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On theorems for weak solutions of nonlinear differential equations with and without delay in Banach spaces

100%

Gomaa A. M.

Commentationes Mathematicae

2007

tom 47

nr 2

In the present work we give an existence theorem for bounded weak solution of the differential equation \[ \dot{x}(t) = A(t)x(t) + f (t, x(t)),\quad t \geq 0 \] where \(\{A(t) : t \in I\mathbb{R}^+ \}\) is a family of linear operators from a Banach space \(E\) into itself, \(B_r = \{x \in E : \|x\| \leq r\}\) and \(f \colon \mathbb{R}^+ \times B_r \to E\) is weakly-weakly continuous. Furthermore, we give existence theorem for the differential equation with delay \[ \dot{x}(t) = \hat{A}(t) x(t) + f^d (t, θ_t x)\quad \text{if}\ t \in [0, T], \] where \(T, d \gt 0\), \(C_{B_r} ([-d, 0])\) is the Banach space of continuous functions from \([-d, 0]\) into \(B_r\), \(f_d\colon [0, T] \times C_{B_r} ([-d, 0]) \to E\) weakly-weakly continuous function, \(\hat{A}(t)\colon [0,T] \to L(E)\) is strongly measurable and Bochner integrable operator on \([0,T]\) and \(θ_t x(s) = x(t + s)\) for all \(s \in [-d, 0]\).

Laplace Adomian decomposition method for solving a fish farm model

100%

Sambath M., Balachandran K.

Nonautonomous Dynamical Systems

2016

tom 3

nr 1

104-111

In this work, a combined form of the Laplace transform method and the Adomian decomposition method is implemented to give an approximate solution of nonlinear systems of differential equations such as fish farm model with three components nutrient, fish and mussel. The technique is described and illustrated with a numerical example.

Ograniczanie wyników

1 Commentationes Mathematicae

1 Nonautonomous Dynamical Systems

1 Balachandran K.

1 Gomaa A. M.

1 Sambath M.

1 2016

1 2007

Wyniki wyszukiwania

On theorems for weak solutions of nonlinear differential equations with and without delay in Banach spaces

Laplace Adomian decomposition method for solving a fish farm model