Solutions Of Nonlinear Partial Differential Equations Research Articles

Entropy Generation and Radiation Analysis on Peristaltic Transport of Hyperbolic Tangent Fluid with Hybrid Nanoparticle Through an Endoscope

The current study explores the impact of entropy generation, thermal jump, radiation, and inclined magnetic field on the peristaltic transport of hyperbolic tangent fluid containing molybdenum disulfide and silver nanoparticles through an endoscope with a long wavelength and low Reynolds number assumptions. Between two coaxial tubes, a non-Newtonian hyperbolic tangent fluid with silver nanoparticles is considered. The Second law of thermodynamics is used to examine the entropy generation. The Homotopy perturbation method (HPM) is applied to describe the solution of nonlinear partial differential equations. We were able to arrive at analytical solutions for velocity, temperature, and nanoparticle concentration. In the end, the impact of various physical parameters on temperature, nanoparticle concentration, velocity, entropy generation, and Bejan number was graphically depicted. The significant outcome of the present study is that the impact of Hartmann number and Brownian motion parameter declines the velocity profile, but the thermal Grashoff number enhances velocity, whereas Platelet-shaped nanoparticles achieve a higher speed as compare to Spherical-shaped nanoparticles.

Bilinear Equation of the Nonlinear Partial Differential Equation and Its Application

The homogeneous balance of undetermined coefficient method is firstly proposed to derive a more general bilinear equation of the nonlinear partial differential equation (NLPDE). By applying perturbation method, subsidiary ordinary differential equation (sub-ODE) method, and compatible condition to bilinear equation, more exact solutions of NLPDE are obtained. The KdV equation, Burgers equation, Boussinesq equation, and Sawada-Kotera equation are chosen to illustrate the validity of our method. We find that the underlying relation among theG′/G-expansion method, Hirota’s method, and HB method is a bilinear equation. The proposed method is also a standard and computable method, which can be generalized to deal with other types of NLPDE.

Wavelet-based lifting scheme for the numerical solution of some class of nonlinear partial differential equations

Partial differential equation (PDE) occurs in the mathematical modeling of many physical phenomena arising in science and engineering. In this paper, we present wavelet-based lifting scheme for the numerical solution of nonlinear PDEs using different wavelet filter coefficients. The numerical results obtained by this scheme are compared with the exact solution to demonstrate the accuracy and also speeds up convergence in lesser computational time as compared with the existing schemes. Some test problems are presented for the validity and applicability of the scheme.

Comparison between the (G’/G) - expansion method and the modified extended tanh method

Abstract In this paper, firstly, we give a connection between well known and commonly used methods called the ( G ' G ) $\left( {{{G'} \over G}} \right)$ -expansion method and the modified extended tanh method which are often used for finding exact solutions of nonlinear partial differential equations (NPDEs). We demonstrate that giving a convenient transformation and formula, all of the solutions obtained by using the ( G ' G ) $\left( {{{G'} \over G}} \right)$ - expansion method can be converted the solutions obtained by using the modified extended tanh method. Secondly, contrary to the assertion in some papers, the ( G ' G ) $\left( {{{G'} \over G}} \right)$ -expansion method gives neither all of the solutions obtained by using the other method nor new solutions for NPDEs. Namely, while the modified extended tanh method gives more solutions in a straightforward, concise and elegant manner without reproducing a lot of different forms of the same solution. On the other hand, the ( G ' G ) $\left( {{{G'} \over G}} \right)$ -expansion method provides less solutions in a rather cumbersome form. Lastly, we obtain new exact solutions for the Lonngren wave equation as an illustrative example by using these methods.

