Method of characteristics for $f f_x + f_y = 1$. Where is the solution valid?
$begingroup$
Suppose we have a PDE that can be solved with the method of characteristics
begin{align}
F(nabla u, u , x) = 0 text{ in $U$}\
u|_Gamma = g text{ on $Gamma$ }
end{align}
Where $Gamma subset partial U$. Suppose the characteristic curves starting on $Gamma$ don't span the whole set $U$, what can we say about the solution in this points outside the span? Do we lack information to solve the problem or can this be somehow avoided?
To have a precise example I solved the following problem,
$$f(x,y)frac{partial f}{partial x} + frac{partial f}{partial y}= 1 qquadtext{with}qquad f(t,t) = frac{t}{2} quadtext{for}quad 0 < t < 1.$$
In the above notation we have $Gamma = {(t,t): 0 < t < 1}$ and $g(t) = t/2$. I got the solution $f(x,y) = z(s) + g(x(0)) =z(s) + frac{x_0}{2} = y - frac{x - 1/2y^2}{2-y}$ where $y neq 2$, what can I say about the validity of my solution? Is it valid for $mathbb{R^2}$ or just for points $x, y$ such that the characteristic curve trough them, $(x(s),y(s))$ lies on $Gamma$ for $s=0$?
pde characteristics
$endgroup$
add a comment |
$begingroup$
Suppose we have a PDE that can be solved with the method of characteristics
begin{align}
F(nabla u, u , x) = 0 text{ in $U$}\
u|_Gamma = g text{ on $Gamma$ }
end{align}
Where $Gamma subset partial U$. Suppose the characteristic curves starting on $Gamma$ don't span the whole set $U$, what can we say about the solution in this points outside the span? Do we lack information to solve the problem or can this be somehow avoided?
To have a precise example I solved the following problem,
$$f(x,y)frac{partial f}{partial x} + frac{partial f}{partial y}= 1 qquadtext{with}qquad f(t,t) = frac{t}{2} quadtext{for}quad 0 < t < 1.$$
In the above notation we have $Gamma = {(t,t): 0 < t < 1}$ and $g(t) = t/2$. I got the solution $f(x,y) = z(s) + g(x(0)) =z(s) + frac{x_0}{2} = y - frac{x - 1/2y^2}{2-y}$ where $y neq 2$, what can I say about the validity of my solution? Is it valid for $mathbb{R^2}$ or just for points $x, y$ such that the characteristic curve trough them, $(x(s),y(s))$ lies on $Gamma$ for $s=0$?
pde characteristics
$endgroup$
add a comment |
$begingroup$
Suppose we have a PDE that can be solved with the method of characteristics
begin{align}
F(nabla u, u , x) = 0 text{ in $U$}\
u|_Gamma = g text{ on $Gamma$ }
end{align}
Where $Gamma subset partial U$. Suppose the characteristic curves starting on $Gamma$ don't span the whole set $U$, what can we say about the solution in this points outside the span? Do we lack information to solve the problem or can this be somehow avoided?
To have a precise example I solved the following problem,
$$f(x,y)frac{partial f}{partial x} + frac{partial f}{partial y}= 1 qquadtext{with}qquad f(t,t) = frac{t}{2} quadtext{for}quad 0 < t < 1.$$
In the above notation we have $Gamma = {(t,t): 0 < t < 1}$ and $g(t) = t/2$. I got the solution $f(x,y) = z(s) + g(x(0)) =z(s) + frac{x_0}{2} = y - frac{x - 1/2y^2}{2-y}$ where $y neq 2$, what can I say about the validity of my solution? Is it valid for $mathbb{R^2}$ or just for points $x, y$ such that the characteristic curve trough them, $(x(s),y(s))$ lies on $Gamma$ for $s=0$?
pde characteristics
$endgroup$
Suppose we have a PDE that can be solved with the method of characteristics
begin{align}
F(nabla u, u , x) = 0 text{ in $U$}\
u|_Gamma = g text{ on $Gamma$ }
end{align}
Where $Gamma subset partial U$. Suppose the characteristic curves starting on $Gamma$ don't span the whole set $U$, what can we say about the solution in this points outside the span? Do we lack information to solve the problem or can this be somehow avoided?
