Solving random mixed heat problems: A random integral transform approach

[EN] This paper develops a random mean square Fourier transform approach to solve random partial differential heat problems with nonhomogeneous boundary value conditions. Random mean square operational rules for the random Fourier sine and cosine transforms are stated and illustrative examples are included.This work has been partially supported by the Spanish Ministerio de Economia y Competitividad grant MTM2013-41765-P.Casabán, MC.; Cortés, JC.; Jódar Sánchez, LA. (2016). Solving random mixed heat problems: A random integral transform approach. Journal of Computational and Applied Mathematics. 291:5-19. https://doi.org/10.1016/j.cam.2014.09.021S51929

Computing probabilistic solutions of the Bernoulli random differential equation

[EN] The random variable transformation technique is a powerful method to determine the probabilistic solution for random differential equations represented by the first probability density function of the solution stochastic process. In this paper, that technique is applied to construct a closed form expression of the solution for the Bernoulli random differential equation. In order to account for the general scenario, all the input parameters (coefficients and initial condition) are assumed to be absolutely continuous random variables with an arbitrary joint probability density function. The analysis is split into two cases for which an illustrative example is provided. Finally, a fish weight growth model is considered to illustrate the usefulness of the theoretical results previously established using real data.This work has been partially supported by the Ministerio de Economía y Competitividad grant MTM2013-41765-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València. Contratos Predoctorales UPV 2014- Subprograma 1.Casabán, M.; Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M.; Villanueva Micó, RJ. (2017). Computing probabilistic solutions of the Bernoulli random differential equation. Journal of Computational and Applied Mathematics. 309:396-407. https://doi.org/10.1016/j.cam.2016.02.034S39640730

Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness

[EN] This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. After studying positivity and conditional random mean square stability, the computation of the expectation and variance of the approximating stochastic process is not performed directly but through using a set of sampling finite difference schemes coming out by taking realizations of the random scheme and using Monte Carlo technique. Thus, the storage accumulation of symbolic expressions collapsing the approach is avoided keeping reliability. Results are simulated and a procedure for the numerical computation is given.This work was supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-PCasabán, M.; Company Rossi, R.; Jódar Sánchez, LA. (2021). Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness. Mathematics. 9(3):1-15. https://doi.org/10.3390/math9030206S1159

Solving linear and quadratic random matrix differential equations: A mean square approach

[EN] In this paper linear and Riccati random matrix differential equations are solved taking advantage of the so called L-p-random calculus. Uncertainty is assumed in coefficients and initial conditions. Existence of the solution in the L-p-random sense as well as its construction are addressed. Numerical examples illustrate the computation of the expectation and variance functions of the solution stochastic process. (C) 2016 Elsevier Inc. All rights reserved.This work has been partially supported by the Spanish Ministerio de Economia y Competitividad grant MTM2013-41765-P and by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement no. 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance).Casabán Bartual, MC.; Cortés López, JC.; Jódar Sánchez, LA. (2016). Solving linear and quadratic random matrix differential equations: A mean square approach. Applied Mathematical Modelling. 40(21-22):9362-9377. https://doi.org/10.1016/j.apm.2016.06.017S936293774021-2

Numerical difference solution of moving boundary random Stefan problems

[EN] This paper deals with the construction of numerical solutions of moving boundary random problems where the uncertainty is limited to a finite degree of randomness in the mean square framework. Using a front fixing approach the problem is firstly transformed into a fixed boundary one. Then a random finite difference scheme for both the partial differential equation and the Stefan condition, allows the discretization. Since statistical moments of the approximate stochastic process solution are required, we combine the sample approach of the difference schemes together with Monte Carlo technique to perform manageable approximations of the expectation and variance of both the approximating stochastic process solution and the stochastic moving boundary solution. Qualitative and reliability properties such as positivity, monotonicity and stability in the mean square sense are treated. Feasibility of the proposed method is checked with illustrative examples of a melting problem and a binary metallic alloys problems.This work was supported by the Spanish Ministerio de EconomIa, Industria y Competitividad (MINECO), Spain, the Agencia Estatal de Investigacion (AEI), Spain and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Casabán, M.; Company Rossi, R.; Jódar Sánchez, LA. (2023). Numerical difference solution of moving boundary random Stefan problems. Mathematics and Computers in Simulation. 205:878-901. https://doi.org/10.1016/j.matcom.2022.10.02687890120

Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique

[EN] This paper presents a full probabilistic description of the solution of random SI-type epidemiological models which are based on nonlinear differential equations. This description consists of determining: the first probability density function of the solution in terms of the density functions of the diffusion coefficient and the initial condition, which are assumed to be independent random variables; the expectation and variance functions of the solution as well as confidence intervals and, finally, the distribution of time until a given proportion of susceptibles remains in the population. The obtained formulas are general since they are valid regardless the probability distributions assigned to the random inputs. We also present a pair of illustrative examples including in one of them the application of the theoretical results to model the diffusion of a technology using real data.This work has been partially supported by the Ministerio de Economia y Competitividad Grants MTM2013-41765-P and TRA2012-36932.Casabán Bartual, MC.; Cortés López, JC.; Romero Bauset, JV.; Roselló Ferragud, MD. (2015). Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation. 24(1):86-97. https://doi.org/10.1016/j.cnsns.2014.12.016S869724

Probabilistic solution of random homogeneous linear second-order difference equations

This paper deals with the computation of the first probability density function of the solution of random homogeneous linear second-order difference equations by the Random Variable Transformation method. This approach allows us to generalize the classical solution obtained in the deterministic scenario. Several illustrative examples are provided.This work was sponsored by "Ministerio de Economa y Competitividad" of the Spanish Government in the frame of the Project with Reference TRA2012-36932.Casabán Bartual, MC.; Cortés López, JC.; Romero Bauset, JV.; Roselló Ferragud, MD. (2014). Probabilistic solution of random homogeneous linear second-order difference equations. Applied Mathematics Letters. 34:27-32. https://doi.org/10.1016/j.aml.2014.03.010S27323

Probabilistic solution of the homogeneous Riccati differential equation: A case-study by using linearization and transformation techniques

[EN] This paper deals with the determination of the first probability density function of the solution stochastic process to the homogeneous Riccati differential equation taking advantage of both linearization and Random Variable Transformation techniques. The study is split in all possible casuistries regarding the deterministic/random character of the involved input parameters. An illustrative example is provided for each one of the considered cases.This work has been partially supported by the Ministerio de Economia y Competitividad grants MTM2013-41765-P and TRA2012-36932.Casabán, MC.; Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M.; Villanueva Micó, RJ. (2016). Probabilistic solution of the homogeneous Riccati differential equation: A case-study by using linearization and transformation techniques. Journal of Computational and Applied Mathematics. 291:20-35. https://doi.org/10.1016/j.cam.2014.11.028S203529

Numerical analysis and computing of a non-arbitrage liquidity model with observable parameters for derivatives

[EN] This paper deals with the numerical analysis and computing of a nonlinear model of option pricing appearing in illiquid markets with observable parameters for derivatives. A consistent monotone finite difference scheme is proposed and a stability condition on the stepsize discretizations is given. © 2010 Elsevier Ltd. All rights reserved.This paper has been supported by the Spanish Department of Science and Education grant DPI2010-C02-01.Casabán Bartual, MC.; Company Rossi, R.; Jódar Sánchez, LA.; Pintos Taronger, JR. (2011). Numerical analysis and computing of a non-arbitrage liquidity model with observable parameters for derivatives. Computers and Mathematics with Applications. 61(8):1951-1956. doi:10.1016/j.camwa.2010.08.009S1951195661

Solving the random diffusion model in an infinite medium: A mean square approach

[EN] This paper deals with the construction of an analytic-numerical mean square solution of the random diffusion model in an infinite medium. The well-known Fourier transform method, which is used to solve this problem in the deterministic case, is extended to the random framework. Mean square operational rules to the Fourier transform of a stochastic process are developed and stated. The main statistical moments of the stochastic process solution are also computed. Finally, some illustrative numerical examples are included.This work has been partially supported by the Ministerio de Economia y Competitividad grant: DPI2010-20891-c0-01, and Universitat Politecnica de Valencia grant: PAID06-11-2070.Casabán, M.; Company Rossi, R.; Cortés, J.; Jódar Sánchez, LA. (2014). Solving the random diffusion model in an infinite medium: A mean square approach. Applied Mathematical Modelling. 38(24):5922-5933. https://doi.org/10.1016/j.apm.2014.04.063S59225933382