Fundamental solutions for second order linear elliptic partial differential equations

This paper is concerned with outlining some fundamental solutions and Green's functions for a system of second order linear elliptic partial differential equations in two independent variables. The fundamental solution and a number of Green's functions are given in relatively elementary closed form for some cases when the coefficients in the equations are constant. When the coefficients are variable the fundamental solution is obtained for some particular classes of equations.     

Squeezing and photon antibunching in second harmonic generation: an analytical approach

With the help of second-order nonlinear interactions and hence Hamiltonian, we construct the equations of motion corresponding to pump and signal (second harmonic) fields in a second harmonic generation. The corresponding coupled differential equations involving the non-commuting field operators are not solvable in closed analytical forms. With the help of a perturbation method, we obtain analytical solutions of these field operators up to cubic orders in the interaction constant. In an appropriate limit (truncating the solution up to the second order in the interaction constant), these solutions lead to the existing solutions under the short-time approximation. The present analytical solutions are exploited to investigate the squeezing and the antibunching of photons of the input coherent light coupled to the said second-order nonlinear medium. We report the squeezing and antibunching effects of the pump field even for solutions up to the second order and hence the present results are consistent with earlier results. However, for the second harmonic mode, we report the squeezing involving the leading cubic term in the interaction constant.     

On the derivation of first integrals for similarity solutions

In two recent papers the authors have obtained a number of first integrals for similarity solutions of nonlinear diffusion and of general high-order nonlinear evolution equation. Such integrals exist only for special parameter values and are obtained via integration of the ordinary differential equation, which results when the functional form of the solution is substituted into the governing partial differential equation. In this paper we show that these special parameter values also occur in a natural way when we utilize the first order partial differential equation instead of the explicit functional form and we ask under what conditions can a first integral with respect to either of the independent variables x or t be deduced. This simple procedure generates all previous results and presents the idea of similarity solutions in an entirely new light. That is, the significant features of similarity solutions for partial differential equations are not necessarily the explicit functional form and subsequent reduction to an ordinary differential equation but rather that the solutions sort are common to two partial differential equations. The process is illustrated with reference to an extensive number of examples including nonlinear diffusion, general diffusion equations containing a number of parameters and high-order nonlinear evolution equations. In addition a new exact solution for nonlinear diffusion is obtained which is illustrated graphically.     

Steady plane magneto-gas-dynamic flows having constant velocity magnitude on each streamline

In this work, well-known methods of solving nonlinear partial differential equations have been used to study steady plane magneto-gas-dynamic orthogonal, aligned, and transverse flow problems when the velocity magnitude is constant on each individual streamline. In the study of the orthogonal case, Monge's method for solving second-order nonlinear partial differential equations is used. The hodograph and Charpits' methods are applied in the other two cases of aligned and transverse flows to investigate the geometries and solutions for the flow problems.     

Group method solutions of the generalized forms of Burgers, Burgers�CKdV and KdV equations with time-dependent variable coefficients

Solutions for the generalized forms of Burgers, Burgers?CKdV, and KdV equations with time-dependent variable coefficients and with initial and boundary conditions are constructed. The analysis rests mainly on the standard group method. Similarity solutions are found which reduce the nonlinear system of partial differential equations to systems of ordinary differential equations to obtain some exact solutions and others as numerical solutions.     

A closed-form solution for the stress field in a rigid-perfectly plastic material under plane-strain conditions

Summary A closed-form solution of the problem of the stress-evaluation in a rigid-perfectly plastic material under plane-strain conditions is presented. The solution is derived from the nonlinear partial differential system consisted of the equations of stress-equilibrium and an appropriate yield condition, in the present case the von Mises-Hencky one. The decoupling of the above system leads to a second order partial differential equation of the Monge type, a closed-form solution of which is obtained.With 1 Figure     

一类变系数偏微分方程的精确解的求法及其计算机机械化实现

The variable-coe?cient partial differential equations are not only used in many physical models, but also fundamentally applied in the field of nonlinear science. In order to solve certain variable-coe?cient partial differential equations, the aux-iliary elliptic-like equation method is introduced in this article by means of the symbolic computation software. The basic idea of the new algorithm is that if certain variable-coe?cient partial differential equation can be converted into the form of elliptic equation, then its solutions are readily obtained. By taking the Kadomtsev-Petviashvili equation for an example, not only the effectiveness of the algorithm is demonstrated, but many new solutions are worked out, including dark solitary wave, bell profile solitary wave solutions and Jacobian elliptic function solutions, which may be useful for depicting nonlinear physical phenomena.     

Domain mappings for the numerical solution of partial differential equations

Independent variable transformations of partial differential equations are examined with regard to their use in numerical solutions. Systems of first order and second order partial differential equations in conservative and nonconservative form are considered. These general equations are transformed using generalized mapping functions and important computational features of the transformed equations are discussed. Examples of mappings which regularize domains are given involving various types of partial differential equations. These mappings are of particular importance in finite difference approximations because of the ease with which a mesh can be adapted to regions formed by co-ordinate lines.     

