finite difference method to the delayed damped wave equation
$begingroup$
I want to do the numerical simulations in MATLAB for the following partial delay differential equations with Dirichlet boundary condition. I need to know the simplest method to do that.
begin{equation}
left{
begin{aligned}
frac{partial ^{2}z}{partial t^{2}}left( x,tright)
&=frac{partial ^{2}z}{partial x^{2}}left( x,tright)
+vleft( tright) frac{partial z}{partial t}left( x,t-rright) , & t&geq 0,~~ xin Omega
\
zleft(x,thetaright)&=sinleft(xthetaright) , &theta&in left[ -r,0right] &
end{aligned}
right.
end{equation}
Here r denoting the delay is a positive real number, and $Omega =left( 0,1right) $ .
N.B: Thank you for your help
numerical-methods delay-differential-equations finite-difference-methods
edited Dec 23 '18 at 13:05
LutzL
60.1k42057
asked Dec 23 '18 at 12:18
zakaria hamidizakaria hamidi
142
$endgroup$
add a comment |
$begingroup$
I want to do the numerical simulations in MATLAB for the following partial delay differential equations with Dirichlet boundary condition. I need to know the simplest method to do that.
begin{equation}
left{
begin{aligned}
frac{partial ^{2}z}{partial t^{2}}left( x,tright)
&=frac{partial ^{2}z}{partial x^{2}}left( x,tright)
+vleft( tright) frac{partial z}{partial t}left( x,t-rright) , & t&geq 0,~~ xin Omega
\
zleft(x,thetaright)&=sinleft(xthetaright) , &theta&in left[ -r,0right] &
end{aligned}
right.
end{equation}
Here r denoting the delay is a positive real number, and $Omega =left( 0,1right) $ .
N.B: Thank you for your help
numerical-methods delay-differential-equations finite-difference-methods
edited Dec 23 '18 at 13:05
LutzL
60.1k42057
asked Dec 23 '18 at 12:18
zakaria hamidizakaria hamidi
142
$endgroup$
$begingroup$
You are missing some condition that excludes the trivial solution $z=0$. Is it to be assumed that $z(x,t)=0$ for all $t<0$? Did you solve similar problems before, esp. DE with delays, when yes, describe the methods used. What would be your first approach and how did it fail?
$endgroup$
– LutzL
Dec 23 '18 at 12:22
$begingroup$
thank you Lutzl. I would like to inform you that I am a beginner on the use of numerical methods,
$endgroup$
– zakaria hamidi
Dec 23 '18 at 12:48
1
$begingroup$
If you are just a beginner in numerical methods, then it is rather courageous to start with (second order) delay-differential equations. The easiest you can do is set $h=r/n$ and apply the Euler method to the first-order (in $t$) system $$∂_tz=w+vz(.,t-r),~~ ∂_tw=∂_x^2z$$ as in $$w_{k+1} = w_k+h∂_x^2z_k,~~ z_{k+1}=z_k+hw_k+v_kz_{k-n}.$$ In $x$ direction you apply some discretization or Fourier expansion and realize $∂_x^2$ according to the framework.
$endgroup$
– LutzL
Dec 23 '18 at 13:14
add a comment |
$begingroup$
I want to do the numerical simulations in MATLAB for the following partial delay differential equations with Dirichlet boundary condition. I need to know the simplest method to do that.
begin{equation}
left{
begin{aligned}
frac{partial ^{2}z}{partial t^{2}}left( x,tright)
&=frac{partial ^{2}z}{partial x^{2}}left( x,tright)
+vleft( tright) frac{partial z}{partial t}left( x,t-rright) , & t&geq 0,~~ xin Omega
\
zleft(x,thetaright)&=sinleft(xthetaright) , &theta&in left[ -r,0right] &
end{aligned}
right.
end{equation}
Here r denoting the delay is a positive real number, and $Omega =left( 0,1right) $ .
N.B: Thank you for your help
numerical-methods delay-differential-equations finite-difference-methods
edited Dec 23 '18 at 13:05
LutzL
60.1k42057
asked Dec 23 '18 at 12:18
zakaria hamidizakaria hamidi
142
$endgroup$
I want to do the numerical simulations in MATLAB for the following partial delay differential equations with Dirichlet boundary condition. I need to know the simplest method to do that.
begin{equation}
left{
begin{aligned}
frac{partial ^{2}z}{partial t^{2}}left( x,tright)
&=frac{partial ^{2}z}{partial x^{2}}left( x,tright)
+vleft( tright) frac{partial z}{partial t}left( x,t-rright) , & t&geq 0,~~ xin Omega
\
zleft(x,thetaright)&=sinleft(xthetaright) , &theta&in left[ -r,0right] &
end{aligned}
right.
end{equation}
Here r denoting the delay is a positive real number, and $Omega =left( 0,1right) $ .
N.B: Thank you for your help
numerical-methods delay-differential-equations finite-difference-methods
numerical-methods delay-differential-equations finite-difference-methods
edited Dec 23 '18 at 13:05
LutzL
60.1k42057
asked Dec 23 '18 at 12:18
zakaria hamidizakaria hamidi
142
edited Dec 23 '18 at 13:05
LutzL
60.1k42057
asked Dec 23 '18 at 12:18
zakaria hamidizakaria hamidi
142
edited Dec 23 '18 at 13:05
LutzL
60.1k42057
edited Dec 23 '18 at 13:05
LutzL
60.1k42057
edited Dec 23 '18 at 13:05
LutzL
60.1k42057
60.1k42057
asked Dec 23 '18 at 12:18
zakaria hamidizakaria hamidi
142
asked Dec 23 '18 at 12:18
zakaria hamidizakaria hamidi
142
asked Dec 23 '18 at 12:18
zakaria hamidizakaria hamidi
142
142
$begingroup$
You are missing some condition that excludes the trivial solution $z=0$. Is it to be assumed that $z(x,t)=0$ for all $t<0$? Did you solve similar problems before, esp. DE with delays, when yes, describe the methods used. What would be your first approach and how did it fail?
