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In this post, we propose two results in the instability of solutions to nonlinear systems. Here, we study ordinary differential equations. Well-know stability theorems are due to Liapunov, based on the linearization of the vector field. Consider a continuous function $f:\Omega\subset \mathbb{R}^d\to\mathbb{R}^d$, and $x_0\in \Omega$. Let the Cauchy problem\begin{align*}\tag{Eq}\dot{u}(t)=f(u(t)),\quad u(0)=x_0.\end{align*} The flow of an autonomous … Read more

In this article, we give an application of the open mapping theorem in functional analysis. This fundamental theorem in functional analysis plays a key role in the study of evolution equations. In the sequel, we propose a nice functional analysis worksheet on some deep applications in different domains of mathematical analysis. An application of the … Read more

Differential calculus in Banach spaces is a very important part of mathematics. In fact, the treatment of partial differential equations strongly depends on this theory. Think about the heat equation. We mention that such equations use function spaces as a workspace. The latter is a Banach space, such as the Lebesgue space, the space of … Read more

In this article, we learn how to solve the heat equation using the Fourier series. The heat equation belongs to the class of partial differential equations widely used in physics. It describes the diffusion of heat over a region of space. Of course, there are many approaches to studying this equation. Here we mainly use … Read more

In this article, we state and prove Gronwall lemma and give some of its applications. In fact, we will use this lemma for the stability of the solution to the differential equations. In particular, the Lyapunov stability of nonlinear systems. The proof of Gronwall’s lemma Many proofs of known theorems in mathematics are based on … Read more

The stability analysis of solutions of differential equations is one of the most important axes in ODE. Here w gives a concise course on the stability of equilibrium (critical) points of differential equations. The stability analysis of solutions of differential equations The existence of solutions is guaranteed by two fundamental theorems, Peano’s theorem which gives … Read more

We propose a nice proof of Peano existence theorem. This theorem shows that the continuity of the vector field suffices for the existence of solutions to the ODE; ordinary nonlinear differential equations. We notice that this theorem does not guarantee the uniqueness of the solution. Local solutions Let $I$ be an interval of $\mathbb{R}$ and … Read more

In this post, we study Fourier transform properties and give some applications. In fact, this transformation helps in converting partial differential equations to ODE. A simple method to solve the heat equation is the Fourier transform technique. The Riemann-Lebesgue Lemma We denote by $L^1(\mathbb{R})$ the Lebesgue space of measurable functions $f:\mathbb{R}\to \mathbb{R} $ such that\begin{align*} … Read more