• Abstract: 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

• Variable Mesh Polynomial Spline Discretization for Solving Higher Order
Nonlinear Singular Boundary Value Problems
• Abstract: 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

• Asymptotic Behavior for Eigenvalues of the Singular Sturm-Liouville
Problem with Transmission Conditions
• Abstract: 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

• Simulation of Fluid Particle Cutting: Validation and Case Study
• Abstract: 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

• Magnetic Trajectories on Lightlike Surfaces
• 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

• CFD Modelling of Casson Fluid Flow and Mass Transport Through
Atherosclerotic Vessels
• 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

• Dynamics of a Predator–Prey-Competition System with Pure Delays
• 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

• Adaptive Huber Scheme for Weakly Singular Fractional Integro-differential
Equations
• 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

• Hilfer Fractional Differential Equations with Almost Sectorial Operators
• 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

• Infinitely Many Solutions for Some Fourth Order Elliptic Equations of p (
x )-Kirchhoff Type
• 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

• Comparative Study on Sixth Order Boundary Value Problems with Application
to Linear Hydrodynamic Stability Problem and Benard Layer Eigenvalue
Problem
• 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

• Basins of Attraction and Stability of Nonlinear Systems’ Equilibrium
Points
• 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

• Dynamical Behavior of Two Toxic Releasing Competing Species in Presence of
Predator
• 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

• The Nonexistence of Positive Solutions for A Coupled System of
Non-separated Boundary Value Problems
• 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

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