Read about partial differential equations | 46 Discussions | Page 1

  1. P

    Heat Diffusion Equation - Using BCs to model as an orthonormal system

    I've tried to show b) by using the sine fourier series on ##[0,2a]##, to get ##g_k = \Sigma_{n=0}^{2a} \sqrt\frac{2}{a} Sin(q_k x)## Therefore ##\sqrt\frac{2}{a} = \frac{1}{a} \int_0^{2a} Sin(q_kx)g_k dx## These are equal therefore it is an orthonomal basis. I'm not sure if this is correct so...
  2. S

    Heat Equation with Periodic Boundary Conditions

    I'm solving the heat equation on a ring of radius ##R##. The ring is parameterised by ##s##, the arc-length from the 3 o'clock position. Using separation of variables I have found the general solution to be: $$U(s,t) = S(s)T(t) = (A\cos(\lambda s)+B\sin(\lambda s))*\exp(-\lambda^2 kt)$$...
  3. currently

    Partial Derivative of a formula based on the height of a cylinder

    The function should use (r,z,t) variables The domain is (0,H) Since U is not dependent on angle, then theta can be ignored in the expression for Laplacian in cylindrical coordinates(?)
  4. E

    A How to get a converging solution for a second order PDE?

    I have been struggling with a problem for a long time. I need to solve the second order partial differential equation $$\frac{1}{G_{zx}}\frac{\partial ^2\phi (x,y)}{\partial^2 y}+\frac{1}{G_{zy}}\frac{\partial ^2\phi (x,y)}{\partial^2 x}=-2 \theta$$ where ##G_{zy}##, ##G_{zx}##, ##\theta##...
  5. Safder Aree

    How to apply the Fourier transform to this problem?

    I am struggling to figure out how to approach this problem. I've only solved a homogenous heat equation $$u_t = u_{xx}$$ using a fourier transform, where I can take the fourier transform of both sides then solve the general solution in fourier terms then inverse transform. However, since this...
  6. T

    A Determine PDE Boundary Condition via Analytical solution

    I am trying to determine an outer boundary condition for the following PDE at ##r=r_m##: $$ \frac{\sigma_I}{r} \frac{\partial}{\partial r} \left(r \frac{\partial z(r,t)}{\partial r} \right)=\rho_D gz(r,t)-p(r,t)-4 \mu_T \frac{\partial^2z(r,t)}{\partial r^2} \frac{\partial z(r,t)}{\partial t} $$...
  7. H

    A Partial Differential Equation Mathematical Modelling

    Salutations, I have been trying to approach a modelling case about organism propagation which reproducing with velocity $$\alpha$$ spreading randomly according these equations: $$\frac{du(x,t)}{dt}=k\frac{d^2u}{dx^2} +\alpha u(x,t)\\\ \\ u(x,0)=\delta(x)\\\ \lim\limits_{x \to \pm\infty}...
  8. Rupul Chandna

    I Why is separation constant l(l+1) instead of +-l^2?

    While separating variables in the Schrodinger Equation for hydrogen atom, why are we taking separation constant to be l(l+1) instead of just l^2 or -l^2, is it just to make the angular equation in the form of Associated Legendre Equation or is there a deeper meaning to it?
  9. M

    MATLAB Chromatography PDE MOL

    Hello all I am using the method of lines to solve the following PDE: ## \frac {\partial C} {\partial t} + F\frac {\partial q} {\partial t} + u \frac {dC} {dz} = D_{ax} \frac{\partial^2 C} {\partial z^2} ## ## \frac {\partial q} {\partial t} = k (q^{*}-q) ## With these initial conditions: ##...
  10. Peter Alexander

    Solving Second Order Partial Derivative By Changing Variable

    1. The problem statement, all variables, and given/known data Given is a second order partial differential equation $$u_{xx} + 2u_{xy} + u_{yy}=0$$ which should be solved with change of variables, namely ##t = x## and ##z = x-y##. Homework Equations Chain rule $$\frac{dz}{dx} = \frac{dz}{dy}...
  11. Peter Alexander

    Solving Partial Differential Equation

    1. The problem statement, all variables, and given/known data Task requires you to solve a partial differential equation $$u_{xy}=2yu_x$$ for ##u(x,y)##. A hint is given that a partial differential equation can be solved in terms of ordinary differential equations. According to the solution...
  12. A

    A How to simplify the solution of the following linear homogeneous ODE?

    During solution of a PDE I came across following ODE ## \frac{d \bar h}{dt} + \frac{K}{S_s} \alpha^2 \bar h = -\frac{K}{S_s} \alpha H h_b(t) ## I have to solve this ODE which I have done using integrating factor using following steps taking integrating factor I=\exp^{\int \frac{1}{D} \alpha^2...
  13. B

    A Difficult partial differential Problem

    Problem: $${\frac {\partial }{\partial t}}A\left( y,t \right) +6\,\Lambda\,\Omega\, \left( {y}^{2}-y \right) \sin \left( t \right) ={\frac {\partial ^{2}}{\partial {y}^{2}}}A \left( y,t \right)$$ $${\frac{\partial }{\partial y}}A \left( t,0 \right) ={\frac {\partial }{\partial y}}A \left( t,1...
  14. J6204

    Calculating the Fourier integral representation of f(x)

    Homework Statement Considering the function $$f(x) = e^{-x}, x>0$$ and $$f(-x) = f(x)$$. I am trying to find the Fourier integral representation of f(x). Homework Equations $$f(x) = \int_0^\infty \left( A(\alpha)\cos\alpha x +B(\alpha) \sin\alpha x\right) d\alpha$$ $$A(\alpha) =...
  15. J6204

