Analytic solutions of a class of nonlinear partial differential equations

An approach is presented for computing the adjoint operator vector of a class of nonlinear （that is, partial-nonlinear） operator matrices by using the properties of conjugate operators to generalize a previous method proposed by the author. A unified theory is then given to solve a class of nonlinear （partial-nonlinear and including all linear） and non-homogeneous differential equations with a mathematical mechanization method. In other words, a transformation is constructed by homogenization and triangulation, which reduces the original system to a simpler diagonal system. Applications are given to solve some elasticity equations.     

NONOSCILLATION RESULTS FOR A CLASS OF THIRD ORDER NONLINEAR DIFFERENTIAL EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＦｉｎｄｉｎｇｓｕｆｆｉｃｉｅｎｔｃｏｎｄｉｔｉｏｎｓｆｏｒｎｏｎｏｓｃｉｌｌａｔｏｒｙｏｆｓｏｌｕｔｉｏｎｓｉｓａｐｒｏｂｌｅｍｏｆｇｅｎｅｒａｌｉｎｔｅｒｅｓｔｉｎｔｈｅｔｈｅｏｒｙｏｆｏｒｄｉｎａｒｙａｎｄｄｅｌａｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ.Ｉｎｔｈｉｓｗｏｒｋｗｅｃｏｎｓｉｄｅｒｄｄｓ σ(ｓ) ｄｄｓ ｒ(ｓ) ｄｘｄｓ +ｑ(ｓ) ｄｘｄｓβ +ｐ(ｓ)ｘα =ｆ(ｓ) ,(1)ｗｈｅｒｅσ ,ｒ,ｑ,ｐａｎｄｆａｒｅｒｅａｌ＿ｖａｌｕｅｄｃｏｎｔｉｎｕｏｕｓｆｕｎｃｔｉ…     

Initial value problems for a class of nonlinear integro-partial differential equations

１ＰｒｏｂｌｅｍｓａｎｄＭａｉｎＲｅｓｕｌｔｓＩｎｔｈｉｓｐａｐｅｒ,ｗｅｓｔｕｄｙｔｈｅｎｏｎｌｉｎｅａｒｖｉｂｒａｔｉｏｎｓｏｆｉｎｆｉｎｉｔｅｒｏｄｓｗｉｔｈｖｉｓｃｏｅｌａｓｔｉｃｉｔｙ．Ｔｈｅｃｏｎｓｔｉｔｕｔｉｏｎｌａｗｏｆｔｈｅｒｏｄｓ...     

The first integral method for solving some important nonlinear partial differential equations

Exact solutions of some important nonlinear partial differential equations are obtained by using the first integral method. The efficiency of the method is demonstrated by applying it for two selected equations.     

INITIAL BOUNDARY VALUE PROBLEMS FOR A CLASS OF NONLINEAR INTEGRO－PARTIAL DIFFERENTIAL EQUATIONS

ＩＮＩＴＩＡＬＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＳＦＯＲＡＣＬＡＳＳＯＦＮＯＮＬＩＮＥＡＲＩＮＴＥＧＲＯ－ＰＡＲＴＩＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＣｕｉＳｈａｎｇ－ｂｉｎ（崔尚斌）（Ｄｅｐｔ．ｏｆＭａｔｈ．,ＬａｎｚｈｏｕＵｎｉ...     

Oscillation of solutions for a class of nonlinear neutral parabolic differential equations boundary value problem

ＩｎｔｒｏｄｕｃｔｉｏｎＤｕｅｔｏｔｈｅａｃｃｕｒｅｎｃｅｏｆａｇｒｅａｔｄｅａｌｏｆｍａｔｈｅｍａｔｉｃｍｏｄｅｌｌｉｎｇｓｄｅｓｃｒｉｂｅｄｉｎｐａｒｔｉａｌｆｕｎｃｔｉｏｎａｌｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｉｎｔｈｅａｐｐｌｉｃａｔｉｏｎｆｉｅｌｄｓｏｆｍｅｃｈａｎｉｃｓａｎｄｃｏｎｔｒｏｌｔｈｅｏｒｙ[1],ｇｒｅａｔｉｎｔｅｒｅｓｔｗａｓａｒｏｕｓｅｄａｓｔｏｔｈｅｓｔｕｄｉｅｓｏｎｔｈｅｐｒｏｐｅｒｔｙｏｆｔｈｅｓｏｌｕｔｉｏｎｓｏｆｔｈｉｓｔｙｐｅｏｆｅｑｕａｔｉｏｎｓ.Ｇ…     

