On the Explicit Numerical Solution of Linear Homogeneous Differential Equations with Constant Coefficients

SECTION 6 - Solutions of Linear Equations

Exponential polynomials, i.e. solutions of linear homogeneous differential equations with constant coefficients, are characterized as continuous solutions of linear homogeneous difference equations with constant coefficients. Namely, denote by C the \({\Bbb C}\)-linear space of all continuous functions on \({\Bbb R}\), and define for \( \omega \in {\Bbb R}_+\) the shift operator \(z_{\omega }:C\to C\) by \(\eta \mapsto z_{\omega }\eta \) such that \(z_{\omega }\eta (t)=\eta (t+{\omega })\) for all \(t\in {\Bbb R}\). Every ideal \({\mathfrak a}\) of \(R:={\Bbb C}[x,y]\) determines a subspace¶¶\( C_{\frak a}=\{\eta \in C:f(z_{{\omega }_1},z_{{\omega }_2})\eta =0\) for all \(f(x,y)\in {\frak a}\}\,.\)¶¶For incommensurable \({\omega }_1,{\omega }_2\in {\Bbb R}_+\) and ideals \({\frak a}\) in R with a factor ring \(R/{\frak a}\) , which is finite dimensional as a \({\Bbb C}\)-linear space, the set \(C_{\mathfrak a}\) is characterized. The main idea consists in applying the Kronecker approximation theorem to the analytic transition from the explicit representation of the solutions of recurrence sequences to that of the continuous solutions of difference equations. For an alternative proof, the Kronecker approximation theorem is combined with the Jordan decomposition theorem.

The aim of this paper is to derive the dependence of the nature of a solution of a class of differential equations of n-th order with polynomial coefficients on the solutions of the corresponding characteristic algebraic equation of n-th degree.

A definite theory exists for solving linear systems of ordinary differential equations with constant coefficients. “This theory is essentially a branch of linear algebra, and allows us to solve all autonomous linear equations”, ([4], p. 95). Furthermore, “In view of the fact that the solution of these equations does not, in principle, present any great difficulties, they are often considered to be of no great interest for theory, and in textbooks they are usually relegated to the position of simple exercises appended to the general theory of linear equations”, ([81], p. 41). Modern textbooks [8, 21, 26, 44, 45, 46, 50, 90] give a detailed and rigorous treatment of the standard methods available for linear problems, supplemented with numerous mechanical, electrical, and engineering applications. The voluminous literature accumulated on the general properties of the solutions of linear differential equations must therefore no doubt leave the impression that no further contribution could be made to the standard normal linear dynamic system with real and constant coefficients. This impression, however, will fade rather quickly as we go on. Paradoxically, some fundamental aspects of this simple dynamic system are still unnoticed.

In this paper, we derive new method for solving particular solution of linear second order ordinary differential equations whenever one solution of their associated homogeneous differential equations is given. Also, we construct second solution of the associated homogeneous differential equation from this new method. Moreover, we have general solution method of linear second order ordinary differential equations without applying the two famous methods, undetermined coefficients method and variation of parameters method, for solving their particular solution

The eigentensors of an arbitrary second-order tensor and their quality analyses

Solution of Homogeneous Linear Sequential Fractional Differential Equations with Constants Coefficients

A note on non-separable solutions of linear partial differential equations

In this paper, we study the growth of solutions of the second order linear complex differential equations insuring that any nontrivial solutions are of infinite order. It is assumed that the coefficients satisfy the extremal condition for Yang’s inequality and the extremal condition for Denjoy’s conjecture. The other condition is that one of the coefficients itself is a solution of the differential equation .

This work is closed to [2] where a dense linear subspace \(\mathbb{E}\)(E) of the space ℰ(E) of the Silva C ∞ functions on E is defined; the dual of \(\mathbb{E}\)(E) is described via the Fourier transform by a Paley-Wiener-Schwartz theorem which is formulated exactly in the same way as in the finite dimensional case. Here we prove existence and approximation result for solutions of linear partial differential difference equations in \(\mathbb{E}\)(E) with constant coefficients. We also obtain a Hahn-Banach type extension theorem for some C∞ functions defined on a closed subspace of a DFN space, which is analogous to a Boland’s result in the holomorphic case [1].

In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates of the solutions depending on the inhomogeneous terms. This will be done by combining general ideas of the Tikhonov regularization and discretization of bounded linear operator equations on reproducing kernel Hilbert spaces. Furthermore, we will provide approximate solutions for the related inverse source problems.

CHAPTER II - LINEAR DIFFERENTIAL EQUATIONS. SUPPLEMENTARY REMARKS ON THE THEORY OF DIFFERENTIAL EQUATIONS

We study Hyers–Ulam stability for linear differential equations in the sense of quaternion-valued framework. This shows that Laplace transformation is also valid for finding the approximate solution for linear quaternion-valued differential equations.

: Operators of numerical integration of functions are constructed. Also considered is the question of estimating the total error. The operators are applied to the solution of a problem with initial conditions for ordinary linear differential equations with constant coefficients. It was determined that the process of finding the solution of a differential equation of the type being examined by means of numerical integration operators reduces to arithmetic operations with numerical matrices and, consequently, is highly suitable for computer realization. An example is given, illustrating the advantages of the proposed method.

On the multisummability of divergent solutions of linear partial differential equations with constant coefficients