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  1. M

    A Boundary Value Problem

    You are right Wrobel. I overlooked that U is negative (the assumption). Sorry. The result now is consistent with an example that I have. u" + 4u = x^2 , u(0)=u(1)=0 Solution: u(x) = (2x^2 + sin(1-2x)/sin(1) - 1)/8 Lower bound solution F=4, G=1 U" + 4U = 1 , U(0)=U(1)=0 Solution...
  2. M

    A Boundary Value Problem

    Sorry to come back again to this thread. If I understand correctly, the mapping \mathcal F : W \rightarrow W where W=\{u\in C[0,1]\mid U\le u\le 0\} required the statement if h\le 0 then Ph=x\int_0^1d\xi\int_0^\xi h(s)ds-\int_0^xd\xi\int_0^\xi h(s)ds \le 0 . Also in the equation -U''=F...
  3. M

    A Boundary Value Problem

    Wow! What a solution. Thank you very much Wrobel. I need some time to properly understand the solution. Hope you don't mind if I ask again in case I do have problem understanding the argument. :smile:
  4. M

    A Boundary Value Problem

    Thanks. Interesting suggestion wrobel. So to which DE should I compare my equation ? Although I would prefer f(x) and g(x) to be general, I would still be content if f(x) is a monotonic increasing function since I'll be solving later on a specific DE with f(x) known.
  5. M

    A Boundary Value Problem

    I have a BVP of the form u" + f(x)u = g(x) , u(0)=u(1)= 0 where f(x) and g(x) are positive functions. I suspect that u(x) < 0 in the domain 0 < x < 1. How do I go proving this. I have try proving by contradiction. Assuming first u > 0 but I can't deduce that u" > 0 which contradict that u has...
  6. M

    Laplace transform to solve a nonhomogeneous equation

    Looks like you have a boundary value problem not initial value problem. If your DE is a linear constant coefficient, I think you still can solve it with Laplace transform.
  7. M

    Runge-Kutta Method - Need help with the calculus

    That explain partly. But why can't we also write the first term as f''(f(f)) and the second term as (f',f')f ?
  8. M

    Runge-Kutta Method - Need help with the calculus

    I got the expression from Prof. J.L. Butcher note, an authoritative person in Runge-Kutta method. That term is related to a rooted tree. Just google rooted tree Runge-Kutta for detail.
  9. M

    Runge-Kutta Method - Need help with the calculus

    When deriving the Runge-Kutta Method to solve y'=f(x) we need to use Taylor expansion. Hence we need to differentiate the function many times. y'(x)=f(y(x)) y''(x)=f'(y(x))y'(x) = f'(y(x))f(y(x)) y''' = f''(y(x))(f(y(x)),y'(x)) + f'(y(x))f'(y(x))y'(x) I can understand the second...
  10. M

    Role of eigenvalues in phase portraits

    Euler's formula e^{it}=\cos{t }+ i \sin{t}. So if the eigenvalue is a complex number, the solution will have sinusoidal functions I think.
  11. M

    Help with solving system of DE's

    Assuming that your system can be written as the matrix form \dot{X}=AX+F(t). , e.g. X=[x1 x2]t etc Then the general solution for this equation should be (if I'm not mistaken) X(t)=e^{At}C+e^{At}\int_0^t e^{-As}F(s) ds Your particular solution in this case is then X_p(t)=e^{At}\int_0^t...
  12. M

    MATLAB Matlab: Suppressing the output

    OK get it already from the other thread Sorry about this.
  13. M

    MATLAB Matlab: Suppressing the output

    I get this example from >>help eval and add a semicolon. for n = 1:12 eval(['M' num2str(n) ' = magic(n)']); end The above commands display all 12 magic square. How do I suppress the output? I only want matlab to assign the variables not display them.
  14. M

    Solution for Sylvester Equation

    I don't think it is possible to express it as AX=B. Even to solve the Sylvester equation you have to diagonalize the matrices.
  15. M

    Differential equation in simple RC-circuit

    Laplace transform of your equation (*) should be sI+\frac{1}{RC}I=0 Solve for I and invert it. You should obtain the same answer.
  16. M

