话说齐权方程

齐次方程作为可化为可分离变量的方程,在一般高等数学教材中都有介绍.齐次方程稍加推广即得齐权方程.齐权方程的可积简化了大量一阶方程的求解过程,拓宽了方程的可积范围.定义1　设t为任意非零的量,若f(x,y)满足f(tx,ty)≡trf(x,y)则称函数f(x,y)为r次齐次函数.特别地,若令t=1x,上式变为f(1,yx)≡1xrf(x,y)或f(x,y)=xrf(1,yx)=xrφ(yx)当r=0时,f(x,y)=φ(yx)　　方程dydx=φ(yx)(1)　称为齐次方程.经变换yx=u(或xy=v)可将(1)化为可分离变量的方程积出.定义2　若存在数m,当分别以tx、tmy、tm-1y′顺次代替函数f(x,y,y′)中的x、y、y′时成立f…     

Bernoulli方程解法的启迪

众所周知 ,Bernoulli方程dydx=P( x) y +Q( x) yn( n≠ 0 ,1 ) ( 1 )是可用初等积分法求解的一类非线性方程 ,其解法是用函数变换 z=y1- n,则方程 ( 1 )就化为关于未知函数 z的一阶性方程dzdx=( 1 -n) P( x) z +( 1 -n) Q( x)上述解法启迪我们提出一般的问题 :非线性微分方程dydx=P( x) f ( y) +Q( x) g( y) ( 2 )经函数变换化的一阶线性微分方程的充要条件是什么 ?又方程 ( 2 )经函数变换化为 Bernoulli方程的充要条件是什么 ?其中 P( x) ,Q( x)和 f( y) ,g( y)都分别是 x和 y的连续函数 ,且它们都不为零。定理 1　方程 ( 2 )经未知函…     

函数方程f(x+y)=f(x)+f(y)的非连续解

讨论了函数方程f(x y)=f(x) f(y)解的性质,给出了方程的一个非连续解及其图像特点     

关于求解第二类Volterra积分方程的Picard迭代收敛性的一点注记

考虑第二类Volterra积分方程: φ(x)+integral from n=0 to x(K(x,y)φ(y)dy)=f(x),x∈[0,L],(1)其中f(x)∈C([0,L]),核函数 K(x,y)对y可积,且     

三阶线性非齐次微分方程的Hyers-Ulam稳定性

讨论了三阶线性非齐次微分方程y′′′+p(x)y″+q(x)y′+r(x)y+f(x)=0的Hyers-Ulam稳定性,即若函数f是它的一个近似解,则该方程一定存在与f是任意接近的精确解,并给出了简单实例.     

对流扩散方程的一类迎风格式

这里Ω为R~2中的有界区域,?Ω为其边界;a为正常数,c(x,y)和b(x,y)=(b_1(x,y),b_2(x,y))τ分别是?上的光滑函数和向量函数,且0相似文献   

奇异积分方程的逼近解法

对于奇异积分方程a(x)y(x)+((b(x))/π)integral from n=-1 to 1 ((y(t))/(t-x))dt+λ integral from n=-1 to 1 K(x,t)y(t)dt=f(x) -1≤x≤1本文通过对核函数K(x,t)进行二元样条插逼近,利用退化核的Fredbolm方程的基本理论,给出了奇异积分方程的逼近解,证明了其收敛性,本文给出的方法克服了用配位法和伽辽金方法须对b(x)所加的限制(b(x)为多项式),同时克服了的方法在计算过程中的不稳定性,便于实际应用。     

一类随机最优控制问题的单调控制解

讨论了一类可允许控制策略满足单调非降条件的随机最优控制问题,给出了值函数v(t,x,y,)满足一类受梯度限制的Hamilton-Jacobi-Bellman(HJB)方程:max{Lv(t,x,y), v(t,x,y)/ y}=0,其中Lv(t,x,y)= v/ t b(t,x,y,) v/ x 1/2σ2(t,x,y) 2v/ x2 f(t,x,y).借助粘性解的思想,定义了该类HJB方程的粘性解并在此意义下证明了v(t,x,y)是唯一粘性解,这类方程在随机控制,金融数学等领域内有重要应用.     

用导数三探零点问题

方程f(x)=0的根也称为函数f(x)的零点,研究方程f(x)=0的根就是研究函数y=f(x)的图象与x轴交点的横坐标.对零点问题的研究几乎汇聚了函数的所有知识点和数学思想方法,因而往往被压轴.在2011年高考冲刺复习中,如     

一道例题的补正

同济大学编《高等数学》(第四版 )上册第 1 0 3— 1 0 4页有一道例 8“求曲线 y=x32 的通过点 (5,1 1 )的切线方程。”书中的解过程为 :“解 :设节点为 (x0 ,y0 ) ,则切线的斜率为y′| x=x0 =32 x | x=x0 =32 x0于是所求切线方程可设为y -y0 =32 x0 (x -x0 ) (1 )切点 (x0 ,y0 )在曲线 y=x32 上 ,故有y0 =x032 (2 )切线 (1 )通过点 (5,1 1 ) ,故有1 1 -y0 =32 x0 (5-x0 ) (3)　　求解方程 (2 )及 (3)组成的方程组的解为 x0 =4,y0 =8,代入 (1 )式并化简 ,即得所求切线方程为3x -y -4 =0 .”　　该题是过曲线外一点求切线问题。显然 ,过曲线 y…     

