Solution diffusion equation delta function pdf

We expect the solution of this equation to be the limit of the. Notes on the fundamental solution of the di usion equation. We proceed to solve this pde using the method of separation of variables. When the diffusion equation is linear, sums of solutions are also solutions. These equations become the defining property of the dirac delta function in. Pdf analytical solution of the nonlinear diffusion equation. It is probably helpful to rewrite the argument of the radial bessel function j0 in eq. Notice that for a linear equation, if u is a solution, then so is cu, and if v is another. This is the solution of the heat equation for any initial data we derived the same formula last quarter, but notice that this is a much quicker way to nd it. In the present paper, we derive the solution of the nonlinear fractional partial differential equations using an efficient approach based on the qhomotopy analysis transform method qhatm. The quantitative treatment of nonsteady state diffusion processes is formulated as a partial differential equation. We then derive the onedimensional diffusion equation, which is a pde for the diffusion of a dye in a pipe. A fundamental solution of this 2d diffusion equation in rectangular coordinates is diracdeltax xodiracdeltay yo, which can be further expanded as an explicit function of space and time as.

To satisfy this condition we seek for solutions in the form of an in nite series of. Here is an example that uses superposition of errorfunction solutions. The transport equation for this system is given in 2. Similarity solutions of the diffusion equation the diffusion equation in onedimension is u t. This second order ordinary differential equation has for its solutions the following. We first consider solutions to the heat equation on the real line, i. Analytical solution of the nonlinear diffusion equation. Pdf solution of the smoluchowski equation with a coulomb. Solution of heat or diffusion equation ii partial differential equation duration. Lectures on partial differential equations arizona math. There are several complementary ways to describe random walks and di. We obtain the greens function solution of the smoluchowski equation with a coulomb potential and an electric field, corresponding to a general boundary condition at the origin. Partial differential equations yuri kondratiev fakultat fur. Delta functions the pde problem defining any green.

Finding a solution to the diffusion equation youtube. The dirac delta function is briefly discussed in the textbook for this course. Dirichlet boundary conditions find all solutions to the eigenvalue problem. The heat equation and convectiondiffusion c2006 gilbert strang. What is the fundamental solution to the diffusion equation. Chapter 7 the diffusion equation the diffusionequation is a partial differentialequationwhich describes density. One dimensional wave equation fundamental solution. The heat equation and convectiondiffusion c 2006 gilbert strang the fundamental solution for a delta function ux, 0. Plugging a function u xt into the heat equation, we arrive at the equation. We will show in equation 7 that this special solution is a bellshaped curve. If we are looking for solutions of 1 on an infinite domainxwhere there is no natural length scale, then we can use the dimensionless variable. Convection diffusion equation and its applications. Two step functions, properly positioned, can be summed to give a solution for finite layer placed between two semiinfinite bodies.