Exact Solvability Conditions for the Non-Local Initial Value Problem for Systems of Linear Fractional Functional Differential Equations

Nonoscillations for odd-dimensional systems of linear retarded functional differential equations

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

In this paper, a Haar wavelets based numerical method to solve a system of linear or nonlinear fractional differential equations has been proposed. Numerous nontrivial test examples along with practical problems from fluid dynamics and chemical engineering have been considered to illustrate applicability of the proposed method. We have derived a theoretical error bound which plays a crucial role whenever the exact solution of the system is not known and also it guarantees the convergence of approximate solution to exact solution.

A Stability Criterion for Linear Autonomous Functional Differential Equations

Some sufficient conditions for exponential stability of linear neutral functional differential equations

The present paper deals with special solutions for linear functional differential equations with small delays. The importance of special solutions relies on the fact that, under some conditions, they can be used to describe the asymptotic behaviour of all solutions. Sufficient conditions for the existence of special solutions for nonautonomous equations in $\mathbb R^n$ are given. For autonomous equations in $\mathbb R^n$, results relating special solutions and characteristic values are presented. We also consider linear autonomous functional differential equations in Banach spaces, and use special solutions to study the asymptotic behaviour of their solutions. Several applications are given.

In this paper we describe a method to solve the linear non-homogeneous fractional differential equations (FDE), composed with Jumarie type Fractional Derivative, and describe this method developed by us, to find out Particular Integrals, for several types of forcing functions. The solutions are obtained in terms of Mittag-Leffler functions, fractional sine and cosine functions. We have used our earlier developed method of finding solution to homogeneous FDE composed via Jumarie fractional derivative, and extended this to non-homogeneous FDE. We have demonstrated these developed methods with few examples of FDE, and also applied in fractional damped forced differential equation. This method proposed by us is useful as it is having conjugation with the classical methods of solving non-homogeneous linear differential equations, and also useful in understanding physical systems described by FDE.

This paper introduces the concept of first-order systems of linear fuzzy differential equations for \(\mathcal {S}\)-linearly correlated fuzzy processes. These fuzzy processes have range embedded in Banach spaces of fuzzy numbers, and the fuzzy initial value problems studied are given in terms of the Fréchet derivative of these fuzzy functions. An equivalence between the first-order systems of linear fuzzy differential equations and a family of classical first-order systems of linear differential equations is established. Also, conditions on the existence and uniqueness of the solutions are presented. Lastly, an application on the multiple mass-spring system is provided.KeywordsBanach spacesStrongly linearly independence\(\mathcal {S}-\)linearly correlated fuzzy processesSystems of fuzzy differential equationsMass-spring multiple system

The problem of existence and uniqueness of solutions defined on the whole real line and satisfying given initial point data for general abstract linear functional differential equations is considered. The equation is not assumed to be of the delay type. The essence of the method presented here consists in the representation of a solution in the form analogous to the variation of constants formula known for linear ordinary differential equations. It is shown that such an approach can be effectively applied to the problem of existence and uniqueness of solutions satisfying an exponential growth estimate, provided that the deviation of the argument is sufficiently small. The proofs are based on the Banach fixed point principle. Detailed comparison and discussion of the hypotheses ensuring the existence and uniqueness of solutions are presented.

The problem on solutions with specified growth for linear functional differential equations with negative coefficients is treated by using two-sided monotone iterations. New theorems on the existence and localisation of such solutions are established.

Conditions guaranteeing the solvability of certain three-point boundary value problems for a system of linear functional differential equations are obtained by using a special successive approximation scheme. We also establish some conditions necessary for a certain set belonging to the domain of the space variables to contain a point determining the initial value of the solution. An algorithm for selecting such points is also indicated.

In this paper we consider the well-posedness of linear functional differential equations on product spaces. Let L and D be linear $\mathbb{R}^n $-valued functions with domains $\mathfrak{D}(L)$ and $\mathfrak{D}(D)$ subspaces of the Lebesgue measurable $\mathbb{R}^n $-valued functions on $[ - r,0]$ and such that $W^{1,p} ([ - r,0];\mathbb{R}^n ) \subseteq \mathfrak{D}(L) \cap \mathfrak{D}(D)$. Under weak conditions on D and L we establish the equivalence between generalized solutions to the functional differential equation \[ \frac{d}{{dt}}Dx_t = Lx_t + f(t)\] and mild solutions to the Cauchy problem in $\mathbb{R}^n \times L_p ([ - r,0];\mathbb{R}^n )$\[ \dot z(t) = \mathfrak{a} z(t) + (f(t),0), \] where $\mathfrak{a}$ is the operator defined on \[ \mathfrak{D}(\mathfrak{a}) = \left\{ {(\eta ,\varphi ) \in \mathbb{R}^n \times L_p {{\left( {[ - r,0];\mathbb{R}^n } \right)} / \varphi } \in W^{1,p} \left( {[ - r,0];\mathbb{R}^n } \right),D\varphi = \eta } \right\},\] by \[ \mathfrak{a}(\eta ,\varphi ) = (L\varphi ,\dot \varphi ). \] The results are applicable to neutral functional differential equations and certain singular integral equations.

System of fractional differential algebraic equations with applications

Solving systems of fractional differential equations using differential transform method

General linear non-autonomous functional differential equations are considered. Explicit criteria for exponential stability are given. Furthermore, the authors present an explicit robust stability bound for systems subject to time-varying perturbations. Two examples are given to illustrate the obtained results.