**Preface to "Nonlinear Differential Equations and Dynamical Systems"**

Nonlinear differential equations, dynamical systems, and related topics are particularly trendy topics currently, as they have had wide and significant applications in many fields of Physics, Chemistry, Engineering, Biology, or even Economics, in general, and Mathematics in particular.

They can be approached using several different methods and techniques. As examples, we can refer to variational and topological methods, fractional derivatives, fixed point theory, initial and boundary value problems, qualitative theory, stability theory, existence and control of chaos, the existence of attractors and periodic orbits, among others.

This Special Issue contains original results and recent developments in some of the above fields, such as fractional differential and integral equations and applications, non-local optimal control, inverse, and higher-order nonlinear boundary value problems, distributional solutions in the form of a finite series of the Dirac delta function and its derivatives, asymptotic properties oscillatory theory for neutral nonlinear differential equations, the existence of extremal solutions via monotone iterative techniques, and predator–prey interaction via fractional-order models, among others.

These recent results, and the diversity of methods and themes, involving new trends in several areas of mathematical research, allow the reader a glance at the related state-of-the-art, and may provide interested researchers with ideas and techniques that lead to new research and new results.

> **Feliz Manuel Minh ´os, Jo˜ao Fialho** *Editors*

## *Article* **Finite Series of Distributional Solutions for Certain Linear Differential Equations**

#### **Nipon Waiyaworn 1, Kamsing Nonlaopon 1,\* and Somsak Orankitjaroen 2**


Received: 31 August 2020; Accepted: 2 October 2020; Published: 13 October 2020

**Abstract:** In this paper, we present the distributional solutions of the modified spherical Bessel differential equations *t* 2*y*--(*t*) + 2*ty*- (*t*) − [*t* 2 + *ν*(*ν* + <sup>1</sup>)]*y*(*t*) = 0 and the linear differential equations of the forms *t* 2*y*--(*t*) + 3*ty*- (*t*) − (*t* 2 + *ν*2 − <sup>1</sup>)*y*(*t*) = 0, where *ν* ∈ N ∪ {0} and *t* ∈ R. We find that the distributional solutions, in the form of a finite series of the Dirac delta function and its derivatives, depend on the values of *ν*. The results of several examples are also presented.

**Keywords:** Dirac delta function; distributional solution; Laplace transform; power series solution
