31 research outputs found
Analyzing stability of a delay differential equation involving two delays
Analysis of the systems involving delay is a popular topic among applied
scientists. In the present work, we analyze the generalized equation
involving two delays
viz. and . We use the the stability conditions to
propose the critical values of delays. Using examples, we show that the chaotic
oscillations are observed in the unstable region only. We also propose a
numerical scheme to solve such equations.Comment: 10 pages, 7 figure
A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations
In this paper we introduce a numerical method for solving nonlinear Volterra
integro-differential equations. In the first step, we apply implicit trapezium
rule to discretize the integral in given equation. Further, the Daftardar-Gejji
and Jafari technique (DJM) is used to find the unknown term on the right side.
We derive existence-uniqueness theorem for such equations by using Lipschitz
condition. We further present the error, convergence, stability and bifurcation
analysis of the proposed method. We solve various types of equations using this
method and compare the error with other numerical methods. It is observed that
our method is more efficient than other numerical methods
Analysis of solution trajectories of linear fractional order systems
The behavior of solution trajectories usually changes if we replace the
classical derivative in a system by a fractional one. In this article, we throw
a light on the relation between two trajectories and of such a
system, where the initial point is at some point of trajectory
. In contrast with classical systems, trajectories and do not
follow the same path. Further, we provide a Frenet apparatus of both
trajectories in various cases and discuss their effect.Comment: 19 pages, 17 figure
Hybrid functions approach to solve a class of Fredholm and Volterra integro-differential equations
In this paper, we use a numerical method that involves hybrid and block-pulse
functions to approximate solutions of systems of a class of Fredholm and
Volterra integro-differential equations. The key point is to derive a new
approximation for the derivatives of the solutions and then reduce the
integro-differential equation to a system of algebraic equations that can be
solved using classical methods. Some numerical examples are dedicated for
showing efficiency and validity of the method that we introduce
Singular points in the solution trajectories of fractional order dynamical systems
Dynamical systems involving non-local derivative operators are of great
importance in Mathematical analysis and applications. This article deals with
the dynamics of fractional order systems involving Caputo derivatives. We take
a review of the solutions of linear dynamical systems
, where the coefficient matrix is in
canonical form. We describe exact solutions for all the cases of canonical
forms and sketch phase portraits of planar systems.
We discuss the behavior of the trajectories when the eigenvalues of
are at the boundary of stable region i.e.
. Further, we discuss the existence of
singular points in the trajectories of such systems in a region of
viz. Region II. It is conjectured that there exists singular point in the
solution trajectories if and only if Region II.Comment: 12 pages, 22 figure
Nonexistence of invariant manifolds in fractional order dynamical systems
Invariant manifolds are important sets arising in the stability theory of
dynamical systems. In this article, we take a brief review of invariant sets.
We provide some results regarding the existence of invariant lines and
parabolas in planar polynomial systems. We provide the conditions for the
invariance of linear subspaces in fractional order systems. Further, we provide
an important result showing the nonexistence of invariant manifolds (other than
linear subspaces) in fractional order systems.Comment: 27 pages, 15 figure
Can we split fractional derivative while analyzing fractional differential equations?
Fractional derivatives are generalization to classical integer-order
derivatives. The rules which are true for classical derivative need not hold
for the fractional derivatives, for example, we cannot simply add the
fractional orders and in to produce the fractional derivative
of order , in general. In
this article we discuss the details of such compositions and propose the
conditions to split a linear fractional differential equation into the systems
involving lower order derivatives. Further, we provide some examples, which
show that the related results in the literature are sufficient but not
necessary conditions.Comment: 16 page