Mean square solution of Bessel differential equation with uncertainties

[EN] This paper deals with the study of a Bessel-type differential equation where input parameters (coefficient and initial conditions) are assumed to be random variables. Using the so-called Lp-random calculus and assuming moment conditions on the random variables in the equation, a mean square convergent generalized power series solution is constructed. As a result of this convergence, the sequences of the mean and standard deviation obtained from the truncated power series solution are convergent as well. The results obtained in the random framework extend their deterministic counterpart. The theory is illustrated in two examples in which several distributions on the random inputs are assumed. Finally, we show through examples that the proposed method is computationally faster than Monte Carlo method.This work has been partially supported by the Spanish Ministerio de Economía 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) and Mexican Conacyt.Cortés, J.; Jódar Sánchez, LA.; Villafuerte, L. (2017). Mean square solution of Bessel differential equation with uncertainties. Journal of Computational and Applied Mathematics. 309:383-395. https://doi.org/10.1016/j.cam.2016.01.034S38339530

Random differential operational calculus: Theory and applications

A product rule and a chain rule for mean square derivatives are obtained using fourth order properties. Applications to the mean square solution of random differential equations are shown

Solving Riccati time-dependent models with random quadratic coefficient

This paper deals with the construction of approximate solutions of a random logistic differential equation whose nonlinear coefficient is assumed to be an analytic stochastic process and the initial condition is a random variable. Applying p-mean stochastic calculus, the nonlinear equation is transformed into a random linear equation whose coefficients keep analyticity. Next, an approximate solution of the nonlinear problem is constructed in terms of a random power series solution of the associate linear problem. Approximations of the average and variance of the solution are provided. The proposed technique is illustrated through an example where comparisons with respect to Monte Carlo simulations are shown. © 2011 Elsevier Ltd. All rights reserved.This work has been partially supported by the Spanish M.C.Y.T. grants MTM2009-08587, DPI2010-20891-C02-01, Universitat Politecnica de Valencia grant PAID06-09-2588 and Mexican Conacyt.Cortés López, JC.; Jódar Sánchez, LA.; Company Rossi, R.; Villafuerte Altuzar, L. (2011). Solving Riccati time-dependent models with random quadratic coefficient. Applied Mathematics Letters. 24(12):2193-2196. https://doi.org/10.1016/j.aml.2011.06.024S21932196241

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

Extending the deterministic Riemann-Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations

[EN] This paper extends both the deterministic fractional Riemann¿Liouville integral and the Caputo fractional derivative to the random framework using the mean square random calculus. Characterizations and sufficient conditions to guarantee the existence of both fractional random operators are given. Assuming mild conditions on the random input parameters (initial condition, forcing term and diffusion coefficient), the solution of the general random fractional linear differential equation, whose fractional order of the derivative is ¿ ¿ [0, 1], is constructed. The approach is based on a mean square chain rule, recently established, together with the random Fröbenius method. Closed formulae to construct reliable approximations for the mean and the covariance of the solution stochastic process are also given. Several examples illustrating the theoretical results are included.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2013-41765-P. The co-author Prof. L. Villafuerte acknowledges the support by Mexican Conacyt.Burgos, C.; Cortés, J.; Villafuerte, L.; Villanueva Micó, RJ. (2017). Extending the deterministic Riemann-Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations. Chaos, Solitons and Fractals. 102:305-318. https://doi.org/10.1016/j.chaos.2017.02.008S30531810

Solving a class of random non-autonomous linear fractional differential equations by means of a generalized mean square convergent power series

[EN] The aim of this paper is to solve a class of non-autonomous linear fractional differential equations with random inputs. A mean square convergent series solution is constructed in the case that the fractional order a of that Caputo derivative lies in ]0,1] using a random Frobenius approach. The analysis is conducted by using the so-called mean square random calculus. The mean square convergence of the series solution is established assuming mild conditions on random inputs (diffusion coefficient and initial condition). We show that these conditions are satisfied for a variety of unbounded random variables. In addition, explicit expressions to approximate the mean, the variance and the covariance functions of the random series solution are given. Two full illustrative examples are shown. (C) 2017 Elsevier Ltd. All rights reserved.Authors gratefully acknowledge the comments made by reviewers, which have greatly enriched the manuscript. This work has been partially supported by Ministerio de Economia y Competitividad grant MTM2013-41765-P.Burgos-Simon, C.; Calatayud-Gregori, J.; Cortés, J.; Villafuerte, L. (2018). Solving a class of random non-autonomous linear fractional differential equations by means of a generalized mean square convergent power series. Applied Mathematics Letters. 78:95-104. https://doi.org/10.1016/j.aml.2017.11.009S951047

