Pde Definition and 743 Threads

1) Solve $\begin{aligned} {{u}_{t}}&=K{{u}_{xx}},\text{ }0<x<L,\text{ }t>0, \\ {{u}_{x}}(0,t)&=0,\text{ }{{u}_{x}}(L,t)=0,\text{ for }t>0, \\ u(x,0)&=6+\sin \frac{3\pi x}{L} \end{aligned}$ 2) Transform the problem so that the boundary conditions get homogeneous: $\begin{aligned}...

I am trying to solve the following equation in spherical coordinates: \left( \nabla f \right) \cdot \vec{B} = g where g is a known scalar function, \vec{B} is a known vector field, and f is the unknown function. I think the best way to approach this is to expand everything into a...

Homework Statement Is the operator Lu = du/dx + u * du/dy linear? Homework Equations Linearity occurs for L[u+cv] = L[u] + cL[v] The Attempt at a Solution I know this isn't linear because of the second term, but I don't understand why I can't write the operator as L =...

Im new on the forum, so I hope you guys will have some patience with me :-) I have a question about the chain rule and partial differential equations that I can't solve, it's: Write the appropriate version of the chain rule for the derivative: ∂z/∂u if z=g(x,y), where y=f(x) and...

Here I have my PDE: http://desmond.imageshack.us/Himg718/scaled.php?server=718&filename=pde.png&res=medium I have found the solution by using the method of characteristics two times, one for x<0 and the other for x>0. I have: U(x,y) = o for x<0 and U(x,y) = Uo(x-1)/(1+Uo*y) for x>0...

Solve $u_y+f(u)u_x=0,$ $x\in\mathbb R,$ $y>0,$ $u(x,0)=\phi(x).$ What's the easy way to solve this? Fourier Transform? Laplace Transform?

Consider $u_t+u_x=g(x),\,x\in\mathbb R,\,t>0$ and $u(x,0)=f(x).$ Given $f,g\in C^1,$ then show that $u(x,t)$ has the form $u(x,t)=f(x-t)+\sqrt{2\pi}(g*h)(x)$ where $h(x)=\chi_{[0,t]}(x).$ So we just apply the Fourier transform to get $\dfrac{{\partial U}}{{\partial t}} + iwU = U(g)$ and...

Hey, before you read this over I'll mention that I've checked the general solution and it works. So if you don't feel like following my steps to get the general solution just jump down to the end of my attempt, because the real problem for me is figuring out what to do with the side conditions...

For quick reference if you have the text, the question is from "Applied Partial Differential Equations" by J. David Logan. Section 1.9 #4 Show that the equation $$u_{tt} - c^2 u_{xx} + au_t + bu_x + du = f(x,t)$$ can be transformed into an equation of the form $$w_{\xi\tau} + kw = g(\xi,\tau)...

Hi all, I am looking for ways to solve the following system of equations for \vec{B}: \vec{B} \cdot \nabla f = 0 \left( \nabla \times \vec{B} \right) \cdot \nabla f = 0 \nabla \cdot \vec{B} = 0 and f is a known scalar function. I think we can assume there is a solution since we...

(∂^2)(z) / (∂x)(∂y) = (x^2)(y) how do find the general solution of this equation? and how do i find a particular solution for which z(x,0) = x^2 z(1,y) = cosy I have no idea!

Hello, I have 2 questions: First, where can I find the best graduate schools for PDE & Probability? I'm thinking MIT and NYU are really good, although I could be wrong. Second, I was wondering if my expected math background is good enough to get into the 90th percentile on the GRE...

Given \triangle u = f(x,y,z) on a given body with vanishing neumann boundary conditions. I'm asked to interpret it in terms of heat and diffusion. Since heat/diffusion take the form u_t = k \triangle u, I am a little confused as I there is no time term here. I think the answer is that u...

Homework Statement Use the fundamental solution or Green function for the diffusion/heat equation in (-\infty, \infty ) to determine the fundamental solution to \frac{\partial u }{ \partial t } =k^2 \frac{\partial ^2 u }{ \partial x ^2 } in the semi-line (0, \infty ) with initial condition...

Hello all, I will be enrolled in an undergraduate course on partial differential equations. I was hoping a few of you might be able to recommend, in your opinions, the best textbook for the subject at an undergraduate level since I'd like to have a second source outside of my instructor's...

Solution of the good PDE ? Find the solution of u of the equation u_x + x u_y = u + x if u(x,0)=x^2,x>0.

Homework Statement I am learning about PDE classification from a text on CFD (by Anderson). This section is not complete enough to be able to extend his example problems into more general cases. I read that to classify a system of PDEs as being parabolic, elliptic, or hyperbolic, I need to do...

i have been trying to solve a pde problem for 3 days but i couldn't even find the answer,now i feel i m about to have a mental disease,anyone can help me ?the question is u(x,0) = x u(x,2) = 0 u(0,y) = 0 d u(1,y) / dx = 0 [ d^2 u / dx^2 ] + [ d^2 u / dy^2 ] = 0 i will really be...