To have a precise example I solved the following problem,
$$f(x,y)frac{partial f}{partial x} + frac{partial f}{partial y}= 1 qquadtext{with}qquad f(t,t) = frac{t}{2} quadtext{for}quad 0 < t < 1.$$
In the above notation we have $Gamma = {(t,t): 0 < t < 1}$ and $g(t) = t/2$. I got the solution $f(x,y) = z(s) + g(x(0)) =z(s) + frac{x_0}{2} = y - frac{x - 1/2y^2}{2-y}$ where $y neq 2$, what can I say about the validity of my solution? Is it valid for $mathbb{R^2}$ or just for points $x, y$ such that the characteristic curve trough them, $(x(s),y(s))$ lies on $Gamma$ for $s=0$?
pde characteristics
pde characteristics
edited Dec 7 '18 at 18:12
Harry49
6,19331132
6,19331132
asked Dec 5 '18 at 10:41
h3h325h3h325
19410
19410
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
This PDE is a non-homogeneous inviscid Burgers' equation. Let us apply the method of characteristics:
$frac{text d y}{text ds} = 1$, letting $y(0) = t$, we know $y = t + s$
$frac{text d f}{text ds} = 1$, letting $f(0) = t/2$, we know $f = t/2 + s$
$frac{text d x}{text ds} = f$, letting $x(0) = t$, we know $x = t + s(t + s)/2$
Therefore, the characteristics are the curves $x = t + y(y-t)/2$ for $0<t<1$, along which $f$ is given by
$$
f(x,y) = y - frac{x-y^2/2}{2-y} , .
$$
One can note that this expression is not defined at $y=2$, where the denominator vanishes. The solution is only valid over the domain in yellow
$$
(x,y) in lbrace (t + y(y-t)/2, y), 0<t<1, y<2rbrace
$$
Indeed, we start following characteristic curves at some point $(t,t)$ with $0<t<1$. On the one hand, we can decrease $y$ without restriction. But on the other hand, we cannot increase $y$ further than $y=2^-$. At the point $(x,y) = (2,2)$, all characteristic curves coming from $lbrace(t,t), 0<t<1rbrace$ intersect. The classical solution collapses (similar case to this one).
The reason why the domain of validity is restricted to $y<2$ can be compared to the case of the Riccati ODE initial-value problem
$$
x'(t) = -x(t)^2 qquadtext{with}qquad x(0) = -1, .
$$
Indeed, the solution $x(t) = frac{1}{t-1}$ blows up at $t=1$ and cannot be used for larger times (even if the solution function is defined for $t>1$). Note that this simple ODE problem can be linked to the PDE problem by following a derivation by Lax, leading to Eq. (6.10) p. 36 of (1).
(1) P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, 1973 doi:10.1137/1.9781611970562.ch1
$endgroup$
$begingroup$
Why is the solution only valid on the domain in yellow when $f(x,y)$ is defined for all $x$, $y neq 2$ ? It seems to me that the validity of the solution still holds outside although we might lose unicity since the associated system of ODE does not have initial conditions.
$endgroup$
– h3h325
Dec 5 '18 at 15:45
$begingroup$
I understand why the solution is not valid for $y =2$, I also see why the idea behind the method should fail if the characteristic curves intersect. But as I see it for $y>2$ the function $f(x,y)$ satisfies the pde and also the initial condition, so why can we not take it as a solution? Thanks a lot for the answer by the way.
$endgroup$
– h3h325
Dec 5 '18 at 16:12
$begingroup$
I see! So the conclusion is that the solution still holds but we cannot guarantee uniqueness, right?
$endgroup$
– h3h325
Dec 5 '18 at 16:22
$begingroup$
Sorry if im beeing too picky but I still don't get your point. The system of ODEs does have a solution by Picard-Lindelöf (even if it is not unique), since 1, 1 and $f$ are lipschitz, for any $[S, infty)$, for some $S$ corresponding to $y >2$, so why are we losing existence? Also $f(x,y)$ satisfies the PDE everywhere where it is defined, and also the initial conditions. So how is $f$ not a solution? Could you maybe refer me somewhere were this is explained?
$endgroup$
– h3h325
Dec 5 '18 at 16:43
1
$begingroup$
@h3h325 I'm troubled by your statement about Picard Lindelof, please note that it is a local existence and uniqueness theorem
$endgroup$
– qbert
Dec 7 '18 at 18:24
|
show 1 more comment
$begingroup$
$$f(x,y)frac{partial f}{partial x} + frac{partial f}{partial y}= 1 tag 1$$
Charpit-Lagrange equations :
$$frac{dx}{f}=frac{dy}{1}=frac{df}{1}$$
First family of characteristic curves, from $frac{dy}{1}=frac{df}{1}$ :
$$f-y=c_1$$
Second family of characteristic curves, from $frac{dx}{f}=frac{df}{1}$
$$frac12 f^2-x=c_2$$
General solution of the PDE expressed on the form of implicit equation :
$$frac12 f^2-x=Phi(f-y) tag 2$$
Where $Phi$ is an arbitrary function, to be determined according to boundary condition.
