Solving Inhomogeneous heat equation using the Fourier transform?
You also need a boundary condition. I assume U(0,t) = U(L,t) = 0 will do the job. So let´s first solve the homogeneous equation Ut = KUxx by assuming, that U can be expanded by a Fourier-series: U(x,t) = sum over[a_n(t) * sin(n*PI/L * x)] from n=1 to infinity Now let´s also expand G(x) into a Fourier series: G(x) = sum over[b_n(t) * sin(n*PI/L * x)] from n=1 to infinity Where the argument of the sines result from the boundary condition U(0,t) = U(L,t) = 0. Now we insert both expressions into the inhomogeneous differential equation. To satisfy the resulting equation, the following condition needs to be satisfied: a'_n(t) + (n*PI/L)^2 * a_n(t) = b_n(t) This is a linear differential equation of order 1. From the initial value condition U(0,x) = sum over[a_n(0) * sin(n*PI/L * x)] from n=1 to infinity = F(x) we see, that the a_n(0) are the Fourier coefficients of the Fourier series of F(x). I´m sorry, didn´t have much time at my hands to write everything properly. Hope this helped a little. You don´t necessarily need to use the sine-Fourier-series, using a_n(t)e^ikx would work just as well.
Inhomogeneous heat equation using the Fourier transform?
Hi this also needs values to work out or are we talking transformation for a particular value. so t has been removed why and what has happened to G. more information required.
Solving inhomogeneous wave equation?
First, find a particular solution. By inspection, we find that u_p(x, t) = (-1/(6a²)) tx³ is one such solution. So, we have u(x, t) = u_h(x, t) + (-1/(6a²)) tx³. Thus, it remains to solve the homogeneous PDE ∂²u_h/∂t² = a² ∂²u_h/∂x², where: (i) 0 = u(0, t) = u_h(0, t) + (-1/(6a²)) t * 0³ ==> u_h(0, t) = 0 (ii) 0 = u(l, t) = u_h(l, t) + (-1/(6a²)) tl³ ==> u_h(l, t) = (l³/(6a²)) t (iii) 0 = u(x, 0) = u_h(x, 0) + (-1/(6a²)) 0 * x³ ==> u_h(x, 0) = 0 (iv) 0 = ∂u(x, 0)/∂t = ∂u_h(x, 0)/∂t + (-1/(6a²)) x³ ==> 0 = ∂u_h(x, 0)/∂t = (1/(6a²)) x³. --------------------------- See if you can try separation of variables from here...
What is a inhomogeneous differential equation?
Yes...the link given by Matthew Ring describes the concept only for a first order inhomogeneous differential equation but for the higher orders it's not complicated and you can find it on the internet too. There is no more to say... We can sum the solutions because the whole set of solutions forms a vector space...We need informations about boundaries to express the entire solutions... Last year I found this .pdf file : Page on ucsc.eduThe author explains how to solve inhomogeneous differential equations by using Green functions. Green's function.It's an interesting complement. But otherwise we can give you a comprehensive and detailed example.
Solving heat equation, please help.....?
This question can be solved using the Green function for the heat equation in 1D for x in (-∞,∞) and t in [0,∞): g(x,x',t) = exp(-(x-x')^2/(4kt)) / √(4πkt) By definition, the Green function is the solution to: g_t - kg_xx = δ(x-x') so that initial value problems are solved by an integration (infinite limits are implied): u(x,t) = ∫ dx' g(x,x') u(x',0) Plugging in u(x',0) = sinh(x') and g(x,x'), the integrations are done by completing the square on the argument of the exponential and shifting the integration variable to reduce the integral to a simple gaussian of the form: ∫ dy exp(-y^2/a) = √(πa) For the first part then you get: u(x,t) = sinh(x)exp(kt) The second part is trivial now, and one can evaluate the integrals in terms of either the probability integral or the error function.
Solve heat equation maths?
1.Solve the heat equation ∂u/∂t=[1/π^2 . (∂^2 u)/(∂x)^2] 0
SOLVE THE HEAT EQUATION?
Using separation of variables (see link below for the basic idea), we have (with L = π and k = 9): u(x, t) = (1/2) A(0) + Σ(n = 1 to ∞) A(n) cos(nx) e^(-9n^2 t). To find B(n), we use u(x, 0) = e^(8x): e^(8x) = (1/2) A(0) + Σ(n = 1 to ∞) A(n) cos(nx). Now, we compute the Fourier cosine coefficients of e^(8x). A(0) = (2/π) ∫(x = 0 to π) e^(8x) dx ........= (2/π) * (1/8)e^(8x) {for x = 0 to π} ........= (1/(4π)) (e^(8π) - 1). (So, a(0) = (1/(8π)) (e^(8π) - 1).) ---- For n > 1: A(n) = (2/π) ∫(x = 0 to π) e^(8x) cos(nx) dx ........= (2/π) [e^(8x) (n sin(nx) + 8 cos(nx))/(n^2 + 64)] {for x = 0 to π}, via int. by parts ........= (16/π) [((-1)^n * e^(8π) - 1)/(n^2 + 64)]. (a(n) = A(n).) Link: http://tutorial.math.lamar.edu/Classes/D... -------------- I hope this helps!
Is there an equation for the big bang theory?
There is a system of equations that you can use to describe the universe. The Friedmann equation, already mentioned in John Bailey's answer to Is there an equation for the big bang theory?, assumes an homogeneous and isotropic universe. If you want to describe an universe that does not satisfy these properties, you need the Einstein-Boltzmann equations.Consider the perturbed FRW metric:[math] g_{\mu \nu} = \begin{pmatrix} -1 - 2 \Psi(\mathbf{x},t) & 0 \\ 0 & a^2(t) (1 + 2 \Phi(\mathbf{x},t) ) \delta_{ij} \end{pmatrix}[/math],where [math] \Psi [/math] and [math] \Phi [/math] are the gravitational potentials, together with the Einstein equation for GR:[math]R_{\mu \nu} - \frac{1}{2}R g_{\mu \nu} = T_{\mu \nu}[/math]to find the perturbed equations of the expansion of the universe. You, then, combine these equations with the perturbed Boltzmann equations, that are very complicated, to find a system of equations relating each species (matter, dark matter, neutrinos, photons) distribution and combine them all to find a complete description of the inhomogeneous universe. Of course, this is an extremely complicated procedure and there are books that have done it with great detail. For example, Modern Cosmology, chapters 4-8 are dedicated to finding and solving (when possible!) these equations.
Help me with differential equations?
Inhomogeneous Equations x'(t) = Matrix [ 1 3 ] x(t) + [1] '''''''''''''''''''''''''''''''[ 3 1 ]'''''''''''''''''[2] a. Find the general solution using the method of undetermined coefficients b. Find the general solution using the method variation of parameters c. Find the general solution using that matrix [ 1 3 ] is diagonalizable ''''''''''''''''''''''''''''''''''''''... 3 1 ] Sorry if it is hard to see the matrices. It is 1 and 3 on top with 3 and 1 on the bottom and the other one is just a 1 on top and 2 on bottom.
What is the world’s hardest math equation?
Not all of these are equations per se, but some of the world’s hardest math problems right now are the Millennium Prize Problems.These are seven problems so difficult that the Clay Mathematics Institute has offered a one million dollar award to anyone who can solve any of them. So far, only one prize has been claimed for the Poincaré Conjecture.If what you wanted was a ridiculously long string of letters, how about this?This is a description of the Lagrangian of the Standard Model. In essence, it’s a description of the underlying physics of the universe. This is a more compact way of writing it, written out to painful verbosity it looks like: