I'm not used to this method, but your example is pretty close to the one on Wikipedia:chwala said:Homework Statement
how do we solve the ode ## y'+y^2=-2, y(0)=0## using adomian decomposition method?Homework Equations
The Attempt at a Solution
##Ly = -2-y^2##
## y= 0 + L^-1[-2-y^2]##
##y[0]= -2t##
##y[1]= -L^-{1}[4t^2] = -4t^3/3## are my steps correct so far in trying to get the Adomian Polynomials? am i following the correct steps?
The notation used in the wiki article whose link fresh_42 provided is ##y_0##. Your notation confused me into thinking you meant y(0), which is given as 0.chwala said:The Attempt at a Solution
##Ly = -2-y^2##
## y= 0 + L^{-1}[-2-y^2]##
##y[0]= -2t##
It looks good so far. How many terms in the polynomial do you need to get?chwala said:##y[1]= -L^{-1}[4t^2] = -4t^3/3## are my steps correct so far in trying to get the Adomian Polynomials? am i following the correct steps?
i want to try and get up to 5 terms of the polynomials and then probably compare with the analytical method/numerical methods...let me try and get the other polynomials then you can correct where necessary...Mark44 said:I've never heard of the Adomian method, either, but I'm very familiar with using Laplace transforms and inverses to solve diff. equations.
Comments below.
The notation used in the wiki article whose link fresh_42 provided is ##y_0##. Your notation confused me into thinking you meant y(0), which is given as 0.
It looks good so far. How many terms in the polynomial do you need to get?
in life we are all learner's probably its good to admit that we may not know everything in math or rather i may know something which you do not know...now you know Adomian decomposition method has been used in solving first order and second order linear and non linear ode and pde. I am conversant with the laplace transforms...Mark44 said:I've never heard of the Adomian method, either, but I'm very familiar with using Laplace transforms and inverses to solve diff. equations.
Comments below.
The notation used in the wiki article whose link fresh_42 provided is ##y_0##. Your notation confused me into thinking you meant y(0), which is given as 0.
It looks good so far. How many terms in the polynomial do you need to get?
i just made some changes on the notation, sorry...Mark44 said:I've never heard of the Adomian method, either, but I'm very familiar with using Laplace transforms and inverses to solve diff. equations.
Comments below.
The notation used in the wiki article whose link fresh_42 provided is ##y_0##. Your notation confused me into thinking you meant y(0), which is given as 0.
It looks good so far. How many terms in the polynomial do you need to get?
The Adomian Decomposition method is a mathematical technique used to solve ordinary differential equations (ODEs). It involves breaking down the original equation into simpler components and solving each component separately, then combining the solutions to obtain the solution to the original equation.
The Adomian Decomposition method works by decomposing the original equation into simpler components using the Adomian polynomials. These polynomials are then substituted into the original equation, resulting in a system of algebraic equations. The solutions to these equations are then combined to obtain the solution to the original ODE.
One of the main advantages of the Adomian Decomposition method is that it can be applied to a wide range of ODEs, including nonlinear and non-homogeneous equations. It also does not require the use of initial or boundary conditions, making it a more flexible method compared to other techniques.
While the Adomian Decomposition method can be applied to a wide range of ODEs, it may not always yield an exact solution. In some cases, the solution obtained may only be an approximation. Additionally, the method may become more complex and time-consuming for higher order ODEs.
The accuracy of the solution obtained using the Adomian Decomposition method can be evaluated by comparing it to the exact solution, if available. Otherwise, the solution can be compared to numerical methods, such as the Runge-Kutta method, to assess its accuracy. It is also important to check for convergence, as the solution may become less accurate as more terms are included in the decomposition process.