Juan Carlos Cortés

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 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.

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.

Non-standard numerical method for a mathematical model of RSV epidemiological transmission

Respiratory Syncytial Virus (RSV) has long been recognized as the single most important virus causing acute severe respiratory-tract infections with symptoms ranging from rhinitis to bronchitis in children who may require hospitalization. Outbreaks of RSV occur every year and all children become infected within the first two years of life, and that overloads hospital casualty services. The transmission dynamics of RSV are strongly seasonal. Epidemics occur each winter in temperate climates and often coincide with the seasonal rainfall in tropical climates. In this paper we develop a non-standard numerical scheme for a SIRS seasonal epidemiological model for RSV transmission. This non-standard numerical scheme preserves the positivity of the continuous model and is applied to approximate the solution using different sizes of step. Finally this method is compared with some well-known explicit methods and simulations with data from Gambia and Finland are carried out.

Random analytic solution of coupled differential models with uncertain initial condition and source term

This paper deals with the construction of random power series solution of vector initial value problems containing uncertainty in both initial condition and source term. Under appropriate hypothesis on the data, we prove that the random series solution constructed by a random Frobenius method is convergent in the mean square sense. Also, the main statistical functions of the approximating stochastic process solution generated by truncation of the exact series solution are given. Finally, we apply the proposed technique to several illustrative examples.

Solving initial and two-point boundary value linear random differential equations: A mean square approach

Abstract This paper deals with the construction of mean square real-valued solutions to both initial and boundary value problems of linear differential equations whose coefficients are assumed to be stochastic processes and, initial and boundary conditions are random variables. A key result to conduct our study is the extension of the Leibniz integral rule to the random framework taking advantage of the so-called random Fourth Calculus. Exact expressions for the main statistical functions (average and variance) associated to the solutions to both problems are also provided. Illustrative examples computing the average and standard deviation are included.

A probabilistic estimation and prediction technique for dynamic continuous social science models

In this paper, a computational technique to deal with uncertainty in dynamic continuous models in Social Sciences is presented. Considering data from surveys, the method consists of determining the probability distribution of the survey output and this allows to sample data and fit the model to the sampled data using a goodness-of-fit criterion based on the ?2-test. Taking the fitted parameters that were not rejected by the ?2-test, substituting them into the model and computing their outputs, 95% confidence intervals in each time instant capturing the uncertainty of the survey data (probabilistic estimation) is built. Using the same set of obtained model parameters, a prediction over the next few years with 95% confidence intervals (probabilistic prediction) is also provided. This technique is applied to a dynamic social model describing the evolution of the attitude of the Basque Country population towards the revolutionary organisation ETA.

Exact and numerical solution of Black-Scholes matrix equation

In this paper a new method for solving Black-Scholes equation is proposed. The approach is based on the Mellin transform. A numerical procedure for the approximation of the solution is given.

Random mixed hyperbolic models

AbstractThis paper deals with the construction of reliable numerical solutions of mixed problems for hyperbolic second order partial differential models with random information in the variable coefficients of the partial differential equation and in the initial data. Using random difference schemes a random discrete eigenfunctions method is developed in order to construct a discrete approximating stochastic process. Mean square consistency of the random difference scheme is treated and mean square stability of the numerical solution is studied and illustrated with examples. Statistical moments of the numerical solution are also computed.