On nonlinear evolution of convective flow in an active mushy layer

We consider the nonlinear effect of convective flow in a horizontal mushy layer during solidification. The mushy layer that we consider has a permeable mush–liquid interface and is treated as an active porous medium with variable permeability. The nonlinear partial differential equations involved in this system are conservation equations for flow momentum, mass, temperature, and concentration. Numerical solutions to the resulting weakly nonlinear equations are obtained using a fourth-order Runge–Kutta integration scheme via a shooting technique. An evolution equation of Landau type is derived in terms of linear and first-order solutions by introducing an adjoint operator. The Landau constant is calculated for both cases: constant permeability and variable permeability. The analysis reveals that there is a slow transition of the flow to a steady state with smaller amplitude for an active mushy layer.     

Explicit solitary and periodic solutions for optical cascading

Many types of periodic solutions to the pair of nonlinear partial differential equations describing optical cascading are obtained in terms of Jacobian elliptic functions. The choice of appropriate forms for periodic solutions is based on the known solitary-wave solutions of the system. By a straightforward procedure for determining coefficients, it is possible to construct periodic analogues of some classes of solitary wave solutions.     

复合载荷作用下波纹扁壳的非线性振动

应用轴对称旋转扁壳的非线性大挠度动力学方程，研究了波纹扁壳在复合载荷作用下的非线性受迫振动问题。采用格林函数方法，将扁壳的非线性偏微分方程组化为非线性积分微分方程组。再使用展开法求出格林函数，即将格林函数展开为特征函数的级数形式，积分微分方程就成为具有退化核的形式，从而容易得到关于时间的非线性常微分方程组。针对单模态振形，得到了谐和激励作用下的幅频响应。作为算例，研究了正弦波纹扁球壳的非线性受迫振动现象。得到的解答可供波纹壳的设计参考。     

A boundary method of Trefftz type for PDEs with scattered data

This paper presents the application of a Trefftz type method for partial differential equations (PDEs) of the elliptic type with inhomogeneous term given by a set of scattered data. The method of particular solutions is used. Basis functions of a new type were introduced to approximate the scattered data. Using these basis functions, we get the approximation in the form of series over some orthogonal system of eigenfunctions. The particular case of the trigonometric eigenfunctions is considered. The corresponding approximation of the inhomogeneous term allows to get a particular solution for PDEs with constant coefficients or for the systems of such PDEs easily. We test our basis functions on recovering well-known Franke's and PEAKS functions given by scattered data. We also present results of solution Helnholtz PDE, PDE with differential operator of 4th order and system of PDEs arising in shell deflection problems. A comparison of the numerical solutions with analytic solutions is performed for all the problems.     

Bioremediation of contaminated soil beds and groundwater — A simulation study

Pollution has reached levels which demand immediate attention and scientific and technological solutions are required on an urgent basis. We are concerned in this paper with bioremediation of soil and groundwater, i.e. the use of indigenous micro-organisms to clean up soil beds and groundwater contaminated with organic pollutants. To achieve managedin situ bioremediation in practice, treated water is recycled with added nutrients into the ground so that oxygen and nitrogen are carried with the water to the subsurface regions. Sorption, convective-dispersive flow and chemical and biological transformations are the chief processes involved that have to be modelled. Here we discuss a simulation model developed to aid in designing an efficient system that maximizes the rate of biodegradation. Simulation models are a must in this case since laboratory experiments take time periods of the order of months. An unusual feature of this simulation model is that it is governed by coupled partial and ordinary differential equations. Partial differential equations (PDEs) model the diffusion and biodegradation processes occurring in the micropores of soil aggregates while ordinary differential equations (ODEs) describe the bioremediation in the interstitial spaces between soil aggregates, both partial and ordinary differential equations being nonlinear. The model is applied to the case of high initial contaminant concentrations. This work is part of a joint project with Regional Research Laboratory, Jorhat and has been carried out in close cooperation with N N Dutta. Discussions with K S Yajnik have been very useful.     

三维非线性弹性壳体的维数分裂法

本文给出了一个建立在半测地坐标系下的非线性弹性壳体的维数分裂方法,它把一个非线性弹性算子,在这个坐标系下,分裂为一个称为膜弹性算子和弯曲弹性算子之和.假设非线性弹性壳体的解可以展开为关于贯裁变量的Taylor级数,那么本文建立了关于首项的2D-3C非线性偏微分方程组,证明其解的存在性,同时给出了两个关于一阶项和二阶项对于首项的函数,从而无需求解偏微分方程即可得到一阶项和二阶项.     

Galerkin based generalized ANOVA for the solution of stochastic steady state diffusion problems

This work presents a novel approach, referred here as Galerkin based generalized analysis of variance decomposition (GG-ANOVA), for the solution of stochastic steady state diffusion problems. The proposed approach utilizes generalized ANOVA (G-ANOVA) expansion to represent the unknown stochastic response and Galerkin projection to decompose the stochastic differential equation into a set of coupled differential equations. The coupled set of partial differential equations obtained are solved using finite difference method and homotopy algorithm. Implementation of the proposed approach for solving stochastic steady state diffusion problems has been illustrated with three numerical examples. For all the examples, results obtained are in excellent agreement with the benchmark solutions. Additionally, for the second and third problems, results obtained have also been compared with those obtained using polynomial chaos expansion (PCE) and conventional G-ANOVA. It is observed that the proposed approach yields highly accurate result outperforming both PCE and G-ANOVA. Moreover, computational time required using GG-ANOVA is in close proximity of G-ANOVA and less as compared to PCE.     

A novel mean square formulation of stochastic nonlinear dynamic systems based on Adomian decomposition

An Adomian decomposition based mathematical framework to derive the mean square responses of nonlinear structural systems subjected to stochastic excitation is presented. The exact mean square response estimation of certain class of nonlinear stochastic systems is achieved using Fokker–Planck–Kolmogorov (FPK) equations resulting in analytical expressions or using Monte Carlo simulations. However, for most of the nonlinear systems, the response estimation using Monte Carlo simulations is computationally expensive, and, also, obtaining solution of FPK equation is mathematically exhaustive owing to the requirement to solve a stochastic partial differential equation. In this context, the present work proposes an Adomian decomposition based formalism to derive semi-analytical expressions for the second order response statistics. Further, a derivative matching based moment approximation technique is employed to reduce the higher order moments in nonlinear systems into functions of lower order moments without resorting to any sort of linearization. Three case studies consisting of Duffing oscillator with negative stiffness, Rayleigh Van-der Pol oscillator and a Pendulum tuned mass damper inerter system with linear auxiliary spring–damper arrangement subjected to white noise excitation are undertaken. The accuracy of the closed form expressions derived using the proposed framework is established by comparing the mean square responses of the systems with the exact solutions. The results demonstrate the robustness of the proposed framework for accurate statistical analysis of nonlinear systems under stochastic excitation.     

A mathematical model of pulsatile flows of microstretch fluids in circular tubes

Pulsatile flows of micropolar fluids with stretch whose microelements can undergo expansions and contractions besides translations and rotations in straight circular tubes are considered. The governing field equations for such flows of linear microstretch fluids turn out to be a nonlinear coupled partial differential system. Solutions are sought for this system starting with a reasonable initial approximation for microinertia and the consequent linearization of the field equations. One of the coupled equations governing the microstretch and microinertia is solved approximately by the method of Laplace transforms taken with respect to the time variable. Making use of this approximate solution, the other coupled equation is solved leading to explicit higher order approximation solutions for microinertia, microstretch and micropressure. Next, the coupled equations governing the velocity and the microrotation fields are solved by employing the finite Hankel transform operators on a space variable and their inversions, and higher order approximation solutions are determined. All the above-mentioned explicit solutions are obtained in computationally suitable forms. These solutions have the promise of application to many practically important physical situations such as flows of polymeric fluids with deformable springy suspensions and flows of biological fluids including blood with deformable cell suspensions in small arteries.     

Semi-inverse methods for obtaining partial differential equations with chaotic solutions

Semi-inverse methods are used to construct partial differential equations that have chaotic solutions. For first order P.D.E., this comes about by the natural association of the P.D.E. with the orbital equations for the characteristics. Explicit examples are given of linear, quasilinear, nonlinear and systems with the same principal part. The latter kind are used to show that there are 2-dimensional, steady, incompressible, inviscid fluid flows that exhibit chaotic behavior. Second order P.D.E. Are obtained through the requirement that they admit similarity solutions that satisfy O.D.E. With known chaotic behavior. Explicit systems of nonlinear second order P.D.E. That admit chaotic simple wave similarity solutions are constructed.     

广义Tanh函数法中Riccati方程和sine-Gordon方程的新解及其新应用

非线性偏微分方程的显式解析解,特别是行波解,蕴含了方程的丰富信息,对于描述各种现象的发展规律起着至关重要的作用．本文尝试构造 KdV 方程多种形式的新显式行波解．首先,利用试探函数法和 Matlab计算给出了 Riccat 方程的许多新显式解析解．其次,运用广义 Tanh 函数法以及 Riccati 方程的新解得到了 sine-Gordon 方程的许多新显式解析解．最后,作为新的应用,把三角函数法结合 sine-Gordon 方程的新显式解析解并利用简化的变换形式进一步找到了 KdV 方程的许多新显式行波解．这些结果推广和补充了以往的相关研究成果,特别地,这些方法和新的结果可以用于求解许多非线性偏微分方程的新显式行波解．     

Analytic parametric solutions for the HRR nonlinear elastic field with low hardening exponents

Summary In this paper, we restore the already constructed approximate asymptotic solutions extracted in [10] concerning the HRR [1] strongly nonlinear fourth-order ordinary differential equation (ODE) for plane strain conditions in nonlinear elastic (plastic) fracture. It is proved that the above equation, for low strain hardening exponents (0 < N ? 1), is reduced to a strongly nonlinear ODE of the second order. The method of the total differentials is used so that the last equation is reduced to Abels' equations of the second kind of the normal form, that can be analytically solved in parametric form. In addition, the case of rigid perfect-plasticity (N=0) is extensively investigated and several important results are extracted.