Unification of all hyperbolic tangent function methods

Abstract Over the last twenty years, several “different” hyperbolic tangent function methods have been proposed to search solutions for nonlinear partial differential equations (NPDEs). The most common of these methods were the tanh-function method, the extended tanh-function method, the modified extended tanh-function method, and the complex tanh-function method. Besides the excellent sides of these methods, weaknesses and deficiencies of each method were encountered. The authors realized that they did not actually give “very different and comprehensive results”, and some of them are even unnecessary. Therefore, these methods were analysed and significant findings obtained. Firstly, they compared all of these methods with each other and gave the connections between them; and secondly, they proposed a more general method to obtain many more solutions for NPDEs, some of which having never been obtained before, and thus to overcome weaknesses and deficiencies of existing hyperbolic tangent function methods in the literature. This new method, named as the unified method, provides many more solutions in a straightforward, concise and elegant manner without reproducing a lot of different forms of the same solution. Lastly, they demonstrate the effectiveness of the unifed tanh method by seeking more exact solutions of the Rabinovich wave equation which were not obtained before.

New Improvement of the Expansion Methods for Solving the Generalized Fitzhugh-Nagumo Equation with Time-Dependent Coefficients

An improvement of the expansion methods, namely, the improved tan⁡Φξ/2-expansion method, for solving nonlinear second-order partial differential equation, is proposed. The implementation of the new approach is demonstrated by solving the generalized Fitzhugh-Nagumo equation with time-dependent coefficients. As a result, many new and more general exact travelling wave solutions are obtained including periodic function solutions, soliton-like solutions, and trigonometric function solutions. The exact particular solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. We obtained further solutions comparing this method with other methods. The results demonstrate that the new tan⁡Φξ/2-expansion method is more efficient than the Ansatz method and Tanh method applied by Triki and Wazwaz (2013). Recently, this method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. Abundant exact travelling wave solutions including solitons, kink, and periodic and rational solutions have been found. These solutions might play an important role in engineering fields. It is shown that this method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving the nonlinear physics.

A new Green's function Monte Carlo algorithm for the solution of the three-dimensional nonlinear Poisson–Boltzmann equation: Application to the modeling of plasma sheath layers

Abstract. The objective of this paper is the exposition of a recently-developed Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a spherical conducting object, carrying a potential and moving at low speeds through a partially ionized medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson–Boltzmann (NPB) equation. The traditional GFMC-based approaches for the solution of nonlinear equations involve the iterative solution of a series of linear problems. Over the last several years, one of the authors of this paper, K. Chatterjee, has been developing a philosophically different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains, where the random-walker makes its walks. On the other hand, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. However, the premise of the algorithm is novel and rigorous mathematical justification has to be established in the future. The application areas of interest include the communication blackout problem during a space vehicle's re-entry into the atmosphere and electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications.

Algorithms for nonlinear fractional partial differential equations: A selection of numerical methods

Fractional order partial differential equations, as generalization of classical integer order partial differential equations, are increasingly used to model problems in fluid flow, finance and other areas of applications. In this paper we present a collection of numerical algorithms for the solution of nonlinear partial differential equations with space- and time-fractional derivatives. The fractional derivatives are considered in the Caputo sense. Two numerical examples are given to demonstrate the effectiveness of the present methods. Results show that the numerical schemes are very effective and convenient for solving nonlinear partial differential equations of fractional order.

A New Rational Algebraic Approach to Find Exact AnalyticalSolutions to a (2+1)-Dimensional System

In this paper, we present a new rational algebraicapproach to uniformly construct a series of exactanalytical solutions for nonlinear partial differentialequations. Compared with most existing tanhmethods and other sophisticated methods, the proposed method not onlyrecovers some known solutions, but also finds some new andgeneral solutions. The solutions obtained in this paperinclude rational form triangular periodic wave solutions,solitary wave solutions, and elliptic doubly periodic wave solutions.The efficiency of the method can be demonstrated on(2+1)-dimensional dispersive long-wave equation.