General solutions to a class of time fractional partial differential equations

A class of time fractional partial differential equations is considered, which in- cludes a time fractional diffusion equation, a time fractional reaction-diffusion equation, a time fractional advection-diffusion equation, and their corresponding integer-order partial differential equations. The fundamental solutions to the Cauchy problem in a whole-space domain and the signaling problem in a half-space domain are obtained by using Fourier- Laplace transforms and their inverse transforms. The appropriate structures of the Green functions are provided. On the other hand, the solutions in the form of a series to the initial and boundary value problems in a bounded-space domain are derived by the sine- Laplace or cosine-Laplace transforms. Two examples are presented to show applications of the present technique.     

The necessary and sufficient condition of uniformly convergent difference schemes for the elliptic-parabolic partial differential equation with a small parameter

This paper studies the necessary and sufficient condition of uniformly convergent difference scheme for the elliptic-parabolic partial differential equation with a small parameter.Communicated by Lin Zong-chi.     

A method for studying waves with spatially localized envelopes in a class of nonlinear partial differential equations

A methodology for investigating stationary and travelling waves with spatially localized envelopes is presented. The nonlinear governing partial differential equations considered possess a constant first integral of motion, and are separable in space and time when the small parameter of the problem is set to zero. To study stationary waves, a coordinate transformation on the governing nonlinear partial differential equation is imposed which eliminates the time dependence from the problem. An amplitude modulation function is then introduced to express the response of the system at an arbitrary point as a nonlinear function of a reference response. Analytic approximations to the amplitude modulation function are developed by expressing it in power series, and asymptotically solving sets of singular functional equations at the various orders of approximation. Travelling solutions may be computed from stationary ones, by imposing appropriate Lorentz transformations. As an application of the methodology, stationary and travelling breathers of a nonlinear partial differential equation are analytically computed.     

火工品可靠性的感度参数设计法

提出一种能同时满足火工品发火可靠性和安全可靠性的设计方法,把火工品可靠性指标转换成感度分布参数的取值范围,通过对感度的调节使产品满足发火可靠性和安全可靠性要求,为火工品可靠性设计提供了一条新思路。推导出了常见火工品感度在正态分布、逻辑斯谛分布、对数正态分布和对数逻辑斯谛分布等常见分布条件下把可靠性指标转换为感度分布参数的表达式,实例证明这种方法对火工品研制有指导作用。     

GLOBAL ATTRACTIVITY FOR A CLASS OF NONLINEARDELAY DIFFERENCE EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｇｌｏｂａｌａｔｔｒａｃｔｉｖｉｔｙｏｆｔｈｅｎｏｎｌｉｎｅａｒｄｅｌａｙｄｉｆｆｅｒｅｎｃｅｅｑｕａｔｉｏｎΔｘｎ+ａｎｘｎ+ｆ(ｎ ,∑0ｓ=-ｋ ｑｓ,ｎｘｓ+ｎ) =0　 (ｎ=0 ,1 ,2 ,… ) . ( 1 )ＴｈｅｉｎｉｔｉａｌｖａｌｕｅｏｆＥｑ .( 1 )ｉｓｘｓ =φｓ　 (ｓ=-ｋ ,-ｋ+ 1 ,… ,-1 ,0 ) . ( 2 )Ｉｎｔｈｅｓｅｑｕｅｌ,ｗｅａｌｗａｙｓｓｕｐｐｏｓｅｔｈａｔＤ1) ａｎ ｉｓａｎｏｎｎｅｇａｔｉｖｅｒｅａｌｓｅｑｕｅｎｃｅ ,0 ≤ａｎ <1ａｎｄｋｉｓａｐｏｓｉｔｉｖｅ…     

On accurate boundary conditions for a shape sensitivity equation method

This paper studies the application of the continuous sensitivity equation method (CSEM) for the Navier–Stokes equations in the particular case of shape parameters. Boundary conditions for shape parameters involve flow derivatives at the boundary. Thus, accurate flow gradients are critical to the success of the CSEM. A new approach is presented to extract accurate flow derivatives at the boundary. High order Taylor series expansions are used on layered patches in conjunction with a constrained least‐squares procedure to evaluate accurate first and second derivatives of the flow variables at the boundary, required for Dirichlet and Neumann sensitivity boundary conditions. The flow and sensitivity fields are solved using an adaptive finite‐element method. The proposed methodology is first verified on a problem with a closed form solution obtained by the Method of Manufactured Solutions. The ability of the proposed method to provide accurate sensitivity fields for realistic problems is then demonstrated. The flow and sensitivity fields for a NACA 0012 airfoil are used for fast evaluation of the nearby flow over an airfoil of different thickness (NACA 0015). Copyright © 2005 John Wiley & Sons, Ltd.     

