Finite Series of Distributional Solutions for Certain Linear Differential Equations

In this paper, we present the distributional solutions of the modified spherical Bessel differential equations t2y″(t)+2ty′(t)−[t2+ν(ν+1)]y(t)=0 and the linear differential equations of the forms t2y″(t)+3ty′(t)−(t2+ν2−1)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.

Remarks on the Nonlocal Dirichlet Problem

We study translation-invariant integrodifferential operators that generate Lévy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided.

The Generalized Solutions of the nth Order Cauchy–Euler Equation

In this paper, we use the Laplace transform technique to examine the generalized solutions of the nth order Cauchy–Euler equations. By interpreting the equations in a distributional way, we found that whether their solution types are classical, weak or distributional solutions relies on the conditions of their coefficients. To illustrate our findings, some examples are exhibited.

Generalized Solutions of the Third-Order Cauchy-Euler Equation in the Space of Right-Sided Distributions via Laplace Transform

Using the Laplace transform technique, we investigate the generalized solutions of the third-order Cauchy-Euler equation of the form t 3 y ′ ′ ′ ( t ) + a t 2 y ′ ′ ( t ) + b y ′ ( t ) + c y ( t ) = 0 , where a , b , and c ∈ Z and t ∈ R . We find that the types of solutions in the space of right-sided distributions, either distributional solutions or weak solutions, depend on the values of a, b, and c. At the end of the paper, we give some examples showing the types of solutions. Our work improves the result of Kananthai (Distribution solutions of the third order Euler equation. Southeast Asian Bull. Math. 1999, 23, 627–631).

The generalized solutions of a certain nth order Cauchy–Euler equation

In this paper, we propose the generalized solutions of a certain [Formula: see text]th order Cauchy–Euler equation of the form [Formula: see text], where [Formula: see text] and [Formula: see text] are any integers with [Formula: see text], and [Formula: see text] using Laplace transform technique. We find that the types of solutions either distributional solutions or weak solutions, depend on the values of [Formula: see text]. At the end of the paper, we give some examples showing the types of solutions.