A Generalized Method and Exact Solutions in Bose–EinsteinCondensates in an Expulsive Parabolic Potential

In the paper, a generalized sub-equation method ispresented to construct some exact analytical solutions of nonlinearpartial differential equations. Making use of the method, we presentrich exact analytical solutions of the one-dimensional nonlinearSchrödinger equation which describes the dynamics of solitons inBose–Einstein condensates with time-dependent atomic scatteringlength in an expulsive parabolic potential. The solutions obtainedinclude not only non-traveling wave and coefficient function'ssoliton solutions, but also Jacobi elliptic function solutions andWeierstrass elliptic function solutions. Some plots are given todemonstrate the properties of some exact solutions under theFeshbach-managed nonlinear coefficient and the hyperbolic secant function coefficient.

Special Conditional Similarity Reduction Solutionsfor Two Nonlinear Partial Differential Equations

We present a method of special conditionalsimilarity reduction solutions for nonlinear partial differentialequations. As concrete examples of its application, we apply thismethod to the (2+1)-dimensional modified Broer-Kaup equations andthe variable coefficient KdV-mKdV equation, which have extensivephysics backgrounds, and obtain abundant exact solutions derivedfrom some reduction equations.

Extended Fan's Algebraic Method and Its Applicationto KdV and Variant Boussinesq Equations

An extended Fan's algebraic method is used forconstructing exact traveling wave solution of nonlinear partial differentialequations. The key idea of this method is to introduce an auxiliary ordinarydifferential equation which is regarded as an extended elliptic equation andwhose degree r is expanded to the case of r>4. The efficiency of themethod is demonstrated by the KdV equation and the variant Boussinesqequations. The results indicate that the method not only offers allsolutions obtained by using Fu's and Fan's methods, but also some new solutions.

Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems

In this paper, we will compare the differential transformation method DTM and Adomian decomposition method ADM to solve partial differential equations (PDEs). The definition and operations of differential transform method was introduced by Zhou [Zhou JK. Differential transformation and its application for electrical circuits. Wuuhahn, China: Huarjung University Press; 1986 [in Chinese]]. Adomian decomposition method which is given by Adomian [Adomian G. Convergent series solution of nonlinear equation. J Comput Appl Math 1984;11:113–7; Adomian G. Solutions of nonlinear PDE. Appl Math Lett 1989;11:121–3; Adomian G. Solving Frontier problems of physics. The decomposition method, Boston, 1994] for approximate solution of linear and nonlinear differential equations and to the solutions of various scientific models see [Abulwafa EM, Abdou MA, Mahmoud AA. The solution of nonlinear coagulation problem with mass loss. Chaos, Solitons & Fractals 2006;29:313–30; El-Danaf TS, Ramadan MA, Abd Alaal FEI. The use of Adomian decomposition method for solving the regularized long-wave equation. Chaos, Solitons & Fractals 2005;26:747–57; El-Sayed SM. The decomposition method for studying the Klein–Gordon equation. Chaos, Solitions & Fractals 2003;18:1025–30; Helal MA, Mehanna MS. A comparison between two different methods for solving KdV-Burgers equation. Chaos, Solitons & Fractals 2006;28:320–6, Hashim I, Noorani MSM, Ahmad R, Bakar SA, Ismail ES, Zakaria AM. Accuracy of the decomposition method applied to the Lorenz system. Chaos, Solitons & Fractals 2006;28:1149–58; Kaya D, El-Sayed SM. An application of the decomposition method for the generalized KdV and RLW equations, Chaos, Solitons & Fractals 2003;17:869–77; Lesnic D. Blow-up solutions obtained using the decomposition method. Chaos, Solitons & Fractals 2006;28:776–87; Wazwaz AM. Construction of solitary wave solutions and rational solutions for KdV equation by Adomian decomposition method. Chaos, Solitons & Fractals 2001;12:2283–93; Wazwaz AM. Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method. Chaos, Solitons & Fractals 2001;12:1549–56]. A distinctive practical feature of the differential transformation method DTM is ability to solve linear or nonlinear differential equations. Higher-order dimensional differential transformation are applied to a few some initial value problems to show that the solutions obtained by the proposed method DTM coincide with the approximate solution ADM and the analytic solutions.

