editorial analytical and numerical methods for solving...

EditorialAnalytical and Numerical Methods for Solving PartialDifferential Equations and Integral Equations Arising inPhysical Models 2014

Santanu Saha Ray,1 Rasajit K. Bera,2 Adem KJlJçman,3 Om P. Agrawal,4 and Yasir Khan5

1Department of Mathematics, National Institute of Technology, Rourkela 769008, India2Department of Science, National Institute of Technical Teachers’ Training and Research, BK-301, Sector II, Salt Lake City,Kolkata 700091, India3Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang,Selangor, Malaysia4Department of Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL 62901, USA5Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Correspondence should be addressed to Santanu Saha Ray; [email protected]

Received 2 March 2015; Accepted 2 March 2015

Copyright © 2015 Santanu Saha Ray et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Nowadays, partial differential equations (PDEs) have becomea suitable tool for describing the natural phenomena ofscience and engineering models. Most of the phenomenaarising in mathematical physics and engineering fields canbe expressed by PDEs. Many engineering applications aresimulated mathematically as PDEs with initial and boundaryconditions. Most physical phenomena of fluid dynamics,gravitation, chemical reaction, dispersion, nonlinear optics,plasma physics, quantum mechanics, electricity, acoustics,solid mechanics, and many other models are controlledwithin their domain of validity by PDEs. Therefore, itbecomes increasingly important to be familiar with all tradi-tional and recently developed methods for solving PDEs andthe implementations of these methods.

For many years the subject of functional equations hasheld a prominent place in the attention of mathematicians.In recent years this attention has been focused on a specifickind of functional equations, an integral equation, wherethe unknown function occurs under the integral sign. Suchequations occur widely in diverse fields including contin-uum mechanics, potential theory, geophysics, electricity andmagnetism, kinetic theory of gases, hereditary phenomenain physics and biology, renewal theory, quantum mechanics,

radiation, optimization, optimal control systems, communi-cation theory, mathematical economics, population genetics,queuing theory, medicine, mathematical problems of radia-tive equilibrium, the particle transport problems of astro-physics and reactor theory, acoustics, fluid mechanics, steadystate heat conduction, fracture mechanics, and radiative heattransfer problems.Theyoffer a powerful technique for solvinga variety of practical problems.

This special issue is devoted to study the recent works inthe above fields of partial differential equations and integralequations done by the leading researchers. Accordingly,various papers on partial differential equations and integralequations have been included in this special issue aftercompleting a heedful, rigorous, and peer-review process.Theissue contains eight research papers.

The (2 + 1)-dimensional nonlinear Schrodinger equations(NLS) are transformed into the standard (1 + 1)-dimensionnonlinear Schrodinger equation by using appropriate trans-formation and then by using Exp-functionmethod bisolitonsand a series of breather solitons (rogue waves) solutionsare obtained. Solutions of the (2 + 1)-dimensional nonlinearSchrodinger equations, which contain Akhmediev breathersoliton, Ma breather soliton and Peregrine breather soliton

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015, Article ID 193030, 2 pageshttp://dx.doi.org/10.1155/2015/193030

2 Abstract and Applied Analysis

are represented. Also, higher-order rational rogue wave solu-tions are obtained for (2 + 1)-dimensional NLS equations byusing the similarity transformation on the solutions of the (1+ 1)-dimensional NLS equation. Several arbitrary parametersare involved to generate abundant wave structures whichgreatly enriched the diversity of wave structures for the (2 +1)-dimensional nonlinear Schrodinger equations.

By using the central difference, upwind, hybrid, power-law, and exponential scheme, the general transport equationhas been investigated. In the solution, three different gridsystems of 80 × 100, 160 × 200, and 320 × 400 from nodalpoints were used by the authors.The obtained results indicatethat hybrid and power-law yield better results as compared tothe other discretization methods.

