Solving the following linear ODE by a numerical method
$begingroup$
ODE:
$$y'(x)+3y(x)=1$$
Initial condition: $y(0)=0$
We know that the exact solution is:
$y left( t right) =1/3-1/3,{{rm e}^{-3,t}}.$
My Objective:
I want to solve the ODE by Legendre wavelets method. The following solution is right? If else, what' is the wrong?
My try: (shortly)
Let $y'(t)=C^intercalpsi(t)$.
By integrating, we have
$y(t)=C^intercal P psi(t)+y(0).$
Substituting $y(t), y'(t)$ to the ode, we have
$$C^intercal(I+3P)psi(t)=F^intercalpsi(t)$$
where $F$ is legendre coefficient matrix of $1$ which is
$1=F^intercalpsi(t)$
So, equating the coefficients of $psi(t)$, we have
$$C^intercal=F^intercal(I+3P)^{-1}$$
Substituting $C^intercal$ to $y(t)=C^intercal P psi(t),$
the solution is
$$y(t)=F^intercal(I+3P)^{-1}P psi(t).$$
(In here, superscript $^intercal$ represents transpose of the matrices. $C,F,psi$ are $Ntimes1$ matrices.Identity matrix $I$ and $P$ are $Ntimes N$ matrices where $N=2^{k-1}M$. )
Many thanks.
Best regards.
ordinary-differential-equations numerical-methods legendre-polynomials wavelets
asked Dec 20 '18 at 16:50
HD239HD239
436415
$endgroup$
add a comment |
$begingroup$
ODE:
$$y'(x)+3y(x)=1$$
Initial condition: $y(0)=0$
We know that the exact solution is:
$y left( t right) =1/3-1/3,{{rm e}^{-3,t}}.$
My Objective:
I want to solve the ODE by Legendre wavelets method. The following solution is right? If else, what' is the wrong?
My try: (shortly)
Let $y'(t)=C^intercalpsi(t)$.
By integrating, we have
$y(t)=C^intercal P psi(t)+y(0).$
Substituting $y(t), y'(t)$ to the ode, we have
$$C^intercal(I+3P)psi(t)=F^intercalpsi(t)$$
where $F$ is legendre coefficient matrix of $1$ which is
$1=F^intercalpsi(t)$
So, equating the coefficients of $psi(t)$, we have
$$C^intercal=F^intercal(I+3P)^{-1}$$
Substituting $C^intercal$ to $y(t)=C^intercal P psi(t),$
the solution is
$$y(t)=F^intercal(I+3P)^{-1}P psi(t).$$
(In here, superscript $^intercal$ represents transpose of the matrices. $C,F,psi$ are $Ntimes1$ matrices.Identity matrix $I$ and $P$ are $Ntimes N$ matrices where $N=2^{k-1}M$. )
Many thanks.
Best regards.
ordinary-differential-equations numerical-methods legendre-polynomials wavelets
asked Dec 20 '18 at 16:50
HD239HD239
436415
$endgroup$
add a comment |
$begingroup$
ODE:
$$y'(x)+3y(x)=1$$
Initial condition: $y(0)=0$
We know that the exact solution is:
$y left( t right) =1/3-1/3,{{rm e}^{-3,t}}.$
My Objective:
I want to solve the ODE by Legendre wavelets method. The following solution is right? If else, what' is the wrong?
My try: (shortly)
Let $y'(t)=C^intercalpsi(t)$.
By integrating, we have
$y(t)=C^intercal P psi(t)+y(0).$
Substituting $y(t), y'(t)$ to the ode, we have
$$C^intercal(I+3P)psi(t)=F^intercalpsi(t)$$
where $F$ is legendre coefficient matrix of $1$ which is
$1=F^intercalpsi(t)$
So, equating the coefficients of $psi(t)$, we have
$$C^intercal=F^intercal(I+3P)^{-1}$$
Substituting $C^intercal$ to $y(t)=C^intercal P psi(t),$
the solution is
$$y(t)=F^intercal(I+3P)^{-1}P psi(t).$$
(In here, superscript $^intercal$ represents transpose of the matrices. $C,F,psi$ are $Ntimes1$ matrices.Identity matrix $I$ and $P$ are $Ntimes N$ matrices where $N=2^{k-1}M$. )
Many thanks.
Best regards.
ordinary-differential-equations numerical-methods legendre-polynomials wavelets
asked Dec 20 '18 at 16:50
HD239HD239
436415
$endgroup$
ODE:
$$y'(x)+3y(x)=1$$
Initial condition: $y(0)=0$
We know that the exact solution is:
$y left( t right) =1/3-1/3,{{rm e}^{-3,t}}.$
My Objective:
I want to solve the ODE by Legendre wavelets method. The following solution is right? If else, what' is the wrong?
My try: (shortly)
Let $y'(t)=C^intercalpsi(t)$.
By integrating, we have
$y(t)=C^intercal P psi(t)+y(0).$
Substituting $y(t), y'(t)$ to the ode, we have
$$C^intercal(I+3P)psi(t)=F^intercalpsi(t)$$
where $F$ is legendre coefficient matrix of $1$ which is
$1=F^intercalpsi(t)$
So, equating the coefficients of $psi(t)$, we have
$$C^intercal=F^intercal(I+3P)^{-1}$$
Substituting $C^intercal$ to $y(t)=C^intercal P psi(t),$
the solution is
$$y(t)=F^intercal(I+3P)^{-1}P psi(t).$$
(In here, superscript $^intercal$ represents transpose of the matrices. $C,F,psi$ are $Ntimes1$ matrices.Identity matrix $I$ and $P$ are $Ntimes N$ matrices where $N=2^{k-1}M$. )
Many thanks.
Best regards.
ordinary-differential-equations numerical-methods legendre-polynomials wavelets
ordinary-differential-equations numerical-methods legendre-polynomials wavelets
asked Dec 20 '18 at 16:50
HD239HD239
436415
asked Dec 20 '18 at 16:50
HD239HD239
436415
edited Dec 21 '18 at 6:29
HD239
asked Dec 20 '18 at 16:50
HD239HD239
436415
asked Dec 20 '18 at 16:50
HD239HD239
436415
asked Dec 20 '18 at 16:50
HD239HD239
436415
436415
add a comment |
add a comment |
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