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By: Nid`a Rababah
Advisor: Prof. Kamel Al-Khaled
Nonlinear partial differential equations.
Conservation laws.
Adomian decomposition method
Motivation
which can be written equivalently in the following form
ADM
3)Applying the inverse operator to the system and using the given conditions we obtain:.
* Adomian polynomials can be constructed by using the general formula
4) Using the recursive relation:
5) Taking the limit to find the exact solutions for u(x,t) and v(x,t).
1) The main advantage of the method is that it reduces the size of computation work and maintains the high accuracy of the analytical solution in terms of rapidly convergent series.
2) It provided a direct scheme that can be solved by recursive relationship by using a few iterations for both linear, nonlinear deterministic and stochastic equations, without the need for linearization, perturbation, massive computation and any transformation.
3) It‘s very efficient for nonlinear models, and it`s results give evidence that high accuracy can be achieved.
The models solved by the ADM are:
1) The shock wave equation
2) The P-system
3)The Cyclic system
The approximate solution
The exact solution
- We choose a problem with known solution allows for more complete error analysis so to derive an approximate solution for the p- systemwe start with the one-dimensional conservation law
such that:
with the approximation to the Riemann type condition:
- The transformation :
- Now we get a new system given by
- To solve this system we consider the coupled system of nonlinear PDE of the form:
by using the transformation
- By using of aforementioned techniques of ADM we reach the following steps as follow:
1) Applying the inverse operator to find the terms of the decomposition series:
2) Representing the unknown functions as infinite series:
3) Finding the values of H(x,t) by using the initial conditions:
4) Using the recursive relationship:
The approximate solution of u(x,t)
The exact solution of u(x,t)
The approximate solution of v(x,t)
The exact solution of v(x,t)
The problem
has a local solution in the interval (0,T), where T is some small constant.
- It might be probable that the scheme will diverge for that T.
- To run the program in the interval (0,T) we find a smaller time interval in which scheme will converge, and solve the system using the given conditions.
Continuing in this way
solve the system
smaller time
the system has a solution
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