[數] 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.
在物理學中，物理參量隨時間的演化往往是由非線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.
倒向隨機微分方程在隨機控制、偏微分方程、數理金融、經濟等領域都有着廣泛的應用。
The partial differential equation of flow in porous medium has been stu***d with variational finite element method.
用變分有限元方法對描述地下流體滲流的偏微分方程進行了分析。
According to the assumption of slightly compressible fluid, neglected the quadratic gra***nt term in nonlinear partial differential equation is usually lead to error.
在非線性偏微分方程中，根據弱可壓縮液體的假設，忽略二次梯度項，在試井較長時間内将産生誤差。
Axisymmetric elasticity problem of cubic quasicrystal is reduced to a solution of a partial differential equation with higher-order by introducing displacement function.
本文通過引入位移函數使得立方準晶的軸對稱彈性問題化為求解一高階偏微分方程。
And a partial differential equation is some relation between its partial derivatives. Let me see.
而一個偏微分方程就是，函數各個偏導之間的聯繫，看看。
Optimal control problem is considered as soon as we can solve a partial differential equation to simulate certain practical problem.
最優控制問題是在能夠求解偏微分方程進行模拟以後必然要考慮的問題。
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.
本文用多層前向神經網絡求解該非線性偏微分方程，從而逼近非線性系統的中心流形。
Approximately resolving the coupled coagulation partial differential equation with matched asymptotic expansions mathematic theory is discussed in the paper.
研究以匹配漸進展開法來處理耦合碰并的偏微分方程的近似求解問題。
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進行近似離散，形成結點的地下水流代數方程組，并插入優化模型中作為等式約束條件。
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.
本文具體讨論了注水過程中由于固體顆粒侵入造成地層傷害的非線性抛物型偏微分方程反問題模型。
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.
将一個場變量取作為其餘場變量和時間的函數并對這個函數建立基本偏微方程。
A new Weighted Residual Method for solving Partial differential equation is presented in this paper.
本文提出一種求解偏微分方程的加權殘數法。
Partial differential equation(PDE) has become an important tool in image processing and analysis.
偏微分方程（PDE）已成為圖像處理與分析中的重要工具。
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.
運用行波約化法将非線性波動方程化為常微分方程，然後利用位移形函數的系數待定法求出了非線性波動方程的位移孤波解。
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，在這個偏微分方程中，我們還不清楚的是，一個我們稱之為u的函數,這個函數和某些物質的濃度相關。
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（即定價模型）的數值解。
The blow-up theory of the solutions for nonlinear evolution equations is an important aspect of partial differential equation.
非線性發展方程解的爆破理論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）對ESPI圖像進行處理通常能取得比較好的結果。
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.
對氣液兩相非定常一元流動的主控制方程找出了兩套比較完整的可用的代數顯式解析解。
The grid generation technique by means of solving elliptic partial differential equation is used to generate structured grid.
本文選用求解橢圓型微分方程的方法生成結構網格；
Relying on mathematical theories and methods, this paper investigates a new method of SAR images super-resolution processing based on the partial differential equation.
本文以數學理論和方法為依托，利用圖像的先驗信息，研究了一種基于偏微分方程方法的多圖超分辨方法。
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.
在借貸利率不同情形下 ，首先得到隨機壽命的未定權益價格應滿足的偏微分方程 ；
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.
為此在擴散方程的基礎上提出了一種改進的水平集重構算法，通過改進的水平集重構實現邊緣沿切線方向的平滑；
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.
隨機波動率的存在使得由複制組合方法所得的定價方程變得更加複雜，并且沒有封閉的解析解，隻能采用近似算法。
Partial differential diffusion equation is used to remove the noise of medical ultrasonic image.
研究了偏微分擴散方程在醫學超聲圖像去噪中的應用。
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.
它是通過在膛壁邊界層建立非穩質量守恒和動量守恒方程的積分式，而獲得有關動量厚度的偏微分方程；
偏微分方程（Partial Differential Equation，簡稱PDE）是描述多變量函數及其偏導數之間關系的數學方程，廣泛應用于物理、工程、經濟等領域的連續系統建模。其一般形式可表示為： $$ Fleft(x_1, ldots, x_n, u, frac{partial u}{partial x_1}, ldots, frac{partial u}{partial x_n}, frac{partial u}{partial x_1 partial x_2}, ldots right) = 0 $$ 其中$u=u(x_1,x_2,ldots,x_n)$是未知函數，$x_i$為自變量。
多變量依賴性
PDE區别于常微分方程的關鍵在于涉及多個獨立變量。例如，熱傳導方程$frac{partial u}{partial t} = alpha abla u$同時包含時間$t$和空間坐标$(x,y,z)$變量，可描述三維空間中溫度隨時間的擴散過程。
基礎分類
根據數學特性分為三類：
"Partial Differential Equation"（偏微分方程，簡稱PDE）是數學中用于描述多變量函數與其偏導數之間關系的方程。以下是詳細解釋：
PDE是包含未知函數及其對多個獨立變量的偏導數的方程。例如，若函數 ( u(x,y) ) 是未知函數，則方程中可能包含 (frac{partial u}{partial x})、(frac{partial u}{partial y}) 等偏導數。
PDE的标準形式可表示為： $$ Fleft(x_1, x_2, ldots, u, frac{partial u}{partial x_1}, frac{partial u}{partial x_2}, ldots, frac{partial u}{partial x_1}, ldotsright) = 0 $$ 其中 ( u ) 是未知函數，( x_1, x_2, ldots ) 是獨立變量。
PDE廣泛應用于物理學（如流體力學、電磁學）、工程學（結構分析）、金融數學（期權定價的Black-Scholes方程）和生物學（種群擴散模型）等領域。
PDE是建模連續系統動态的核心工具，其解通常需要結合邊界條件和初始條件。研究PDE的解的存在性、唯一性和穩定性是數學分析的重要課題。
如果需要進一步了解具體解法（如分離變量法、有限元分析）或某類PDE的案例，可以提出更具體的問題。
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