International Journal of Differential Equations
[SJR: 0.363] [H-I: 11] [6 followers] Follow
Open Access journal
ISSN (Print) 1687-9643 - ISSN (Online) 1687-9651
Published by Hindawi [333 journals]
- Fractional Variational Iteration Method for Solving Fractional Partial
Differential Equations with Proportional Delay
Abstract: This paper deals with an alternative approximate analytic solution to time fractional partial differential equations (TFPDEs) with proportional delay, obtained by using fractional variational iteration method, where the fractional derivative is taken in Caputo sense. The proposed series solutions are found to converge to exact solution rapidly. To confirm the efficiency and validity of FRDTM, the computation of three test problems of TFPDEs with proportional delay was presented. The scheme seems to be very reliable, effective, and efficient powerful technique for solving various types of physical models arising in science and engineering.
PubDate: Mon, 13 Mar 2017 07:50:09 +000
- Critical Oscillation Constant for Euler Type Half-Linear Differential
Equation Having Multi-Different Periodic Coefficients
Abstract: We compute explicitly the oscillation constant for Euler type half-linear second-order differential equation having multi-different periodic coefficients.
PubDate: Mon, 27 Feb 2017 00:00:00 +000
- Approximate Controllability of Semilinear Control System Using Tikhonov
Abstract: For an approximately controllable semilinear system, the problem of computing control for a given target state is converted into an equivalent problem of solving operator equation which is ill-posed. We exhibit a sequence of regularized controls which steers the semilinear control system from an arbitrary initial state to an neighbourhood of the target state at time under the assumption that the nonlinear function is Lipschitz continuous. The convergence of the sequences of regularized controls and the corresponding mild solutions are shown under some assumptions on the system operators. It is also proved that the target state corresponding to the regularized control is close to the actual state to be attained.
PubDate: Thu, 23 Feb 2017 00:00:00 +000
- An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed
Linear Delay Differential Equations
Abstract: In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.
PubDate: Wed, 08 Feb 2017 00:00:00 +000
- Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional
Abstract: In this paper, we study the existence and uniqueness of solutions for the following boundary value problem of nonlinear fractional differential equation: , , , where , , , and . The main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle.
PubDate: Sun, 29 Jan 2017 00:00:00 +000
- Asymptotics for the Ostrovsky-Hunter Equation in the Critical Case
Abstract: We consider the Cauchy problem for the Ostrovsky-Hunter equation , , , , where . Define . Suppose that is a pseudodifferential operator with a symbol such that , , and . For example, we can take . We prove the global in time existence and the large time asymptotic behavior of solutions.
PubDate: Mon, 23 Jan 2017 12:00:01 +000
- A Trigonometrically Fitted Block Method for Solving Oscillatory
Second-Order Initial Value Problems and Hamiltonian Systems
Abstract: In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nyström method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHTRKNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.
PubDate: Sun, 22 Jan 2017 13:47:16 +000
- A Family of Boundary Value Methods for Systems of Second-Order Boundary
Abstract: A family of boundary value methods (BVMs) with continuous coefficients is derived and used to obtain methods which are applied via the block unification approach. The methods obtained from these continuous BVMs are weighted the same and are used to simultaneously generate approximations to the exact solution of systems of second-order boundary value problems (BVPs) on the entire interval of integration. The convergence of the methods is analyzed. Numerical experiments were performed to show efficiency and accuracy advantages.
PubDate: Sun, 15 Jan 2017 13:29:21 +000
- Modelling the Potential Role of Media Campaigns in Ebola Transmission
Abstract: A six-compartment mathematical model is formulated to investigate the role of media campaigns in Ebola transmission dynamics. The model includes tweets or messages sent by individuals in different compartments. The media campaigns reproduction number is computed and used to discuss the stability of the disease states. The presence of a backward bifurcation as well as a forward bifurcation is shown together with the existence and local stability of the endemic equilibrium. Results show that messages sent through media have a more significant beneficial effect on the reduction of Ebola cases if they are more effective and spaced out.
