Final Project Similarity Solutions of Nonlinear PDE Harvard Case Solution & Analysis

Final Project Similarity Solutions of Nonlinear PDE Case Solution

Objective:Find a method which was published in a journal and is relevant to the material covered in class, examples: Split-Step method for NLS, Accelerated Jacobi Method, etc. Describe (with illustrations and derivations) and implement the method described in the paper. Reproduce some of the figures in the paper.

481:Discuss the novelty of the method with respect to methods covered in class.

Introduction:The issue with the standard application of similarity solution is that the reduction in ODEs equations is very complicated to integrate for tabulated functions and elementary functions, which tends to solve the equation with numerical method.Therefore, we do not have any systematic method’s procedure available for reduced ODE to obtain original partial differential equation solutions.In order to deal with this situation we proposed the practical and tractable method to obtain approximate similarity solutions of initial-boundary value problem.
Description:To demonstrate our approach we selected the transient flow through semi-infinite porous medium problem. We considered the initially filled with gas to study the unsteady flow of gas at initial pressure P0>0 at time=0, In case of diffusion the pressure from outflow reduces to P1>=0, far the case vacuum diffusion P1=0 and this all we will consider in the nonlinear PDEs.

Method of Implementation and Derivation:

In order to implement the method we will consider initial value problem.The idea is to find the solution of ordinary differential equation in the form of function as this could be used with similarity variables to find approximate solution of original partial differential equation it is more efficient , requires less processing period.
Musk at derived the nonlinear PDE equation for the unsteady flow of gas with semi-infinite porous.
〖 ∇〗^2 (P^2 )=2(∅μ/K) ∂P/∂t (1)
Where,
Øis porosity
P is pressure
µ is the viscosity
K is the permeability

For the 1-D medium extending from x=0 to x=infinity, the reduction is given below,
∂/( ∂x) (p∂P/∂x)=A∂p/∂x(2)
The boundary conditions are given below,
p(x,0)=p_0,0<x<∞, (3)
p(0,t)=p_1 (<p_0 ),0≤t<∞
WhereA=ص/K.

Now to illustrate the proposed method we will consider the initial value problem.
∂/∂x (p∂p/∂x)=A∂p/∂t (4)
p(x,0)=p_0,p(0,t)=p_1 (<p_0 ), p(∞,t)=p_0. (5)
Assume P0=1 with no any loss reason is that the change in variable P- (x, t) = p (x, t) /P0 tends to same problem with P- (x, 0) = 1. Therefore,
∂/∂x (p∂p/∂x)=A∂p/∂t (6)
Initial conditions and boundary conditions
p(x,0)=1,0<x<∞,
p(0,t)=p_1,0≤t<∞, (7)
p(∞,t)=1,0<t<∞,
Step 1: we will reduce the Initial value boundary to boundary value problem of ODE. The following transformation requires.
z=x/√t (A/4)^(1/2) , ω(z)=α^(-1) (1-(p(x,t) )^2 , (8)
With the help of α=1-P12, we have reduced the IBVP to BVP as given below,
〖 ω〗^''+ z/√(1-αω) ω^'=0 (9)
ω(z=0)=1, ω(z→∞)=0.....................

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