📘 partial differential equations

1. The heat equation is a partial differential equation that describes how heat diffuses through a given region over time. 2. The standard form of the heat equation in one dimensio

1. **State the problem:** Simplify or analyze the expression $uxx + uxxyy + uyy$ where subscripts denote partial derivatives. 2. **Identify notation:** Here, $uxx$ means $\frac{\pa

1. **State the problem:** Solve the partial differential equation (PDE) given by $$x \frac{\partial u}{\partial x} = y \frac{\partial u}{\partial y}$$ where $u = u(x,y)$. 2. **Assu

1. **Problem Statement:** Solve the partial differential equation (PDE) given by $$n \frac{\partial u}{\partial x} = y \frac{\partial u}{\partial y}$$

1. **State the problem:** Solve the partial differential equation (PDE) given by $$n \frac{\partial u}{\partial m} = y \frac{\partial u}{\partial y}$$

1. **Énoncé du problème** : Nous avons l'équation aux dérivées partielles (EDP) suivante :

1. **Problem statement:** Solve the heat equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$$ with boundary conditions $$u(x,0) = 3 \sin n\pi x,$$ $$u(0,t

1. **Stating the problem:** Solve the partial differential equation $$\frac{\partial^2 z}{\partial x^2} + z = 0$$ with initial conditions at $$x=0$$: $$z = e^y$$ and $$\frac{\parti

1. **بیان مسئله:** حل معادله دیفرانسیل با مشتقات جزئی لاپلاس در مختصات استوانه‌ای برای تابع دما $T(r,\theta,z)$ که معادله آن به صورت زیر است:

1. Let's start by stating the problem: solving partial differential equations (PDEs) which can be nonlinear, semilinear, or quasilinear. 2. A PDE involves unknown multivariable fun

1. **Problem Statement:** Define a PDE and give an example of a linear PDE of order 2. 2. **Definition:** A Partial Differential Equation (PDE) is an equation involving partial der

1. The problem asks about the stability of travelling wave solutions of the radially symmetric Stefan problem in 3 dimensions. 2. The Stefan problem models phase changes, such as m

1. The problem asks about the asymptotic stability of the free boundary in the radial Stefan problem, which is a classical moving boundary problem describing phase changes like mel

1. Muammo: Berilgan tenglama $$\partial^{\beta}_{t}u +(-1)^n\frac{\partial^n}{\partial x^n} \left( x^\alpha \frac{\partial^n u}{\partial x^n} \right) = f(x,t),$$ \n\nbu yerda $\bet

1. **Problem statement:** Find the differential equations of the space curves formed by the intersection of the two families of surfaces: $$u = x^2 + y^2 + z^2 = c_1$$

1. **Problem statement:** Find the differential equations of the space curves formed by the intersection of the two families of surfaces given by $$u = x^2 + y^2 + z^2 = c_1$$

1. **Problem Statement:** (a) Explain why polar coordinates are introduced in solving partial differential equations (PDEs).

1. The problem is to understand the expression for the function $$u(x,y,t) = \sum_{m=1}^\infty \sum_{n=1}^\infty A_{mn} \cos\left(c\pi t \left(\left(\frac{m}{a}\right)^2 + \left(\f

1. **Problem Statement:** Find the deflection $u(x,y,t)$ of a rectangular membrane with sides $a$ and $b$, wave speed $c=1$, initial displacement $f(x,y) = \sin\left(\frac{6\pi x}{

1. **Problem Statement:** (a) Write down the governing equation for the deflection of a rectangular membrane.

1. **Problem Statement:** We need to solve the partial differential equation (PDE) for $v(r, \theta, \varphi, t)$ given by the heat/diffusion equation in spherical coordinates: