editorial mathematical methods and models in the natural to ...editorial mathematical methods and...

EditorialMathematical Methods and Models in the Natural tothe Life Sciences

Rehana Naz,1 Igor Leite Freire,2 Imran Naeem,3 and Mariano Torrisi4

1 Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan2 Centro de Matematica, Computacao e Cognicao, Universidade Federal do ABC, Rua Santa Adelia 166, Bairro Bangu,09.210-170 Santo Andre, SP, Brazil

3 Department of Mathematics, School of Science and Engineering, Lahore University of Management Sciences, Lahore Cantt,Lahore 54792, Pakistan

4Dipartimento di Matematica ed Informatica, Universita di Catania Viale A. Doria 6, 95125 Catania, Italy

Correspondence should be addressed to Rehana Naz; [email protected]

Received 25 August 2014; Accepted 25 August 2014; Published 11 September 2014

Copyright © 2014 Rehana Naz et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The search for mathematical models and their analysis wasborn as a powerful tool for the investigations of nature andas an aid to solve the daily problems that life is offeringus. On the other hand, the last decades have witnessedan increased interest in the growth and progress of thestudy of mathematical methods that are crucial in the quickadvancement of the natural, physical, engineering, social, andlife sciences.

The purpose of this special issue is to bring togetherpapers covering a wide range of scientific interests concernedwith differential equations emanating from mathematicalmodels of classical fields, such as solids and fluid mechanics,or from some fields as biology, economics and finance, andsciences of life that, only in the second part of the last century,have begun to take into consideration the tools offeredfrommathematics.Themathematical techniques are not onlydevoted to the search for numerical and exact solutions ofthe underlying differential equations for general and specialproblems; they also include new theoretical developmentssuitable for novel applications of solving real-world problems.

The response to this special issue was beyond ourexpectation. We received 60 papers to be considered forpublication.Those submissions, from different countries andcontinents, fall in different areas of the above-mentionedresearch fields. All papers submitted to this special issuewent through a rigorous peer-refereeing process. Based onthe reviewers reports, thirty-three original research articles

have finally been accepted for publication. The contentsembrace different qualitative and quantitative techniques, Liesymmetry techniques, numerical methods, stability analysis,and statistical methods to analyze different aspects that areconcerned with differential equations arising in economics,finance, biology, physics, and fluid dynamics. It is certainlyimpossible to provide in this short editorial a more com-prehensive description for all articles of this special issue.However, the guest editors believe that the results includedreflect some recent trends in research and outline new ideasfor future studies. In the following, we briefly describe thesignificance of the key contributions to our special session.

Lie symmetry methods are powerful tools to find conser-vation laws/first integrals, reductions, and exact solutions ofdifferential equations. Some papers in our special issue haveused Lie symmetry methods to study interesting problemslike heat transfer in extended surfaces of different geometries,nonlinear Jaulent-Miodek equation, neutron transport equa-tions in nonhomogeneous media, nonlinear fin equation,and general bond-option pricing equation of mathematicalfinance. Moreover, different approaches to construct firstintegrals for ordinary differential equations are compared, aswell as symmetries, and their associated first integrals anddouble reduction of difference equations are also investigated.

Several authors study some interesting fluid mechanicsmodels. Similarity solutions of MHD heat and mass transferflow of the steady viscous incompressible fluid over a flat

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 706858, 2 pageshttp://dx.doi.org/10.1155/2014/706858

2 Abstract and Applied Analysis

plate are derived and the ordinary differential equationsobtained from similarity transformations are solved by afinite difference scheme known as the Keller Box method.The homotopy analysis method is employed to study themixed convective heat transfer in an incompressible steadytwo-dimensional viscoelastic fluid flow over a wedge in thepresence of buoyancy effects.

Stability analysis plays a vital role in understanding ofthe dynamic structure of complex dynamical systems. Inour special issue several papers focus on stability analysis ofbiological systems and these models include Lotka-Volterapredator-prey systemwith time-delayed feedback, a nutrient-phytoplankton model with delay effect, a model of emerginginfectious disease leading to amphibian decline, a nutrient-algae-zooplankton system with sinking of algae, a reaction-diffusion systemmodeling cancer network, Hodgkin-Huxleynonlinear model with constraint, malaria epidemic inKaronga district, stochastic cooperative predator-prey systemwith Beddington-DeAngelis functional response, a reaction-diffusion phytoplankton-zooplankton system with a doubleAllee effect on prey, and a stochastic delayed competitivesystem with impulsive toxicant input in a polluted environ-ment. Dynamics of the oxygen, carbon dioxide, and waterinteraction across the insect is investigated. A traveling waveanalysis is carried out for a tumour-immune interactionmodel with immunotherapy. Numerical simulations are alsoperformed for few of these papers.

Papers collected in this special issue are also concernedwith statistical methods which are important to study sometools essential for financial analysts. These papers includeanalysis of corporate bond valuation under an infinite dimen-sional compound Poisson framework, the pricing problemfor convertible bondswith higher loan rate, convertible bondsvia backward stochastic differential equations, the forwardcredit default swap with general random loss, and study offinancial time series model by copula analysis.

Our special issue contains few papers in which differentnumerical techniques are employed. Haar wavelet method isused to find numerical solution of the system of Fredholmintegral equations and the system of Volterra integral equa-tions. The solution of a class of Lane-Emden equations isderived by the reproducing kernel method. Two nonstandardfinite difference schemes to solve the regularized long waveequation are reported in our special issue. A new approachbased on the effective combination of Lie symmetry method,homotopy perturbation method, finite element method, andsimulation based error reduction techniques is presentedand employed to transient nonlinear heat conduction prob-lems. Numerical reduced variable optimization methods viaimplicit functional dependence are proposed and applied tonontrivail real-world problem. Amethod is developed for theapproximate computation of the frequency-dependent mag-netic and electric matrix Green’s functions in a rectangularparallelepiped with a perfect conducting boundary. Analyticformulation for the sound absorption of a panel absorberunder the effects ofmicroperforation, air pumping, and linearand nonlinear vibrations is reported.

We hope the papers published in this special issue willprovide a useful guide to a large community of researchers

and will give way to development of new innovative theoriesin the fields of natural, social, and life sciences.

Acknowledgments

We thank all the authors who contributed to this specialissue. Moreover a special gratitude goes to the honorablereviewers for their hard work, time, and efforts to review allthe submitted articles.

Rehana NazIgor Leite FreireImran Naeem

Mariano Torrisi

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