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In this article, we answer the question: what is real analysis about? Apparently the word real comes from the set of real numbers while analysis is what we study with…

In this article, we propose an exercise that describes the optimal method of the step gradient for coercive functions on spaces of finite dimension. These specific problems are used in…

We discuss the properties of the translation operator defined in a Lebesgue space. The latter appears in the study of differential equations, mainly equations with negative memories called also delay…

We discuss the Lyapunov stability for nonlinear systems. Indeed, having a reference solution, a stationary solution, one wonders if the ODE solution is closed to this reference when the time…

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…

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…

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….

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….

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…

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…

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…

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…