Exact Travelling Solutions of Discretesine-Gordon Equation via Extended Tanh-Function Approach

In this paper, we generalize the extended tanh-function approach,which was used to find new exact travelling wave solutions ofnonlinear partial differential equations or couplednonlinear partial differential equations, to nonlineardifferential-difference equations. As illustration, twoseries of exact travelling wave solutions of the discretesine-Gordon equation are obtained by means of the extendedtanh-function approach.

New Generalized Transformation Method and Its Application inHigher-Dimensional Soliton Equation

A new generalized transformation method is presentedto find more exact solutions of nonlinear partial differentialequation. As an application of the method, we choose the (3+1)-dimensionalbreaking soliton equation to illustrate the method. As a resultmany types of explicit and exact traveling wave solutions, whichcontain solitary wave solutions, trigonometric function solutions,Jacobian elliptic function solutions, and rational solutions,are obtained. The new method can be extended to other nonlinearpartial differential equations in mathematical physics.

Computational Partial Differential Equations: Numerical Methods and Diffpack Programming, Second Edition

11R1. Computational Partial Differential Equations: Numerical Methods and Diffpack Programming, Second Edition. - HP Langtangen (Simula Res Lab, Martin Linges vei 17, Fornebu, PO Box 134, Lysaker, 1325, Norway). Springer-Verlag, Berlin. 2003. 855 pp. ISBN 3-540-43416-X. $69.95.Reviewed by RL Huston (Dept of Mech, Indust, and Nucl Eng, Univ of Cincinnati, PO Box 210072, Cincinnati OH 45221-0072).This is the second edition of a popular tutorial on the numerical solution of partial differential equations (PDEs). It is intended for students, researchers, and practitioners interested in developing computer codes for the solution of the equations. The stated aim of the book is to equip the reader with skills for developing simulation software for physical phenomena (particularly, solid and fluid mechanics) governed by PDEs. The flow and style of the book are numeric together with listed computer codes. The software tools are based upon Diffpack–a numerical library using C++ and object oriented modules. Prior familiarity with C++ and Diffpack is thus obviously an advantage for potential readers. However, the book is written so that readers can learn both C++ and the use of Diffpack through a series of simple introductory examples and illustrations. The book is directed toward application in the various areas of solid and fluid mechanics. The book itself is divided into seven large chapters (or sections) together with four appendices spanning over 800 pages. Chapter 1 introduces the concepts of PDE solution using Diffpack. Elements of C++ programming are included. The chapter presents several illustrations of finite difference solution of the Poisson equation and the wave equation. Chapter 2 provides an introduction to the finite-element method starting with a discussion of weighted-residual methods and concluding with the mathematics of variational formulations. The third chapter presents a discussion of the use of Diffpack’s finite element software tools. Applications in heat transfer and the solution of the wave equation are given. Chapter 4 is devoted to nonlinear problems. It discusses discretation and the solution of nonlinear PDEs using both finite-difference and finite-element methods. Chapter 5, 6, and 7 present applications in solid mechanics, fluid mechanics, and coupled solid/fluid and fluid/heat transfer problems. The book concludes with four appendices providing extensive discussions of the underlying mathematics, Diffpack topics, linear systems, and software tools for solving linear systems. In the spirit of being a tutorial and text, Computational Partial Differential Equations: Numerical Methods and Diffpack Programming has over 150 exercises and a comparable number of worked-out examples together with computational code. There is an extensive bibliography of 156 references for further reading. The book is clearly very specialized but still devoted to an important aspect of applied mechanics. Therefore, it should be of interest and use to researchers and practitioners working in computational mechanics and to students aspiring to enter that field. It should make a good text for graduate-level numeric courses. Purchase by libraries is recommended.

Reduction of nonlinear partial differential equation and exact solutions

Nonlinear partial differential equation (NLPDE) is converted into ordinary differential equation (ODE) via a new ansatz. Using undetermined function method, the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica. The exact solutions and solitary solutions of NLPDE are obtained.