    4th order DE

    You need to get the homogeneous solution and the particular solution. To obtain the homogeneous solution use the characteristic equation.
  17. M

    Solution for Sylvester Equation

    Given n by n matrices A, B, C. I know how to solve the Sylvester equation AX + XB + C = 0 using the matlab command >> X=lyap(A,B,C) But how do we solve the extended Sylvester equation AX + XB + CXD + E = 0 ? Either numerical or analytical method I'm willing to learn.
  18. M

    MATLAB Matlab - can't even do a simple arithmatic

    From what I understand the command format won't effect the computation. It is only for display purposes. I show you another session in the default format. >> 1/3-1/2 ans = -0.1667 >> ans + 1/6 ans = -2.7756e-017 >> -1/6 ans = -0.1667 >> -1/6+1/6 ans =...
  19. M

    MATLAB Matlab - can't even do a simple arithmatic

    Thank you for all those information. When we were first introduced to matlab, they said matlab can be used as a calculator. Now I think my desktop scientific calculator or Window Accessories calculator can do a much better job for simple arithmetic calculation. What puzzled me is that when...
  20. M

    MATLAB Matlab - can't even do a simple arithmatic

    This really surprised me. >> format rat >> 1/3-1/2+1/6 ans = -1/36028797018963968 Even school student knows that the answer is 0. Even format short does not give a correct answer. >> format short >> 1/3-1/2+1/6 ans = -2.7756e-017
  21. M

    Why is general solution of homogeneous equation linear

    The general solution is a linear combination of two linearly independent solutions y1(x) and y2(x). p.s. I think something is not right with your notation e^(xt) .
  22. M

    Linear pde of order one

    I want to determine whether u=-x^3_1-x_1-\sqrt{3}x_2 is a stabilizing control for the system \begin{array}{cc}\dot{x}_1=x_2\\ \dot{x}_2=x^3_1+u\end{array} with cost functional \frac{1}{2}\int^{\infty}_0 x^2_1 +x^2_2+u^2 \ dt. After looking at some examples, I understand that I have to find...
  23. M

    Transport phenomenion problem

    That is the original r from the previous equation.
  24. M

    Optimal Control Problem - LQR

    I have gone through the paper again but cannot extract new information about the terminal point other than what I have already written. But I see there is a sentence which claim that this example is for linear unstable system. Why is it unstable? Will it effect the computation?
  25. M

    Y'' + py'+ qy = 0 explain why the value of y''(a) is determined by the values of y(a)

    Re: Y'' + py'+ qy = 0 explain why the value of y''(a) is determined by the values of The DE is linear so it must have a general solution which is a linear combination of two linearly independent solutions y1(x) and y2(x). y(x)=c1y1(x) + c2y2(x) . c1 and c2 can be determined uniquely...
  26. M

    MATLAB Runga-Kutta help needed (Matlab)

    Matlab already has a built-in Runge-Kutta solver, probabaly ode45. So I think you need not have to worry about those ki's.
  27. M

    Optimal Control Problem - LQR

    This is the part that really confuse me, the terminal point, because so far I have been doing by just following examples. Some problem have specific fixed end. Whilst others are free and yet some have infinite time. So I'm not fully understand what I'm doing here whether it is fixed end...
  28. M

    Numerical Integration of Equation

    What is i in the expression? Is it a positive integer or the imaginary number √-1 ?
  29. M

    Optimal Control Problem - LQR

    Thanks Pyrrhus. Probably thats my mistake. My arguement why the control u is a scalar because in the equation \dot{x}=Ax+Bu , B is a column vector. The only way we can compute Bu is when u is a scalar. Bu=\left(\begin{array}{cc}0\\u\end{array}\right). The initial condition x(0) is...
  30. M

    Writing PDEs as differential equations on Hilbert space

    I will be following this thread. :smile: For a diffusion equation, the first derivative is wrt to time. \frac{\partial u(x,t)}{\partial t}= \frac{\partial^2 u(x,t)}{\partial x^2}