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Many conventional physical and engineering phenomena have been identified to be well expressed by making use of the fractional order partial differential equations (FOPDEs). For that reason, a proficient and stable numerical method is needed to find the approximate solution of FOPDEs. This article is designed to develop the numerical scheme able to find the approximate solution of generalized fractional order coupled systems (FOCSs) with mixed partial derivative terms of fractional order. Our main objective in this article is the development of a new operational matrix for fractional mixed partial derivatives based on the orthogonal shifted Legendre polynomials (SLPs). The fractional derivatives are considered herein in the sense of Caputo. The proposed method has the advantage to reduce the considered problems to a system of algebraic equations which are simple in handling by any computational software. Being easily solvable, the associated algebraic system leads to finding the solution of the problem. Some examples are included to demonstrate the accuracy and validity of the proposed method.

We use Lyapunov functions to study stability of the first-order Volterra integro-differential equation with Caputo fractional derivative $$ {}_{t_0}^{C}D^{q}_tx(t) =-a(t)f(x(t)) +\int_{t-r}^tB(t,s)g(s,x(s))ds+h(t,x(t),x(t-\tau(t))) \,. $$ For the Lyapunov functions, we consider three types of fractional derivatives. By means of these derivatives, we obtain new sufficient conditions for stability and uniformly stability of solutions We consider both constant and time variable bounded delays, and illustrated our results with an example.

In this paper, we consider a nonlinear integro-vector differential equation of the second order. We establish sufficient conditions that guarantee the global existence and boundedness of solutions of the equation considered. The method of proof involves constructing a suitable Lyapunov functional that gives meaningful results for the problem to be investigated. The result obtained is new and complements that found in the literature. We give an example to verify the result obtained and for illustration purposes. Using MATLAB Simulink, the behaviors of the orbits of the equation considered are clearly shown.

Abstract  This paper investigates stability and boundedness of solutions to third order nonlinear differential equation with retarded argument: $$\begin{array}{l}x'''(t)+\varphi\bigl(x(t-r),x'(t-r),x''(t-r)\bigr)x''(t)\\\qquad{}+\psi\bigl(x'(t-r)\bigr)+h\bigl(x(t-r)\bigr)\\\quad=p\bigl(t,x(t),x(t-r),x'(t),x'(t-r),x''(t)\bigr).\end{array}$$ By the use of the Lyapunov functional, sufficient conditions for stability and boundedness of solutions to the considered equations are obtained. Examples are introduced throughout the paper for illustrations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic stability of the zero solution and boundedness of all solutions of a certain third order nonlinear ordinary vector differential equation. The results are proved using Lyapunov’s second (or direct method). Our results include and improve some well known results existing in the literature. [ABSTRACT FROM AUTHOR]

In this article, we give sufficient conditions for the boundedness, uniformly asymptotic stability and square integrability of the solutions to a fourth-order non-autonomous differential equation with bounded delay by using Lyapunov's second method.

In this article, study the existence of solutions for the second-order nonlinear coupled system of ordinary differential equations $$\displaylines{ u''(t)=f(t,v(t)),\quad t\in [0,1],\cr v''(t)=g(t,u(t)),\quad t\in [0,1], }$$ with nonlinear coupled boundary conditions $$\displaylines{ \phi(u(0),v(0),u(1),v(1),u'(0),v'(0))=(0,0), \cr \psi(u(0),v(0),u(1),v(1),u'(1),v'(1))=(0,0), }$$ where $f,g:[0,1]\times \mathbb{R}\to \mathbb{R}$ and $\phi,\psi:\mathbb{R}^6\to \mathbb{R}^2$ are continuous functions. Our main tools are coupled lower and upper solutions, Arzela-Ascoli theorem, and Schauder's fixed point theorem.

We establish sufficient conditions for the existence of periodic solutions for a Rayleigh-type equation with state-dependent delay. Our approach is based on the continuation theorem in degree theory, and some analysis techniques. An example illustrates that our approach to this problem is new.

In this article, the author studies the stability and boundedness of solutions for the non-autonomous third order differential equation with a deviating argument, $r$: \begin{equation*} \begin{array}{c} x^{\prime \prime \prime }(t)+a(t)x^{\prime \prime }(t)+b(t)g_{1}(x^{\prime}(t-r))+g_{2}(x^{\prime}(t))+h(x(t-r)) \\ =p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime }(t-r),x^{\prime \prime }(t)), \end{array} \end{equation*} where $r>0$ is a constant. Sufficient conditions are obtained; a stability result in the literature is improved and extended to the preceding equation for the case $p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime}(t-r),x^{\prime \prime }(t))=0,$ and a new boundedness result is also established for the case $p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime}(t-r),x^{\prime \prime }(t))\neq 0.$

Our goal in this paper is to obtain sufficient conditions for instability of the zero solution to the non-linear vector differential equation $$ dddot{X}+F(X,dot{X})ddot{X}+G(dot{X})+H(X)=0. $$ An example illustrates the results obtained.

We give criteria for the asymptotic stability and boundedness of solutions to the nonlinear fourth-order ordinary differential equation $$ x^{(4)}+varphi (ddot{x})dddot{x}+f(x,dot{x}) ddot{x}+g(dot{x})+h(x)=p(t,x,dot{x},ddot{x},dddot{x}),, $$ when $pequiv 0$ and when $p eq 0$. Our results include and improve some well-known results in the literature.

In this paper, we give an instability criteria for the nonlinear sixth-order vector differential equation $$displaylines{ X^{(6)} + AX^{(5)} +B(t)Phi (X,dot {X},ddot {X}, X^{(3)},X^{(4)},X^{(5)})X^{(4)} + C(t)Psi (ddot {X}),X^{(3)}cr +D(t)Omega (X,dot {X},ddot {X},X^{(3)},X^{(4)},X^{(5)}) ddot {X}+E(t)G(dot {X})+H(X) = 0,. }$$ The result extends and includes earlier results [6,16,20].

In this paper, we study the instability properties of solutions of a kind of functional differential equations of the fifth order with constant delay. Using the Lyapunov-Krasovskii functional approach, we obtain certain sufficient conditions to guarantee that the zero solution of the equation is unstable.