Lucas Jódar

On the hypergeometric matrix function

Abstract This paper deals with the study of the hypergeometric function with matrix arguments F(A,B;C;z). Conditions for matrices A, B, C so that the series representation of the hypergeometric function be convergent for ¦z¦ = 1 and satisfies a matrix differential equation are given. After the study of beta and gamma matrix functions, an integral representation of F(A,B;C;z) is obtained for the case where B, C and C - B are positive stable matrices with BC = CB.

Some applications of the Hermite matrix polynomials series expansions

This paper deals with Hermite matrix polynomials expansions of some relevant matrix functions appearing in the solution of differential systems. Properties of Hermite matrix polynomials such as the three terms recurrence formula permit an efficient computation of matrix functions avoiding important computational drawbacks of other well-known methods. Results are applied to compute accurate approximations of certain differential systems in terms of Hermite matrix polynomials.

Laguerre matrix polynomials and systems of second-order differential equations

Abstract In this paper we introduce the class of Laguerre matrix polynomials which appears as finite series solutions of second-order matrix differential equations of the form tX ″ +( A + I − λtI ) X ′ + CX = 0. An explicit expression for the Laguerre matrix polynomials, a three-term matrix recurrence relation, a Rodrigues formula and orthogonality properties are given.

Some Properties of Gamma and Beta Matrix Functions

Abstract In this paper, conditions for matrices P , Q so that the Beta matrix function B ( P , Q ) satisfies B ( P , Q ) = B ( Q , P ) and B ( P , Q ) = Γ ( P ) Γ ( Q ) Γ −1 ( P + Q ) are given. Counter-examples showing that hypotheses cannot be removed are also included. A limit expression for the Gamma function of a matrix is established.

Numerical solution of random differential equations: A mean square approach

This paper deals with the construction of numerical solutions of random initial value differential problems by means of a random Euler difference scheme whose mean square convergence is proved based on conditions expressed in terms of the mean square behavior of the right-hand side of the underlying random differential equation. A random mean value theorem is required and established. The concept of mean square modulus of continuity is also introduced and illustrative examples and possibilities are included. Expectation and variance of the approximating process are computed.

A NEW DIRECT METHOD FOR SOLVING THE BLACK-SCHOLES EQUATION

Abstract Using the Mellin transform a new method for solving the Black–Scholes equation is proposed. Our approach does not require either variable transformations or solving diffusion equations.

An implicit difference method for the numerical solution of coupled systems of partial differential equations

An explicit and computable solution for coupled partial difference systems appearing when one considers difference approximations of coupled systems of partial differential equations is given. Our approach is based on the consideration of a discrete matrix separation of variables method. By using discrete Fourier series we avoid the problem of solving algebraic linear systems.

Random differential operational calculus: Theory and applications

In this article, we obtain a product rule and a chain rule for mean square derivatives. An application of the chain rule to the mean square solution of random differential equations is shown. However, to achieve such mean square differentiation rules, fourth order properties were needed and, therefore, we first studied a mean fourth order differential and integral calculus. Results are applied to solve random linear variable coefficient differential problems.

Mean square numerical solution of random differential equations: Facts and possibilities

This paper deals with the construction of numerical solutions of random initial value differential problems. The random Euler method is presented and the conditions for the mean square convergence are established. Numerical examples show that random Euler method gives good results even if the sufficient convergence conditions are not satisfied.

On Hermite matrix polynomials and Hermite matrix functions

In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix polynomials is proved. Asymptotic behaviour of Hermite matrix polynomials is studied and the relationship of Hermite matrix functions with certain matrix differential equations is developed. A new expression of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.