Poisson Equation with Boundary Value Problem
I am trying to learn about solving boundary value problems, but I stuck when I came to finding BVP of $1$D Poisson equation on $[0,1]$:
$$
begin{cases}
dfrac{mathrm{d}^2}{mathrm{d}x^2}u(x)=-g(x) \
u(0)= a \
u(1)= b
end{cases}
$$
where $g(x)=−6pi cos(3pi x)+9pi^2x sin(3pi x)$ and $a=b=0$.
Should I get the integral of given $g$ two times and then plug in the $u(0)$ and $u(1)$ values to find out the values of the two integration constants $c_1$ and $c_2$? Or am I supposed to do something else? Also, how can I start solving it numerically by discretization for any $h$ value? Any hint or tip is appreciated.
differential-equations boundary-value-problem poissons-equation
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
asked Oct 31 '18 at 20:34
enesenes
245
add a comment |
I am trying to learn about solving boundary value problems, but I stuck when I came to finding BVP of $1$D Poisson equation on $[0,1]$:
$$
begin{cases}
dfrac{mathrm{d}^2}{mathrm{d}x^2}u(x)=-g(x) \
u(0)= a \
u(1)= b
end{cases}
$$
where $g(x)=−6pi cos(3pi x)+9pi^2x sin(3pi x)$ and $a=b=0$.
Should I get the integral of given $g$ two times and then plug in the $u(0)$ and $u(1)$ values to find out the values of the two integration constants $c_1$ and $c_2$? Or am I supposed to do something else? Also, how can I start solving it numerically by discretization for any $h$ value? Any hint or tip is appreciated.
differential-equations boundary-value-problem poissons-equation
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
asked Oct 31 '18 at 20:34
enesenes
245
add a comment |
I am trying to learn about solving boundary value problems, but I stuck when I came to finding BVP of $1$D Poisson equation on $[0,1]$:
$$
begin{cases}
dfrac{mathrm{d}^2}{mathrm{d}x^2}u(x)=-g(x) \
u(0)= a \
u(1)= b
end{cases}
$$
where $g(x)=−6pi cos(3pi x)+9pi^2x sin(3pi x)$ and $a=b=0$.
Should I get the integral of given $g$ two times and then plug in the $u(0)$ and $u(1)$ values to find out the values of the two integration constants $c_1$ and $c_2$? Or am I supposed to do something else? Also, how can I start solving it numerically by discretization for any $h$ value? Any hint or tip is appreciated.
differential-equations boundary-value-problem poissons-equation
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
asked Oct 31 '18 at 20:34
enesenes
245
I am trying to learn about solving boundary value problems, but I stuck when I came to finding BVP of $1$D Poisson equation on $[0,1]$:
$$
begin{cases}
dfrac{mathrm{d}^2}{mathrm{d}x^2}u(x)=-g(x) \
u(0)= a \
u(1)= b
end{cases}
$$
where $g(x)=−6pi cos(3pi x)+9pi^2x sin(3pi x)$ and $a=b=0$.
Should I get the integral of given $g$ two times and then plug in the $u(0)$ and $u(1)$ values to find out the values of the two integration constants $c_1$ and $c_2$? Or am I supposed to do something else? Also, how can I start solving it numerically by discretization for any $h$ value? Any hint or tip is appreciated.
differential-equations boundary-value-problem poissons-equation
differential-equations boundary-value-problem poissons-equation
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
asked Oct 31 '18 at 20:34
enesenes
245
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
asked Oct 31 '18 at 20:34
enesenes
245
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
12.4k31026
asked Oct 31 '18 at 20:34
enesenes
245
asked Oct 31 '18 at 20:34
enesenes
245
asked Oct 31 '18 at 20:34
enesenes
245
245
add a comment |
add a comment |
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