PubDate: Thu, 12 Jan 2017 14:34:31 +000
- Numerical Solution of Piecewise Constant Delay Systems Based on a Hybrid
Abstract: An efficient numerical scheme for solving delay differential equations with a piecewise constant delay function is developed in this paper. The proposed approach is based on a hybrid of block-pulse functions and Taylor’s polynomials. The operational matrix of delay corresponding to the proposed hybrid functions is introduced. The sparsity of this matrix significantly reduces the computation time and memory requirement. The operational matrices of integration, delay, and product are employed to transform the problem under consideration into a system of algebraic equations. It is shown that the developed approach is also applicable to a special class of nonlinear piecewise constant delay differential equations. Several numerical experiments are examined to verify the validity and applicability of the presented technique.
PubDate: Thu, 29 Dec 2016 14:07:02 +000
- Some Comparison of Solutions by Different Numerical Techniques on
Mathematical Biology Problem
Abstract: We try to compare the solutions by some numerical techniques when we apply the methods on some mathematical biology problems. The Runge-Kutta-Fehlberg (RKF) method is a promising method to give an approximate solution of nonlinear ordinary differential equation systems, such as a model for insect population, one-species Lotka-Volterra model. The technique is described and illustrated by numerical examples. We modify the population models by taking the Holling type III functional response and intraspecific competition term and hence we solve it by this numerical technique and show that RKF method gives good results. We try to compare this method with the Laplace Adomian Decomposition Method (LADM) and with the exact solutions.
PubDate: Wed, 07 Dec 2016 11:33:01 +000
- Existence of Solutions for Fractional Impulsive Integrodifferential
Equations in Banach Spaces
Abstract: We investigate the existence of solutions for a class of impulsive fractional evolution equations with nonlocal conditions in Banach space by using some fixed point theorems combined with the technique of measure of noncompactness. Our results improve and generalize some known results corresponding to those obtained by others. Finally, two applications are given to illustrate that our results are valuable.
PubDate: Tue, 06 Dec 2016 08:48:47 +000
- The Dynamical Analysis of a Prey-Predator Model with a Refuge-Stage
Structure Prey Population
Abstract: We proposed and analyzed a mathematical model dealing with two species of prey-predator system. It is assumed that the prey is a stage structure population consisting of two compartments known as immature prey and mature prey. It has a refuge capability as a defensive property against the predation. The existence, uniqueness, and boundedness of the solution of the proposed model are discussed. All the feasible equilibrium points are determined. The local and global stability analysis of them are investigated. The occurrence of local bifurcation (such as saddle node, transcritical, and pitchfork) near each of the equilibrium points is studied. Finally, numerical simulations are given to support the analytic results.
PubDate: Thu, 24 Nov 2016 07:48:20 +000
- About a Problem for Loaded Parabolic-Hyperbolic Type Equation with
Abstract: An existence and uniqueness of solution of local boundary value problem with discontinuous matching condition for the loaded parabolic-hyperbolic equation involving the Caputo fractional derivative and Riemann-Liouville integrals have been investigated. The uniqueness of solution is proved by the method of integral energy and the existence is proved by the method of integral equations. Let us note that, from this problem, the same problem follows with continuous gluing conditions (at ); thus an existence theorem and uniqueness theorem will be correct and on this case.
PubDate: Wed, 23 Nov 2016 06:07:26 +000
- Piecewise Approximate Analytical Solutions of High-Order Singular
Perturbation Problems with a Discontinuous Source Term
Abstract: A reliable algorithm is presented to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The algorithm is based on an asymptotic expansion approximation and Differential Transform Method (DTM). First, the original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion of the solution is constructed. Then a piecewise smooth solution of the terminal value reduced system is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented. The results show that the method is a reliable and convenient asymptotic semianalytical numerical method for treating high-order singular perturbation problems with a discontinuous source term.
