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

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    Convergence Ode Pde
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Discussion Overview

The discussion revolves around the relationship between convergence solutions and asymptotic solutions of ordinary differential equations (ODEs) and partial differential equations (PDEs). Participants explore whether these two types of solutions can be considered equivalent or complementary, particularly in the context of the Adomian decomposition method.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Chwala questions whether convergence solutions of ODEs/PDEs are equivalent to their asymptotic solutions, particularly in the context of the Adomian decomposition method.
  • Some participants express confusion regarding the definitions of "convergence solution" and "asymptotic solution," suggesting that a solution is simply a function that satisfies the differential equation, regardless of the method used to obtain it.
  • Chwala elaborates on the Adomian decomposition method, noting that it emphasizes convergence as a condition for obtaining specific solutions, which may also be approached through analytical or numerical methods.
  • Chwala mentions that asymptotic methods can yield solutions that can then be tested for convergence or divergence, indicating a potential relationship between the two methods.
  • There is a suggestion that asymptotic and convergence methods may complement each other in solving differential equations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the equivalence or complementarity of convergence and asymptotic solutions. There is ongoing confusion and debate regarding the definitions and implications of these terms.

Contextual Notes

Participants highlight the importance of definitions in discussing convergence and asymptotic solutions, indicating that the lack of clarity may hinder the discussion. The relationship between the two methods remains unresolved.

chwala
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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
 
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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.
 
HallsofIvy said:
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 referring 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 said:
Hallsoflvy,
thanks for your quick response...i am referring 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
 

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