[数] 偏微分方程
In physics, the physical parameter evolution along with the time is often controlled by the non-linear partial differential equation.
在物理学中，物理参量随时间的演化往往是由非线性的偏微分方程支配的。
From then on, BSDE is further stu***d and applied widely in stochastic control, partial differential equation (PDE), mathematical finance and economics.
倒向随机微分方程在随机控制、偏微分方程、数理金融、经济等领域都有着广泛的应用。
The partial differential equation of flow in porous medium has been stu***d with variational finite element method.
用变分有限元方法对描述地下流体渗流的偏微分方程进行了分析。
According to the assumption of slightly compressible fluid, neglected the quadratic gra***nt term in nonlinear partial differential equation is usually lead to error.
在非线性偏微分方程中，根据弱可压缩液体的假设，忽略二次梯度项，在试井较长时间内将产生误差。
Axisymmetric elasticity problem of cubic quasicrystal is reduced to a solution of a partial differential equation with higher-order by introducing displacement function.
本文通过引入位移函数使得立方准晶的轴对称弹性问题化为求解一高阶偏微分方程。
And a partial differential equation is some relation between its partial derivatives. Let me see.
而一个偏微分方程就是，函数各个偏导之间的联系，看看。
Optimal control problem is considered as soon as we can solve a partial differential equation to simulate certain practical problem.
最优控制问题是在能够求解偏微分方程进行模拟以后必然要考虑的问题。
In this paper, multi-layer feedforward neural networks are used to solve the nonlinear partial differential equation, and approach the centre manifold of the nonlinear system.
本文用多层前向神经网络求解该非线性偏微分方程，从而逼近非线性系统的中心流形。
Approximately resolving the coupled coagulation partial differential equation with matched asymptotic expansions mathematic theory is discussed in the paper.
研究以匹配渐进展开法来处理耦合碰并的偏微分方程的近似求解问题。
A ground-water management model was developed by using finite-difference and finite element approximations to the partial differential equation governing groundwater flow in a confined aquifer.
地下水偏微分方程用有限差分和有限元法进行近似离散，形成结点的地下水流代数方程组，并插入优化模型中作为等式约束条件。
In this paper, an inverse problem of a nonlinear parabolic partial differential equation of formation damage caused by the solids invading during the course of injection is presented.
本文具体讨论了注水过程中由于固体颗粒侵入造成地层伤害的非线性抛物型偏微分方程反问题模型。
One of the field variables is considered as a function of other variables and time, and the fundamental partial differential equation for the function is established.
将一个场变量取作为其余场变量和时间的函数并对这个函数建立基本偏微方程。
A new Weighted Residual Method for solving Partial differential equation is presented in this paper.
本文提出一种求解偏微分方程的加权残数法。
Partial differential equation(PDE) has become an important tool in image processing and analysis.
偏微分方程（PDE）已成为图像处理与分析中的重要工具。
By using travelling wave reduced form method, the nonlinear partial differential equation of wave propagation in a functionally graded beam was transformed into an ordinary differential equation.
运用行波约化法将非线性波动方程化为常微分方程，然后利用位移形函数的系数待定法求出了非线性波动方程的位移孤波解。
I mean the unknown, in this partial differential equation, is a function that we call u that corresponds to the concentration of some substance.
我的意思是，在这个偏微分方程中，我们还不清楚的是，一个我们称之为u的函数,这个函数和某些物质的浓度相关。
In solving partial differential equation(i. e. pricing model), the author adopted the finite difference method to get its numerical solution.
本文采用有限差分法求解偏微分方程（即定价模型）的数值解。
The blow-up theory of the solutions for nonlinear evolution equations is an important aspect of partial differential equation.
非线性发展方程解的爆破理论是偏微分方程的重要内容。
To the ESPI fringe patterns and phase patterns, always we could get better effects with partial differential equation (PDE) filter methods.