PubDate: Tue, 15 Nov 2016 06:04:00 +000
- Positive Solutions to Periodic Boundary Value Problems of Nonlinear
Fractional Differential Equations at Resonance
Abstract: By Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima, we discuss the existence of positive solutions to fractional order with periodic boundary conditions at resonance. At last, an example is presented to demonstrate the main results.
PubDate: Wed, 09 Nov 2016 09:26:56 +000
- Multiplicity Results for the -Laplacian Equation with Singular
Nonlinearities and Nonlinear Neumann Boundary Condition
Abstract: We investigate the singular Neumann problem involving the -Laplace operator: , in , where is a bounded domain with boundary, is a positive parameter, and , and are assumed to satisfy assumptions (H0)–(H5) in the Introduction. Using some variational techniques, we show the existence of a number such that problem has two solutions for one solution for , and no solutions for .
PubDate: Mon, 07 Nov 2016 14:15:31 +000
- Dirichlet Boundary Value Problem for the Second Order Asymptotically
Abstract: We consider the second order system with the Dirichlet boundary conditions , where the vector field is asymptotically linear and . We provide the existence and multiplicity results using the vector field rotation theory.
PubDate: Thu, 03 Nov 2016 12:45:41 +000
- Error Analysis of an Implicit Spectral Scheme Applied to the
Abstract: We develop error estimates of the semidiscrete and fully discrete formulations of a Fourier-Galerkin numerical scheme to approximate solutions of a coupled nonlinear Schrödinger-Benjamin-Ono system that describes the motion of two fluids with different densities under capillary-gravity waves in a deep water regime. The accuracy of the numerical solver is checked using some exact travelling wave solutions of the system.
PubDate: Thu, 03 Nov 2016 06:28:50 +000
- A Numerical Computation of a System of Linear and Nonlinear Time Dependent
Partial Differential Equations Using Reduced Differential Transform Method
Abstract: This paper deals with an analytical solution of an initial value system of time dependent linear and nonlinear partial differential equations by implementing reduced differential transform (RDT) method. The effectiveness and the convergence of RDT method are tested by means of five test problems, which indicates the validity and great potential of the reduced differential transform method for solving system of partial differential equations.
PubDate: Thu, 27 Oct 2016 12:49:12 +000
- Multiple Solutions for the Asymptotically Linear Kirchhoff Type Equations
Abstract: The multiplicity of positive solutions for Kirchhoff type equations depending on a nonnegative parameter on is proved by using variational method. We will show that if the nonlinearities are asymptotically linear at infinity and is sufficiently small, the Kirchhoff type equations have at least two positive solutions. For the perturbed problem, we give the result of existence of three positive solutions.
PubDate: Thu, 13 Oct 2016 12:44:41 +000
- On Accuracy and Stability Analysis of the Reproducing Kernel Space Method
for the Forced Duffing Equation
Abstract: It is attempted to provide the stability and convergence analysis of the reproducing kernel space method for solving the Duffing equation with with boundary integral conditions. We will prove that the reproducing space method is stable. Moreover, after introducing the method, it is shown that it has convergence order two.
PubDate: Tue, 11 Oct 2016 07:32:49 +000
- The Maximal Strichartz Family of Gaussian Distributions: Fisher
Information, Index of Dispersion, and Stochastic Ordering
Abstract: We define and study several properties of what we call Maximal Strichartz Family of Gaussian Distributions. This is a subfamily of the family of Gaussian Distributions that arises naturally in the context of the Linear Schrödinger Equation and Harmonic Analysis, as the set of maximizers of certain norms introduced by Strichartz. From a statistical perspective, this family carries with itself some extrastructure with respect to the general family of Gaussian Distributions. In this paper, we analyse this extrastructure in several ways. We first compute the Fisher Information Matrix of the family, then introduce some measures of statistical dispersion, and, finally, introduce a Partial Stochastic Order on the family. Moreover, we indicate how these tools can be used to distinguish between distributions which belong to the family and distributions which do not. We show also that all our results are in accordance with the dispersive PDE nature of the family.