Exploring the interdependencies of research funders in the UK

Investment in medical research is vital to the continuing improvement of the UK's health and wealth. It is through research that we expand our understanding of disease and develop new treatments for patients. Medical research charities currently contribute over £1 billion annually to medical research in the UK, of which over £350 million is provided by Cancer Research UK. Many charities, including Cancer Research UK, receive no government funding for their research activity. Cancer Research UK is engaged in a programme of work in order to better understand the medical research funding environment and demonstrate the importance of sustained investment. A key part of that is the Office of Health Economics‟ (OHE) 2011 report “Exploring the interdependency between public and charitable medical research”. This study found that there are substantial benefits, both financial and qualitative, from the existence of a variety of funders and that reductions in the level of government financial support for medical research are likely to have broader negative effects. This contributed to other evidence which found that the activities and funding of the charity, public and private sectors respectively are complementary, i.e. mutually reinforcing, rather than duplicative or merely substituting for one another. “Exploring the interdependencies of research funders in the UK” by the Office of Health Economics (OHE) and SPRU: Science and Technology Policy Research at the University of Sussex, represents a continued effort to build the evidence base around the funding of medical research. This report uncovers the extent to which funders of cancer research are interdependent, nationally and internationally. Key figures show that two thirds of publications acknowledging external support have relied on multiple funders, while just under half benefited from overseas funding, and almost a fifth are also supported by industry. In addition the analysis shows that the general public would not want tax funding of cancer research to be reduced, but would not donate enough to charities to compensate for any such reduction

Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus

[EN] The aim of this paper is to study a generalization of fractional Airy differential equations whose input data (coefficient and initial conditions) are random variables. Under appropriate hypotheses assumed upon the input data, we construct a random generalized power series solution of the problem and then we prove its convergence in the mean square stochastic sense. Afterwards, we provide reliable explicit approximations for the main statistical information of the solution process (mean, variance and covariance). Further, we show a set of numerical examples where our obtained theory is illustrated. More precisely, we show that our results for the random fractional Airy equation are in full agreement with the corresponding to classical random Airy differential equation available in the extant literature. Finally, we illustrate how to construct reliable approximations of the probability density function of the solution stochastic process to the random fractional Airy differential equation by combining the knowledge of the mean and the variance and the Principle of Maximum Entropy.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2017-89664-P. The authors express their deepest thanks and respect to the editors and reviewers for their valuable comments.Burgos-Simon, C.; Cortés, J.; Debbouche, A.; Villafuerte, L.; Villanueva Micó, RJ. (2019). Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus. Applied Mathematics and Computation. 352:15-29. https://doi.org/10.1016/j.amc.2019.01.039S152935

Experimental study on the effect of cover and vaccination on the survival of juvenile European rabbits

et al.In Mediterranean ecosystems, the European rabbit is a keystone species that has declined dramatically, with profound implications for conservation and management. Predation and disease acting on juveniles are considered the likely causes. In the field, these processes are managed by removing predators, increasing cover to reduce predation risk and by vaccinating against myxomatosis. These manipulations can be costly and, when protected predators are killed, they can also be damaging to conservation interests. Our goal was to test the effectiveness of cover and vaccination on juvenile survival in two large enclosures, free of mammalian predators, by adding cover and vaccinating juveniles. Rabbit warrens were our experimental unit, with nine replicates of four treatments: control, cover, vaccination, and cover and vaccination combined. Our results showed that improved cover systematically increased juvenile rabbit survival, whereas vaccination had no clear effect and the interactive effect was negligible. Our experimental data suggest that improved cover around warrens is an effective way of increasing rabbit abundance in Mediterranean ecosystems, at least when generalist mammalian predators are scarce. In contrast the vaccination programme was of limited benefit, raising questions about its efficacy as a management tool.Funding was provided by Confederación Hidrográfica del Guadalquivir and the projects CGL2009-11665/BOS, PEII 09-0097-4363, POII09-0099-2557. C.F. was supported by a PhD grant (Ref. SFRH/BD/22084/2005) funded by the Fundaçao para a Ciência e Tecnologia of the Ministério da Ciência, Tecnologia e Ensino Superior, Portuguese government. S.R. was supported by the Centre for Ecology and Hydrology and a grant from NERC.Peer Reviewe