$endgroup$
– LutzL
Dec 23 '18 at 12:22
$begingroup$
thank you Lutzl. I would like to inform you that I am a beginner on the use of numerical methods,
$endgroup$
– zakaria hamidi
Dec 23 '18 at 12:48
1
$begingroup$
If you are just a beginner in numerical methods, then it is rather courageous to start with (second order) delay-differential equations. The easiest you can do is set $h=r/n$ and apply the Euler method to the first-order (in $t$) system $$∂_tz=w+vz(.,t-r),~~ ∂_tw=∂_x^2z$$ as in $$w_{k+1} = w_k+h∂_x^2z_k,~~ z_{k+1}=z_k+hw_k+v_kz_{k-n}.$$ In $x$ direction you apply some discretization or Fourier expansion and realize $∂_x^2$ according to the framework.
$endgroup$
– LutzL
Dec 23 '18 at 13:14
add a comment |
$begingroup$
You are missing some condition that excludes the trivial solution $z=0$. Is it to be assumed that $z(x,t)=0$ for all $t<0$? Did you solve similar problems before, esp. DE with delays, when yes, describe the methods used. What would be your first approach and how did it fail?
$endgroup$
– LutzL
Dec 23 '18 at 12:22
$begingroup$
thank you Lutzl. I would like to inform you that I am a beginner on the use of numerical methods,
$endgroup$
– zakaria hamidi
Dec 23 '18 at 12:48
1
$begingroup$
If you are just a beginner in numerical methods, then it is rather courageous to start with (second order) delay-differential equations. The easiest you can do is set $h=r/n$ and apply the Euler method to the first-order (in $t$) system $$∂_tz=w+vz(.,t-r),~~ ∂_tw=∂_x^2z$$ as in $$w_{k+1} = w_k+h∂_x^2z_k,~~ z_{k+1}=z_k+hw_k+v_kz_{k-n}.$$ In $x$ direction you apply some discretization or Fourier expansion and realize $∂_x^2$ according to the framework.
$endgroup$
– LutzL
Dec 23 '18 at 13:14
$begingroup$
You are missing some condition that excludes the trivial solution $z=0$. Is it to be assumed that $z(x,t)=0$ for all $t<0$? Did you solve similar problems before, esp. DE with delays, when yes, describe the methods used. What would be your first approach and how did it fail?
$endgroup$
– LutzL
Dec 23 '18 at 12:22
$begingroup$
You are missing some condition that excludes the trivial solution $z=0$. Is it to be assumed that $z(x,t)=0$ for all $t<0$? Did you solve similar problems before, esp. DE with delays, when yes, describe the methods used. What would be your first approach and how did it fail?
$endgroup$
– LutzL
Dec 23 '18 at 12:22
$begingroup$
thank you Lutzl. I would like to inform you that I am a beginner on the use of numerical methods,
$endgroup$
– zakaria hamidi
Dec 23 '18 at 12:48
$begingroup$
thank you Lutzl. I would like to inform you that I am a beginner on the use of numerical methods,
$endgroup$
– zakaria hamidi
Dec 23 '18 at 12:48
1
1
$begingroup$
If you are just a beginner in numerical methods, then it is rather courageous to start with (second order) delay-differential equations. The easiest you can do is set $h=r/n$ and apply the Euler method to the first-order (in $t$) system $$∂_tz=w+vz(.,t-r),~~ ∂_tw=∂_x^2z$$ as in $$w_{k+1} = w_k+h∂_x^2z_k,~~ z_{k+1}=z_k+hw_k+v_kz_{k-n}.$$ In $x$ direction you apply some discretization or Fourier expansion and realize $∂_x^2$ according to the framework.
$endgroup$
– LutzL
Dec 23 '18 at 13:14
$begingroup$
If you are just a beginner in numerical methods, then it is rather courageous to start with (second order) delay-differential equations. The easiest you can do is set $h=r/n$ and apply the Euler method to the first-order (in $t$) system $$∂_tz=w+vz(.,t-r),~~ ∂_tw=∂_x^2z$$ as in $$w_{k+1} = w_k+h∂_x^2z_k,~~ z_{k+1}=z_k+hw_k+v_kz_{k-n}.$$ In $x$ direction you apply some discretization or Fourier expansion and realize $∂_x^2$ according to the framework.
$endgroup$
– LutzL
Dec 23 '18 at 13:14
add a comment |
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$begingroup$
You are missing some condition that excludes the trivial solution $z=0$. Is it to be assumed that $z(x,t)=0$ for all $t<0$? Did you solve similar problems before, esp. DE with delays, when yes, describe the methods used. What would be your first approach and how did it fail?
$endgroup$
– LutzL
Dec 23 '18 at 12:22
$begingroup$
thank you Lutzl. I would like to inform you that I am a beginner on the use of numerical methods,
$endgroup$
– zakaria hamidi
Dec 23 '18 at 12:48
1
$begingroup$
If you are just a beginner in numerical methods, then it is rather courageous to start with (second order) delay-differential equations. The easiest you can do is set $h=r/n$ and apply the Euler method to the first-order (in $t$) system $$∂_tz=w+vz(.,t-r),~~ ∂_tw=∂_x^2z$$ as in $$w_{k+1} = w_k+h∂_x^2z_k,~~ z_{k+1}=z_k+hw_k+v_kz_{k-n}.$$ In $x$ direction you apply some discretization or Fourier expansion and realize $∂_x^2$ according to the framework.
$endgroup$
– LutzL
Dec 23 '18 at 13:14