Homework Statement From a previous exercise (https://www.physicsforums.com/showthread.php?t=564520), I obtained u(r,\phi) = \frac{1}{2}A_{0} + \sum_{k = 1}^{\infty} r^{k}(A_{k}cos(k\phi) + B_{k}sin(k\phi)) which is the general form of the solution to Laplace equation in a disk of radius a. I...

Suppose you start with a function f(x,y,t) which satisfies some partial differential equation in the variables x,y,t. Suppose you make a change of variables x,y,t \to \xi,z,\tau, where \tau = g_\tau(x,y,t) and similarly for \xi and z. If you want to know what the differential operators...

Dear All, I got some trobule in solving the following simple-looking PDE's. Can anyone give a hint about how to solve it? thanks a lot! I guess the solution is of the form y(u,v)=A[\cos(k(u-f(v))-B]\cosh(v)-C. But I don't know a formal way to solve. \frac{\partial^4y(u,v)}{\partial u^2 \partial...

what is the general solution of the poisson equation : ∂2A/∂r2 + 1/r ∂A/∂r + 1/r2 ∂2A/∂θ2 = f(r,θ) the function f(r,θ) is : f(r,θ)=1/r (Ʃ Xncos(nθ) + Ynsin(nθ)) where the boundary is : I(a<r<b, 0<θ<2pi) the boundary condition is the netural boundary on (r=a) expressed as ...

This may be more of a MATLAB question, and if so, I do apologize for posting this in the wrong place. I am doing a project on the Buttke scheme, which is a numerical approximation to the Biot-Savart Law. I am almost finished, but I am having trouble writing the code. The scheme is...

Hello to everyone. I am new with this forum and I am asking help with PDE. I have a linear PDE: L f(x,y,t) = 0 where L is a second order linear operator depending on x, y, their partial derivatives, and t, but not on derivatives with respect to t. The question...

Hello I want to resolve a nonlinear partial differential equation of second order with finite difference method in matlab. the equation is in the pdf file attached. Thanks

Hello, I have a problem in the form \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}+e^{x}u=0 with conditions u(x,0)=u_0(x) u(0,t)=\int_{0}^{\infty}f(x)u(x,t)dx Im confused, because in first order PDE i require only 1 condition. How to solve this for two conditions?

Hi, I'm new here and I'm really hoping that someone could answer my question. i am trying to design a PDE but the problem about this is that the temperature caused by detonation (3461 K) is so high that it is really difficult to find a suitable material for the detonation chamber and thermal...

I have a pde set as following: parameters: γ, ω, α, β, c, η variables: z,t; x,y want: S = S(z,t;x,y) A = A(z,t) ∂S/∂t = -γ*S - i ω*A*exp{-i*[(-θ-α*t)*x+β*t*y]} [∂/∂t + (1/c)*∂/∂t] A = -i η*∫∫dxdy S*exp{i*[(-θ-α*t)*x+β*t*y]} The integral range is angle:(0,2Pi), radius: (0,R) How to...

Hi, I have a problem solving this PDE: (y^2)*u(x,y)''+2*y*u(x,y)'-2*u(x,y) = 0 Every derivate of u is in fonction of y. What I tried: I said that (y^2)*u(x,y)''+2*y*u(x,y)' = (u(x,y)'*y^2)' and make v=u(x,y)'*y^2 then I tried to isolate u(x,y) and I arrive to u(x,y)=-v/y+C(X)...

Homework Statement u_{t} = ku_{xx} u_{x}(0, t) = 0 u_{x}(L, t) = B =/= 0 u(x, 0) = f(x) Homework Equations The Attempt at a Solution I believe that no equilibrium solution exists because we can't solve u_{xx} = 0 with our boundary conditions. I'm a little lost as to where...

I've always heard the phrase "exact solution," but was never really sure what it meant. If I find a particular solution (not a general solution) to a PDE, is that solution considered an "exact solution"? (The solution satisfies given b.c. and i.c.)

Hi! I'm implementing a scheme to solve the following equation \frac{\partial \psi}{\partial t}=-c_{s} \cdot \frac{\partial \phi}{\partial x} \frac{\partial \phi}{\partial t}=-c_{s} \cdot \frac{\partial \psi}{\partial x} c_{s} is just the isothermal velocity of sound. The equations are for a...