RESEARCH OF THE PERIODIC SOLUTION FOR A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS

ＲＥＳＥＡＲＣＨＯＦＴＨＥＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＦＯＲＡＣＬＡＳＳＯＦ　ＮＯＮＬＩＮＥＡＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＲＥＳＥＡＲＣＨＯＦＴＨＥＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＦＯＲＡＣＬＡＳＳＯＦＮＯＮＬＩＮＥＡＲＤＩＦＦＥ...     

EXISTENCE THEOREMS OF EXTREMAL SOLUTIONS FOR A CLASS OF NONLINEAR INTEGRO－DIFFERENTIAL EQUATIONS

ＥＸＩＳＴＥＮＣＥＴＨＥＯＲＥＭＳＯＦＥＸＴＲＥＭＡＬＳＯＬＵＴＩＯＮＳＦＯＲＡＣＬＡＳＳＯＦＮＯＮＬＩＮＥＡＲＩＮＴＥＧＲＯ－ＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＳｏｎｇＧｕａｎｇｘｉｎｇ（宋光兴）（ＲｅｃｅｉｖｅｄＭａｙ４，１９９５；Ｃｏｍ...     

Initial value problem for a class of fourth-order nonlinear wave equations

In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.     

GLOBAL SOLUTION OF THE INVERSE PROBLEM FOR ACLASS OF NONLINEAR EVOLUTION EQUATIONSOF DISPERSIVE TYPE

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｐｓｅｕｄｏ＿ｐａｒａｂｏｌｉｃｅｑｕａｔｉｏｎｗｉｔｈｐｒｉｎｃｉｐａｌｐａｒｔｕｔ -ｕｘｘｔｈａｓｂｅｅｎｓｔｕｄｙｉｎｇｒｅｃｅｎｔｌｙ ,ｂｅｃａｕｓｅｔｈｅｒｅｅｘｉｓｔｓｗｉｄｅｐｈｙｓｉｃａｌｂａｃｋｇｒｏｕｎｄｆｏｒｔｈｉｓｃｌａｓｓｏｆｅｑｕａｔｉｏｎｓ.[1 ]ｓｔｕｄｉｅｄｔｈｅｍｕｌｔｉ＿ｄｉｍｅｎｓｉｏｎｉｎｖｅｒｓｅｐｒｏｂｌｅｍｆｏｒｔｈｅｆｏｌｌｏｗｉｎｇｃｌａｓｓｏｆｎｏｎｌｉｎｅａｒｅｖｏｌｕｔｉｏｎｅｑｕ…     

Global solution of the inverse problem for a class of nonlinear evolution equations of dispersive type

The inverse problem for a class of nonlinear evolution equations of dispersive type was reduced to Cauchy problem of nonlinear evolution equation in an abstract space. By means of the semigroup method and equipping equivalent norm technique, the existence and uniqueness theorem of global solution was obtained for this class of abstract evolution equations, and was applied to the inverse problem discussed here. The existence and uniqueness theorem of global solution was given for this class of nonlinear evolution equations of dispersive type. The results extend and generalize essentially the related results of the existence and uniqueness of local solution presented by YUAN Zhong-xin. Contributed by Chen Yu-shu Foundation item: the National Natural Science Foundation of China (Significance 199990510); the National Key Basic Research Special Foundation of China (G1998020316); Liuhui Center for Applied Mathematics, Nankai University & Tianjin University Biography: Chen Fang-qi (1963-)     

Instability of solution for a class of the third order nonlinear differential equation

ＩＮＳＴＡＢＩＬＩＴＹＯＦＳＯＬＵＴＩＯＮＦＯＲＡＣＬＡＳＳＯＦＴＨＥＴＨＩＲＤＯＲＤＥＲＮＯＮＬＩＮＥＡＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＬｕＤｅｙｕａｎ卢德渊（ＲｅｃｅｉｖｅｄＮｏｖ１０１９９４ＣｏｍｍｕｎｉｃａｔｅｄｂｙＺｈａｎｇＳｈｉ...     

Computer computation of the method of multiple scales—dirichlet problem for a class of system of nonlinear differential equations

The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations. The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales. Contributed by Jiang Fu-ru, Original Member of Editorial Committe, AMM Biography: Xie La-bing (1976∼); Jiang Fu-ru(1927∼)