Some explicit travelling wave solutions are presented toconstruct the exact solutions of nonlinear partial differen-tial equations of mathematical physics. Applying Frobeniustransformation to the coupled Burgers, combined Korteweg-de Vries- (cKdV-) modified KdV (mKdV) and Schrodinger-KdV equations can be written as bilinear ordinary differentialequations and two travelling wave solutions are generated.The properties of the multiply travelling wave solutions areshown in figures. The solutions thus obtained are stable andare used to describe nonlinear interaction of travelling waves.

The authors have studied the influence of heat transferon thin film flow of a reactive third order fluid with variableviscosity and slip boundary condition.The problem is formu-lated in the form of coupled nonlinear equations governingthe flow together with appropriate boundary conditions.Approximate analytical solutions for velocity and temper-ature are obtained by using optimal homotopy asymptoticmethod (OHAM) and are compared with Adomian decom-position method (ADM) solutions. Both of these solutionsare found identical as demonstrated in graphs and tables.The graphical results for embedded flow parameters are alsoshown by the authors.

The nonlinear density temperature variations in two-dimensional nanofluid flow over heated vertical surface witha sinusoidal wall temperature are examined in detail. Themodel includes the effects of Brownian motion and ther-mophoresis. Two-dimensional momentum, heat, and masstransfer equations are transferred to nonlinear partial dif-ferential equations using the boundary layer approximationand solved numerically using spectral local linearizationmethod. The effects of the governing parameters on the fluidproperties as well as on the heat and nanomass transfercoefficients are determined and are shown graphically.

An efficient fourth-order numerical algorithm that com-bines the Pade approximation in space and fourth-orderaccurate Rosenbrock method in time is proposed to solvea nonlinear parabolic partial differential equation. Thesemidiscrete ordinary differential equation (ODE) systemresulting from compact higher-order finite difference spatialdiscretization of the reaction-diffusion equation is highlystiff.Therefore, numerical time integrationmethods with stiffstability such as implicit Runge-Kutta methods and implicitmultistep methods are required to solve the large scale stiffODE system. In this work the authors construct a new fourth-order Rosenbrock method to solve the nonlinear parabolic

partial differential equation supplemented with Dirichlet orNeumann boundary condition. Two numerical examples aresolved to demonstrate the efficiency, stability, and accuracy ofthe proposed method.

Themodeling of unidirectional propagation of long waterwaves in dispersive media is considered. The authors havederived the Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations from water wave models. Byimplementing the extended direct algebraic and modifiedsech-tanhmethods, the authors have presented new travelingwave solutions of the KdV and BBM equations. The stabilityof the obtained traveling wave solutions are analyzed anddiscussed.

Newtonian fluids described by Navier Stokes equationsare extensively studied in the literature for the past fewdecades. The unsteady free flow of a Casson fluid over anoscillating vertical plate with constant wall temperature hasbeen studied. The Casson fluid model is used to distinguishthe non-Newtonian fluid behaviour. The governing partialdifferential equations corresponding to the momentum andenergy equations are transformed into linear ordinary dif-ferential equations by using nondimensional variables. Thedimensionless governing equations are solved by using theLaplace transform technique. Expressions for shear stressin terms of skin-friction and the rate of heat transfer interms of Nusselt number are also obtained. Numerical resultsof velocity and temperature profiles with various values ofembedded flow parameters are shown graphically and theireffects are discussed in detail.

At the present, the use of partial differential equationsand integral equations are commonly encountered in thefields of science and engineering. This issue has addressedsome efficient computational tools, recent trends, and devel-opments regarding the analytical and numerical methods forthe solutions of partial differential equations and integralequations arising in physical models. Eventually, it may beexpected that the present special issue would certainly help toexplore the researchers with their new arising problems andstimulate the efficiency and accuracy of the solutionmethodsfor those problems in use nowadays.

Santanu Saha RayRasajit K. BeraAdem KılıcmanOm P. Agrawal

Yasir Khan

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

editorial analytical and numerical methods for solving...

Documents