Do Convergence Solutions of ODE/PDEs Match Their Asymptotic Solutions?

[chwala]

Hi,
well let me put the question a bit clear...my concern area is on ode and pde...my question is when you solve a pde/ode analytically and get a solution by asymptotic means does this mean that if solution exists then ...when using convergence as an alternative way of getting solution of the pde/ode ...read adomian decomposition...

Does it therefore mean that convergence solution of an ode/pde EQUATION = Asymptotic solution of an ode/pde EQUATION ?
regards,
chwala

 

[HallsofIvy]

It's not at all clear what you mean by "convergenced solution" or "asymptotic solution". A "solution" to a differential equation by is a function that satisfies the differential equation. How you got that function does not matter.

 

[chwala]

It's not at all clear what you mean by "convergenced solution" or "asymptotic solution". A "solution" to a differential equation by is a function that satisfies the differential equation. How you got that function does not matter.

Hallsoflvy,
thanks for your quick response...i am reffering to adomian decomposition method of solving linear and non linear partial and ordinary differential equation...where the author has emphasised on the convergence of the equations(pde & ode) as a condition in the working of the problemto a specific solution that may also be solved by other means i.e thro..analytical means or numerical means e.g runge kutta ,finite difference method and so on...now on the analytic method..one may get a solution that is asymptotic in nature.

Are the two methods I.E asymptotic and convergence complementing each other..kindly look at adomian decomposition in general as you make a conclusion,
regards,
chwala

 

[chwala]

Hallsoflvy,
thanks for your quick response...i am reffering to adomian decomposition method of solving linear and non linear partial and ordinary differential equation...where the author has emphasised on the convergence of the equations(pde & ode) as a condition in the working of the problemto a specific solution that may also be solved by other means i.e thro..analytical means or numerical means e.g runge kutta ,finite difference method and so on...now on the analytic method..one may get a solution that is asymptotic in nature.

Are the two methods I.E asymptotic and convergence complementing each other..kindly look at adomian decomposition in general as you make a conclusion,
regards,
chwala

i have researched on the two- asymptotic and convergence...a differential equation can be solved analytically by use of asymptotic method...once solved and a solution found ...the same solution can be tested for convergence or divergence as a property of solution...thank you all...on the other hand convergence can be used as a method in solving differential equation particularly of second order...by use of what we call adomian decomposition...where a solution is found and in most cases the solutions are convergent in nature.
regards

 

[blue_raver22]

Hello Chwala,

Convergence of ODE and PDE refers to the process of finding a solution to a differential equation using numerical methods. This can be done by approximating the solution with a series of simpler equations and gradually improving the accuracy of the solution. Asymptotic means that the solution approaches a certain value as the independent variable approaches infinity.

In terms of your question, using convergence as an alternative way of solving an ODE/PDE does not necessarily mean that the solution will be the same as the asymptotic solution obtained analytically. Convergence methods provide approximations of the solution, which may not be as accurate as the asymptotic solution. However, in some cases, the convergence solution may closely match the asymptotic solution.

The Adomian decomposition method is one such convergence method that can be used to solve ODEs and PDEs. It involves breaking down the original equation into a series of simpler equations and solving them sequentially. The final solution is then obtained by combining the solutions of these simpler equations.

In conclusion, while convergence methods provide an alternative way of obtaining solutions to ODEs and PDEs, the accuracy of these solutions may vary and may not always match the asymptotic solution obtained analytically. It is important to carefully consider the convergence method being used and the level of accuracy required for the specific problem at hand.