PubDate: Thu, 29 Sep 2016 09:49:30 +000
- A Hybrid Natural Transform Homotopy Perturbation Method for Solving
Fractional Partial Differential Equations
Abstract: A hybrid analytical method for solving linear and nonlinear fractional partial differential equations is presented. The proposed analytical approach is an elegant combination of the Natural Transform Method (NTM) and a well-known method, Homotopy Perturbation Method (HPM). In this analytical method, the fractional derivative is computed in Caputo sense and the nonlinear term is calculated using He’s polynomial. The proposed analytical method reduces the computational size and avoids round-off errors. Exact solution of linear and nonlinear fractional partial differential equations is successfully obtained using the analytical method.
PubDate: Tue, 27 Sep 2016 15:42:46 +000
- Symmetry Classification and Exact Solutions of a Variable Coefficient
Space-Time Fractional Potential Burgers’ Equation
Abstract: We investigate the symmetry properties of a variable coefficient space-time fractional potential Burgers’ equation. Fractional Lie symmetries and corresponding infinitesimal generators are obtained. With the help of the infinitesimal generators, some group invariant solutions are deduced. Further, some exact solutions of fractional potential Burgers’ equation are generated by the invariant subspace method.
PubDate: Sun, 25 Sep 2016 12:24:40 +000
- Homogeneous-Like Generalized Cubic Systems
Abstract: We consider properties and center conditions for plane polynomial systems of the forms , where , and , are polynomials of degrees and , respectively, for integers . We restrict our attention to those systems for which . In this case the system can be transformed to a trigonometric Abel equation which is similar in form to the one obtained for homogeneous systems . From this we show that any center condition of a homogeneous system for a given can be transformed to a center condition of the corresponding generalized cubic system and we use a similar idea to obtain center conditions for several other related systems. As in the case of the homogeneous system, these systems can also be transformed to Abel equations having rational coefficients and we briefly discuss an application of this to a particular Abel equation.
PubDate: Mon, 05 Sep 2016 11:26:55 +000
- On Oscillatory and Asymptotic Behavior of a Second-Order Nonlinear Damped
Neutral Differential Equation
Abstract: This paper discusses oscillatory and asymptotic properties of solutions of a class of second-order nonlinear damped neutral differential equations. Some new sufficient conditions for any solution of the equation to be oscillatory or to converge to zero are given. The results obtained extend and improve some of the related results reported in the literature. The results are illustrated with examples.
PubDate: Mon, 29 Aug 2016 16:25:02 +000
- Estimates on the Lower Bound of the Eigenvalue of the Smallest Modulus
Associated with a General Weighted Sturm-Liouville Problem
Abstract: We obtain a lower bound on the eigenvalue of smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. The main motivation for this study is the result obtained by Mingarelli (1988).
PubDate: Mon, 29 Aug 2016 09:28:25 +000
- Interval Oscillation Criteria for Forced Second-Order Nonlinear Delay
Dynamic Equations with Damping and Oscillatory Potential on Time Scales
Abstract: We are concerned with the interval oscillation of general type of forced second-order nonlinear dynamic equation with oscillatory potential of the form , on a time scale . We will use a unified approach on time scales and employ the Riccati technique to establish some oscillation criteria for this type of equations. Our results are more general and extend the oscillation criteria of Erbe et al. (2010). Also our results unify the oscillation of the forced second-order nonlinear delay differential equation and the forced second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our results.
PubDate: Wed, 24 Aug 2016 17:53:57 +000
- Semianalytic Solution of Space-Time Fractional Diffusion Equation
Abstract: We study the space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. The series solution of this problem is obtained via the optimal homotopy analysis method (OHAM). Numerical simulations are presented to validate the method and to show the effect of changing the fractional derivative parameters on the solution behavior.
PubDate: Mon, 08 Aug 2016 09:59:49 +000