利用偏微分方程（PDE）对ESPI图像进行处理通常能取得比较好的结果。
Two sets of algebraically explicit analytical solution are derived for the partial differential equation set describing the one-dimensional unsteady gas-liquid two-phase flow.
对气液两相非定常一元流动的主控制方程找出了两套比较完整的可用的代数显式解析解。
The grid generation technique by means of solving elliptic partial differential equation is used to generate structured grid.
本文选用求解椭圆型微分方程的方法生成结构网格；
Relying on mathematical theories and methods, this paper investigates a new method of SAR images super-resolution processing based on the partial differential equation.
本文以数学理论和方法为依托，利用图像的先验信息，研究了一种基于偏微分方程方法的多图超分辨方法。
In this paper we first obtain a partial differential equation described by price of contingent claim when borrowing rate is not equal to lending rate.
在借贷利率不同情形下 ，首先得到随机寿命的未定权益价格应满足的偏微分方程 ；
We present a new subpixel edge location algorithm based on partial differential equation(PDE), where level-set reconstruction(LSR) is introduced to smooth edge along their tangents.
为此在扩散方程的基础上提出了一种改进的水平集重构算法，通过改进的水平集重构实现边缘沿切线方向的平滑；
At the same time we can make sure that volatility is positive. Now, it will make the partial differential equation of pricing become more complex and hasn't any closed-form solution.
随机波动率的存在使得由复制组合方法所得的定价方程变得更加复杂，并且没有封闭的解析解，只能采用近似算法。
Partial differential diffusion equation is used to remove the noise of medical ultrasonic image.
研究了偏微分扩散方程在医学超声图像去噪中的应用。
A partial differential equation on momentum thickness is derived from unsteady mass and momentum conservation equations in integrating form on boundary layer of the tube.
它是通过在膛壁边界层建立非稳质量守恒和动量守恒方程的积分式，而获得有关动量厚度的偏微分方程；
偏微分方程（Partial Differential Equation，简称PDE）是描述多变量函数及其偏导数之间关系的数学方程，广泛应用于物理、工程、经济等领域的连续系统建模。其一般形式可表示为： $$ Fleft(x_1, ldots, x_n, u, frac{partial u}{partial x_1}, ldots, frac{partial u}{partial x_n}, frac{partial u}{partial x_1 partial x_2}, ldots right) = 0 $$ 其中$u=u(x_1,x_2,ldots,x_n)$是未知函数，$x_i$为自变量。
多变量依赖性
PDE区别于常微分方程的关键在于涉及多个独立变量。例如，热传导方程$frac{partial u}{partial t} = alpha abla u$同时包含时间$t$和空间坐标$(x,y,z)$变量，可描述三维空间中温度随时间的扩散过程。
基础分类
根据数学特性分为三类：
"Partial Differential Equation"（偏微分方程，简称PDE）是数学中用于描述多变量函数与其偏导数之间关系的方程。以下是详细解释：
PDE是包含未知函数及其对多个独立变量的偏导数的方程。例如，若函数 ( u(x,y) ) 是未知函数，则方程中可能包含 (frac{partial u}{partial x})、(frac{partial u}{partial y}) 等偏导数。
PDE的标准形式可表示为： $$ Fleft(x_1, x_2, ldots, u, frac{partial u}{partial x_1}, frac{partial u}{partial x_2}, ldots, frac{partial u}{partial x_1}, ldotsright) = 0 $$ 其中 ( u ) 是未知函数，( x_1, x_2, ldots ) 是独立变量。
PDE广泛应用于物理学（如流体力学、电磁学）、工程学（结构分析）、金融数学（期权定价的Black-Scholes方程）和生物学（种群扩散模型）等领域。
PDE是建模连续系统动态的核心工具，其解通常需要结合边界条件和初始条件。研究PDE的解的存在性、唯一性和稳定性是数学分析的重要课题。
如果需要进一步了解具体解法（如分离变量法、有限元分析）或某类PDE的案例，可以提出更具体的问题。
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