An Analytical Approach for Quasi-Linear Equation in Second Order

AbstractThis paper is devoted to the studies of the properties of the solutions of non-linear partial differential equation, being basic ones in differential formulation of the magnetostatic problem of finding the magnetic field dis- tribution. The question of the existence of solutions, possessing an unlimited gradient, for this equation is of particular interest. Previous works dealt with the linear equation type, and also a boundary value problem was con- sidered for certain requirements for the µ function, as well as a more general non-linear case was studied. It was shown that such solutions exist, and their properties will be investigated. The difference scheme for the boundary value problem was built in the domain with corner and numerical calculations were given.

On a general method for investigation of finite difference schemes

Recently sufficiently deep knowledge has been accumulated about the numerical solution of special types of nonlinear partial differential equations (PDE) and also convergence analysis methods of discrete algorithms have been developed [1, 2]. An important problem is to work out general methods for the investigation of finite difference and variational algorithms for the solution of nonlinear PDE. Construction of such methodology is based on generalization of stability and convergence results for the solution of nonlinear ordinary differential equations (ODE) and the linear PDE. In the case of ODE we will mention the books [3, 4], in which a deep analysis of the basic numerical analysis concepts of approximation, stability, and convergence is given. Recent results of stability theory are comprehensively surveyed in [5]. A fairly complete theory has been already constructed for the numerical solution of the linear PDE. The theorems on necessary and sufficient stability conditions of finite-difference schemes are proved and efficient methods for verification these conditions are developed [6, 7]. Therefore attempts to relate investigation of the convergence of nonlinear finite difference schemes to the convergence analysis of some classes of linear problems seems to be fruitful. Usually we take the equation linearized around a given solution of the nonlinear PDE as an example of such a linear problem. Let us mention the most important works related to this trend. The results of [3] are generalized in [8, 9] and a quite general convergence theory for the nonlinear finite difference schemes is obtained. Similar results are also proved in [10], whose ideas are based on the analysis of nonlinear iterative methods developed in [11]. We point out also a rather general methodology for investigation of twoand threestep finite difference schemes developed in [12, 13]. The results given in [14] for the solution of the nonlinear operator equations are used substantially in this methodology. In this article we generalize the methodology for investigation of nonlinear difference schemes, which was proposed in our papers [15-19] for the solution of diffusion-reaction and nonlinear Schrdinger type problems. Our aim is to obtain sufficient stability conditions of finite difference schemes which will enable us to use the well-known investigation scheme "approXimation + stability = convergence" in the case of nonlinear difference schemes and to prove the existence and uniqueness of the finite difference scheme solution at the same time. First we consider thoroughly the case when the neighborhood of a solution of a differential problem is independent of discrete mesh parameters. The efficiency of the proposed methodology is demonstrated for a concrete finite difference scheme, which approximates the well-known Kuramoto-Tsuzuki problem. After this the given investigation scheme is generalized for the case when radius of the region, in which stability of nonlinear difference scheme is proved, depends on the discrete mesh parameters r, h, and also when, generally speaking,

A multiprocessor architecture for solving nonlinear partial differential equations

This paper presents the architecture of a special-purpose multiprocessor system, which we call the Broadcast Cube System (BCS), for solving non-linear Partial Differential Equations (PDEs). The BCS has the following important features: (a) Being based on the conceptually simple bus interconnection scheme it is easily understood. The use of homogeneous Processing Elements (PEs) which can be realized as standard VLSI chips makes the hardware less costly. (b) The interconnection network is simple and regular. The network can easily be extended to vast number of PEs by adding buses with new PEs on them and by slightly increasing the number of PEs on existing buses. The interconnection pattern is highly redundant to support fault tolerance in the event of PE failures. (c) In terms of the switching delays, the delay a message undergoes between a pair of PEs connected to a common bus is zero. The maximum delay between any pair of PEs is one unit and thus a strong localization of communicating tasks is not needed to avoid long message delays even in networks of thousands of PEs. The effectiveness of the BCS has been demonstrated by both analytical and simulation methods using heat transfer and fluid flow simulation, which requires solution of non-linear PDEs, as a benchmark program.