How can I solve two non-linear coupled second order ODE numerically and analytically? My equation of systems are autonomous and they have interface continuity conditions as well. Can anyone suggest me a good numerical method along with some sources/paper ? I have attached a file displaying the problem statement and conditions .Your help will be much appreciated. Thank you.
Dear Vivekananthan, using MAPLE, the solution of an ODE of this form:
d2x/dt2 = k*exp(m*x(t)))
is the following:
t = ln((1/2)*C1*(tan((1/2)*t*sqrt(C1*m)+(1/2)*C2*sqrt(C1*m))^2+1)/k)/m
where C1 and C2 are constants.
The system of differential equations is only "coupled" through the continuity conditions (iii-iv). Thus you can solve the differential equations separately for each domain and then apply the boundary conditions. To solve this numerically I believe you could apply a finite difference or finite element discretization along the x-direction. Also, the shooting method may work since you are basically solving the same ODE with different parameters in both domains.
Dear Vivekananthan Balakrishnan
read "Using Matlab to integrate Ordinary Differential Equations (ODEs)" and "Numerical Methods for Differential
Equations " in site http://faculty.olin.edu/bstorey/Notes/DiffEq.
may be help you
I am unable to answer your question because I don't know what the condition (ii) means. Specifically, what is theta_L? If it means theta2(x=L)=theta_0, then the system is one which could be solved numerically using a shooting method, as suggested earlier. One would need to ensure that the interface lies on a grid point so that the new value of theta' is used when first moving away from the interface.
to solve complex coupled odes system,
you can convert this system into real system by assuming for example
y=y1+i y2 where y1 and y2 are real functions, separate real and imaginary parts to end with real coupled system, then after that you can solve your real system using any odes solver .
best wishes
You can use the Runge-Kutta's method (RK4) implemented in Matlab, Octave or Scilab. These packages work with complex numbers
The original problem is a real problem because theta and x are real. This is not a problem which has complex numbers. Perhaps it might have been better to call this a complicated problem, rather than a complex one!
The problem which is being solved is a second order real ODE in the range 0
Dear Professor V. Balakrishnan,
You want analytic solution of system of nonlinear second order ODE with given conditions.The following Five steps will help you to solve your problem.
- The equation (1) with initial condition (i) can be solved by " Adomian Decomposition Method " .Thus the solution theta1 is known for all x.
- It is also known at x = L0
- Using condition (iv) ,we get initial condition for theta_2
- This initial condition for theta_2 together with your equation (2) can be solved again by " Adomian Decomposition Method " Thus the solution theta2 is known for all x.
- Check whether these solutions satisfies the compatibility conditions or not. You must be very attentive for the choice of constant theta_0My dear (south Indian ?) friend inform me about your success.
BOOK: Adomian G(1994).Solving Frontier Problems of Physics:The Decomposition Method, Kluwer Academic Publisher, Boston.
For numerical solution you can choose the finite differences method or Runge Kutta methods or ... coupled may be with the shooting method.
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