Ok, so I got this equation: y^2 \frac{∂^2 u}{∂x^2} + 2xy \frac{∂^2 u}{∂x∂y} + x^2 \frac{∂^2 u}{∂y^2} = 0 A = y^2 B = xy C = x^2 Now I want to see what type it is, so I compute B^2 - A C = 0 which by definition is parabolic. However, according to an earlier statement in my book a...

i have this 2 equation to be solved, can someone help me with it please? 1- let m= m(x;y;z) solve2((d^3/dx^3)+3(d/dx)(d^2/dy^2)-(d/dx)(d/dy))*m(x;y;z)=0 2-let m=m(x;y;z) solve (ydx+ydz)*m=m

PDE Harmonic Function help! Homework Statement A bounded harmonic function u(x,y) in a semi-infinite strip x>0, 0<y<1 is to satisfy the boudary conditions: u(x,0)=0, uy(x,1)=-hu(x,1), u(0,y)=u0, Where h (h>0) and u0 are constants. Derive the expression...

Suppose you have a PDE with an arbitrary number of independent variables (not necessarily two), and of order n. Is there a nice classification akin to the hyperbolic, parabolic, etc. Thanks

I need to solve three coupled differential equations. The equations are as follows: dE1/dz = f(E2,E3) dE2/dz = f(E1,E3) dE3/dz = f(E1,E2) Where E1,E2,E3 represents field amplitudes. In case of plane waves these amplitudes will be constant in the transverse direction therefore i can...

Hello all: I'm a newbie, trying to write/use code for solving a 2D advection-diffusion problem. I'm not sure how many boundary conditions I should have for the property that is being transported. In my problem, I have diffusion switched off (advection only). The property being...

Hi, I am struggling with the heat equation ut = kuxx with the boundary conditions u(0,t) = u'(L,t) = 0 and initial condition u(x,0) = f(x) 0 ≤ x ≤ L 0 ≤ t I want to derive it's eigenvalue using complex analysis. After separating the variables...

What method can I use to analytically solve the following 2nd order PDE? u=u(x,t) ∂u/∂t - a*x*∂u/∂x-D*∂^{2}u/∂t^{2} = 0 I.C.: u(x,t=0)=u_i B.C.: u(x=+∞)=0 u(x=-∞)=1 Is self-similar the only way to solve it, or is there any other method can be used to solve it? How to set the...

In an open domain in the plane (xU_{x}-yU_{y}-U)/U = (xV_{x} - y V_{y}+V)/V

Homework Statement Using the method of separation of variables, obtain a solution of the following PDE subject to the given conditions \frac{du}{dx}+y\frac{du}{dy}=(2x+y)u u(x,1)=5e^{x^{2}-x} Homework Equations The Attempt at a Solution See my attached working the...

1. Show that, if the velocity field (V) is a fixed (spatially constant) vector, then the characteristic curves will be a family of parallel-straight lines. 2. ut+V1ux+V2uy=f f=S-[dell dotted with V]u characteristic curves: dX/dt=V1(X,Y) & dY/dt=V2(X,Y) 3. really looking for...

Homework Statement I have found the general solution to a second order pde to be U(x,t) = f(3x + t) + g(-x + t) where f and g are arbitrary functions I have initial conditions U(x,0) = sin(x) Du/dt (x,0) = cos (2x) The Attempt at a Solution I have found that U(x,0) = f(3x) +...

Homework Statement I have a pde, 16d2u/dxdy + du/dx + du/dy + au = 0 where a is constant. Homework Equations The Attempt at a Solution I have tried to solve this pde using the substitutions x=e^t and y=e^s so t=ln(x) and s=ln(y) then finding Du/dx= 1/x du/dt and du/dy= 1/y...

Homework Statement Suppose that the Celsius temperature at the point (x, y) in the xy plane is T(x,y) = xsin(2y) and that the distance in the xy plane is measured in meters. A particle moving clockwise around the circle of radius 1m centered at the origin at the constant rate of 2 m/s a...

Homework Statement I have a PDE for which i have found the general solution to be u(x,y) = f1(3x + t) + f2(-x + t) where f1 and f2 are arbitrary functions. I have initial conditions u(x,0) = sin (x) and partial derivative du/dt (x,0) = cos (2x)Homework Equations u(x,y) = f1(3x + t) + f2(-x +...

Is the following relationship true?: \frac{\partial (ln(k))}{\partial P}=\frac{1}{k}\frac{\partial k}{\partial P} I am getting both of these terms from a paper on mineral physics and they seem to use both terms interchangeably. If so, how are these related?

Simple PDE... I'm trying to solve the PDE: \frac{\partial^2 f(x,t)}{\partial x^2}=\frac{\partial f(x,t)}{\partial t} with x \in [-1,1] and boundary conditions f(1,t)=f(-1,t)=0. Thought that e^{i(kx-\omega t)} would work, but that obviously does not fit with the boundary conditions. Has...

Hello, It has taken me a long time to try and figure out what a system of coupled PDEs actually IS-and I still can't get a straight answer. For example I have a system: \dot{M}=-LvM \dot{N}=-Lv+wN where here ,L, represents the lie derivative and M, N , v, w, are all elements of...