    Extending function to determine Fourier series

    In the following question I need to find the Fourier cosine series of the triangular wave formed by extending the function f(x) as a periodic function of period 2 $$f(x) = \begin{cases} 1+x,& -1\leq x \leq 0\\ 1-x, & 0\leq x \leq 1\\\end{cases}$$ I just have a few questions then I will be able...
  16. S

    A How to find the partial derivatives of a composite function

    Hello, dear colleague. Now I'm dealing with issues of modeling processes of heat and mass transfer in frozen and thawed soils. I am solving this problems numerically using the finite volume method (do not confuse this method with the finite element method). I found your article: "Numerical...
  17. grquanti

    I Substitution in partial differential equation

    Hello everybody. Consider $$\frac{\partial}{\partial t}f(x) + ax\frac{\partial }{\partial x}f(x) = b x^2\frac{\partial^2}{\partial x^2}f(x)$$ This is the equation (19) of...
  18. R

    A Convert to frequency domain

    Hello All, I would like to convert a partial diff equation in time domain into frequency domain, however there is a term of the form: Re(∇(E1.E2*) exp(j[ω][/0]t)) where E1 and E2 are the magnitudes of the electric field and [ω][/0] is the angular frequency. Can someone please help me to...
  19. FranciscoSili

    I Simple PDE

    Hello everybody. I'm about to take a final exam and i've just encountered with this exercise. I know it's simple, but i tried solving it by Separation of variables, but i couldn't reach the result Mathematica gave me. This is the equation: ∂u/∂x = ∂u/∂t Plus i have a condition...
  20. Conservation

    Inverse Fourier Transform

    Homework Statement Solve ut+3ux=0, where -infinity < x < infinity, t>0, and u(x,0)=f(x). Homework Equations Fourier Transform where (U=fourier transform of u) Convolution Theorem The Attempt at a Solution I've used Fourier transform to get that Ut-3iwU=0 and that U=F(w)e3iwt. However, I'm...
  21. K

    Courses Maths for Experimental Physics?

    Hi all, I'm currently a second year Applied Maths and Physics student. I will be specialising in Experimental physics next year. With this, I will have no more core (mandatory) mathematics modules. So far, I have taken, Linear Algebra I, Calculus, Differential Equations, Linear Algebra II...
  22. Elvis 123456789

    Courses Partial Differential Equations vs Classical Mechanics 2?

    Hello everyone. So I wanted to get some opinions on what some of you thought was a better choice, as far taking PDE's or classical mechanics 2 goes. First let me start off by giving a little info; I've already taken calc 1-3 and ordinary differential equations, physics 1 & 2...
  23. M

    A Nonlinear differential equation

    ρCp (∂T/∂t) + k (∂2T/∂x2) = exp(-σt2)exp(-λx2)φo i have this equation... i was thinking of taylor series expansion to solve it and make it easier... any ideas on how to solve?
  24. dykuma

    Partial Differential equation, Temp in a Cylinder

    Homework Statement Homework Equations The Attempt at a Solution Because we are only looking at a cross section, I tried to reduce 5.3 down to just being a function of R and Theta. However I reasoned that there should be, based on this problem, no dependence on Theta either, so I figured I...
  25. S

    Graphing solutions to PDEs at various times

    Homework Statement Graph snapshots of the solution in the x-u plane for various times t if \begin{align*} f(x) = \begin{cases} & 3, \text{if } -4 \leq x \leq 0 \\ & 2, \text{if } 4 \leq x \leq 8 \\ & 0, \text{otherwise} \end{cases} \end{align*} Homework Equations Assuming that c=1 and g(x)...
  26. A

    I Are the derivatives of eigenfunctions orthogonal?

    We know that modes of vibration of an Euler-Bernoulli beam are given by eigenfunctions, with the natural frequency of each mode being given by its eigenvalue. Thus these modes are all mutually orthogonal. Can anything be said of the derivatives of these eigenfunctions? For example, I have the...
  27. S

    A Nonlinear first order Differential equation

    I need to solve the well known momentum equation in 3D cylindrical coordinates: ρ(∂v/∂t +(v.∇)v)=A where A and the velocity v are both local vector variables. I am actually looking for the stationary solution to the equation, i.e. no ∂/∂t term) I have tried evolving the velocity and tried...
  28. P

    Nonhomog heat equation that's piecewise

    Homework Statement $$u_{t}=u_{xx}+f(x) \\ u(0,t)=50 \\ u(\pi , t)=0 \\ u(x,0)=g(x)$$ $$0<x<\pi \\ t>0$$ $$f(x)=\begin{cases} 50 & 0<x<\frac{\pi}{2} \\ 0 & \frac{\pi}{2}\leq x< \pi \end{cases}$$ $$g(x)=\begin{cases} 0 & 0<x<\frac{\pi}{2} \\ 50 & \frac{\pi}{2}\leq...
  29. M

    A Obtain parameter derivatives solving PDE

    I have a PDE which is the following: $$\frac {\partial n}{\partial t} = -G\cdot\frac {\partial n}{\partial L}$$ with boundary condition: $$n(t,0,p) = \frac {B}{G}$$ , where G is a constant, L is length and t is time. G and B depend on a set of parameters, something like $$B = k_1\cdot C^a$$...
  30. A

    Understanding the method of Green's function

    I'm trying to understand the derivation for methods of Greens functions for PDEs but I can't get my head around some parts. I'm starting to feel comfortable with the method itself but I want to understand why it works. The thing I have problem with is quite crucial and it is the following: I...
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