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In this paper we consider Bessel equations of the type $t^2{X}^{\left(2\right)}\left(t\right)+t{X}^{\left(1\right)}\left(t\right)+(t^{2}I-A^2)X\left(t\right)=0$, where A is an n$\times$n complex matrix and X(t) is an n$\times$m matrix for t > 0. Following the ideas of the scalar case we introduce the concept of a fundamental set of solutions for the above equation expressed in terms of the data dimension. This concept allows us to give an explicit closed form solution of initial and two-point boundary value problems related to the Bessel equation.

In this paper we obtain existence conditions and a closed form of the general solution of higher order singular regular boundary value problems. The approach is based on the concept of co-solution of algebraic matrix equations of polynomial type that permits the treatment of the problem without considering an extended first order system as it has been done in the known literature.

Cauchy problem, boundary value problems with a boundary value condition and Sturm-Liouville problems related to the operator differential equation $X^{\left(2\right)}-AX=0$ are studied for the general case, even when the algebraic equation $X^2-A=0$ is unsolvable. Explicit expressions for the solutions in terms of data problem are given and computable expressions of the solutions for the finite-dimensional case are made available.

In this paper explicit expressions for solutions of Cauchy problems and two-point boundary value problems concerned with the generalized Riccati matrix differential equation are given. These explicit expressions are computable in terms of the data and solutions of certain algebraic Riccati equations related to the problem. The interplay between the algebraic and the differential problems is used in order to obtain approximate solutions of the differential problem in terms of those of the algebraic...

In this paper we construct analytic-numerical solutions for initial-boundary value systems related to the equation $u_t-A{u}_{xx}-Bu=0$, where $B$ is an arbitrary square complex matrix and $A$ ia s matrix such that the real part of the eigenvalues of the matrix $\frac{1}{2}(A+A^H)$ is positive. Given an admissible error $\epsilon$ and a finite domain $G$, and analytic-numerical solution whose error is uniformly upper bounded by $\epsilon$ in $G$, is constructed.