Semi-implicit finite volume level set method for advective motion of interfaces in normal direction

In this paper a semi-implicit finite volume method is proposed to solve the applications with moving interfaces using the approach of level set methods. The level set advection equation with a given speed in normal direction is solved by this method. Moreover, the scheme is used for the numerical solution of eikonal equation to compute the signed distance function and for the linear advection equation to compute the so-called extension speed [1]. In both equations an extrapolation near the interface is used in our method to treat Dirichlet boundary conditions on implicitly given interfaces. No restrictive CFL stability condition is required by the semi-implicit method that is very convenient especially when using the extrapolation approach. In summary, we can apply the method for the numerical solution of level set advection equation with the initial condition given by the signed distance function and with the advection velocity in normal direction given by the extension speed. Several advantages of the proposed approach can be shown for chosen examples and application. The advected numerical level set function approximates well the property of remaining the signed distance function during whole simulation time. Sufficiently accurate numerical results can be obtained even with the time steps violating the CFL stability condition.     

One-Dimensional Symmetry and Liouville Type Results for the Fourth Order Allen-Cahn Equation in \RN

In this paper, the authors prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R~N, as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. The authors also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities.     

Split fields and direction of propagation for the solution to first-order systems of equations

The standard wave-splitting approach for the wave equation in inhomogeneous media is first reexamined. Next, by analogy with the theory of wave propagation through singular surfaces, a characterization is given for a function in space-time to represent a wave propagating in a direction. The condition is applied in connection with a simple example and found to be quite restrictive. The same problem is then considered in the Fourier-transform domain where the unknown function is an n-tuple satisfying a system of ordinary differential equations. The condition for propagation in a direction is established for the Fourier components. Next, some physical problems are considered which are expressed by partial differential equations or by integro-differential equations. The associated first-order system of equations is examined in terms of the eigenvalues of a matrix. This shows that, for any eigenvalue, the direction of propagation may change with the frequency and that arguments about the dominance of the principal part of the operator may cease to hold.     

Some Properties of Meromorphic Solutions to Systems of Complex Differential-Difference Equations

Applying Nevanlinna theory of the value distribution of meromorphic functions, the author studies some properties of Nevanlinna counting function and proximity function of meromorphic solutions to a type of systems of complex differential-difference equations. Specifically speaking, the estimates about counting function and proximity function of meromorphic solutions to systems of complex differential-difference equations can be given.     

Evaluation of a continuous wavelet transform by solving the Cauchy problem for a system of partial differential equations

It is shown that the problem of evaluating the continuous Morlet wavelet transform can be stated as the Cauchy problem for a system of two partial differential equations. The initial conditions for the desired functions, i.e., for the real and imaginary parts of the wavelet transform, are the analyzed function and a vanishing function, respectively. Numerical examples are given.     

Non-monotonic solutions and continuously differentiable solutions of conjugacy equations

In this paper solutions of conjugacy equation φ(f(x))=g(φ(x)) for a strictly decreasing continuous given function f and a continuous given function g (maybe non-monotonic) are constructed by piecewise defining. We determine the conditions for piecewise continuously differentiable solutions of conjugacy equations with a strictly decreasing continuously differentiable given function f and a continuously differentiable given function g. Finally, the recursive algorithm is implemented in MATLAB software and two examples are respectively presented for a non-monotonic solution and a continuously differentiable one.     

Note on exact solutions for a class of nonlinear diffusion equations

An effective characterization is given for a class of generalized nonlinear diffusion equations with power law dependent terms. Further, a new auxiliary equation ansatz is derived. Consequently, new exact traveling wave trigonometric function, solitary-like and Weierstrass elliptic solutions to a subclass are obtained by means of an auxiliary equation method and a generalized Riccati equation expansion method.     

On the behavior of the solutions to periodic linear delay differential and difference equations

Some new results on the behavior of the solutions to periodic linear delay differential equations as well as to periodic linear delay difference equations are given. These results are obtained by the use of two distinct roots of the corresponding (so called) characteristic equation.     

一类求行波解的线性方法

基于齐次平衡法和李志斌的 tanh函数法 ,本文得到一类简单有效的求解非线性发展方程的线性方法 .这类方法利用非线性发展方程孤立波的局部性特点 ,适当地选取函数 f 和 g,将孤波表示为 f,g的多项式 ,从而将非线性发展方程求解问题转化为非线性代数方程组的求解问题 ,再利用吴消元法求解方程组从而得到非线性发展方程的行波解     

一类迭代方程C~1类解的讨论

本文讨论一类非多项式形式的函数迭代方程，作者给出了这类方程Ｃ１类解的存在性．唯一性和稳定性条件．