Condition : $f(x,x)=frac{x}{2}$
$frac12 (frac{x}{2})^2-x=Phi(frac{x}{2}-x)quad;quad frac{x^2}{8}-x=Phi(-frac{x}{2})$
Let $X=-frac{x}{2}quad;quad x=-2X$
$ frac{(-2X)^2}{8}-(-2X)=Phi(X)$
$$Phi(X)=frac12 X^2+2X$$
So, the function $Phi$ is determined. We put it into the general solution Eq.$(2)$ where $X=f-y$ :
$$frac12 f^2-x=frac12 (f-y)^2+2(f-y)$$
After simplification :
$$f(x,y)=frac{y^2-4y+2x}{2y-4}qquad yneq 2.$$
This is the solution of the PDE which satisfies the boundary condition.
One can check that this solution is valid in putting it into the PDE.
$endgroup$
add a comment |
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2 Answers
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2 Answers
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active
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$begingroup$
This PDE is a non-homogeneous inviscid Burgers' equation. Let us apply the method of characteristics:
$frac{text d y}{text ds} = 1$, letting $y(0) = t$, we know $y = t + s$
$frac{text d f}{text ds} = 1$, letting $f(0) = t/2$, we know $f = t/2 + s$
$frac{text d x}{text ds} = f$, letting $x(0) = t$, we know $x = t + s(t + s)/2$
Therefore, the characteristics are the curves $x = t + y(y-t)/2$ for $0<t<1$, along which $f$ is given by
$$
f(x,y) = y - frac{x-y^2/2}{2-y} , .
$$
One can note that this expression is not defined at $y=2$, where the denominator vanishes. The solution is only valid over the domain in yellow
$$
(x,y) in lbrace (t + y(y-t)/2, y), 0<t<1, y<2rbrace
$$
Indeed, we start following characteristic curves at some point $(t,t)$ with $0<t<1$. On the one hand, we can decrease $y$ without restriction. But on the other hand, we cannot increase $y$ further than $y=2^-$. At the point $(x,y) = (2,2)$, all characteristic curves coming from $lbrace(t,t), 0<t<1rbrace$ intersect. The classical solution collapses (similar case to this one).
The reason why the domain of validity is restricted to $y<2$ can be compared to the case of the Riccati ODE initial-value problem
$$
x'(t) = -x(t)^2 qquadtext{with}qquad x(0) = -1, .
$$
Indeed, the solution $x(t) = frac{1}{t-1}$ blows up at $t=1$ and cannot be used for larger times (even if the solution function is defined for $t>1$). Note that this simple ODE problem can be linked to the PDE problem by following a derivation by Lax, leading to Eq. (6.10) p. 36 of (1).
(1) P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, 1973 doi:10.1137/1.9781611970562.ch1
$endgroup$
$begingroup$
Why is the solution only valid on the domain in yellow when $f(x,y)$ is defined for all $x$, $y neq 2$ ? It seems to me that the validity of the solution still holds outside although we might lose unicity since the associated system of ODE does not have initial conditions.
$endgroup$
– h3h325
Dec 5 '18 at 15:45
$begingroup$
I understand why the solution is not valid for $y =2$, I also see why the idea behind the method should fail if the characteristic curves intersect. But as I see it for $y>2$ the function $f(x,y)$ satisfies the pde and also the initial condition, so why can we not take it as a solution? Thanks a lot for the answer by the way.
$endgroup$
– h3h325
Dec 5 '18 at 16:12
$begingroup$
I see! So the conclusion is that the solution still holds but we cannot guarantee uniqueness, right?
$endgroup$
– h3h325
Dec 5 '18 at 16:22
$begingroup$
Sorry if im beeing too picky but I still don't get your point. The system of ODEs does have a solution by Picard-Lindelöf (even if it is not unique), since 1, 1 and $f$ are lipschitz, for any $[S, infty)$, for some $S$ corresponding to $y >2$, so why are we losing existence? Also $f(x,y)$ satisfies the PDE everywhere where it is defined, and also the initial conditions. So how is $f$ not a solution? Could you maybe refer me somewhere were this is explained?
$endgroup$
– h3h325
Dec 5 '18 at 16:43
1
$begingroup$
@h3h325 I'm troubled by your statement about Picard Lindelof, please note that it is a local existence and uniqueness theorem
$endgroup$
– qbert
Dec 7 '18 at 18:24
|
show 1 more comment
$begingroup$
This PDE is a non-homogeneous inviscid Burgers' equation. Let us apply the method of characteristics:
$frac{text d y}{text ds} = 1$, letting $y(0) = t$, we know $y = t + s$
$frac{text d f}{text ds} = 1$, letting $f(0) = t/2$, we know $f = t/2 + s$
$frac{text d x}{text ds} = f$, letting $x(0) = t$, we know $x = t + s(t + s)/2$
Therefore, the characteristics are the curves $x = t + y(y-t)/2$ for $0<t<1$, along which $f$ is given by
$$
f(x,y) = y - frac{x-y^2/2}{2-y} , .
$$
One can note that this expression is not defined at $y=2$, where the denominator vanishes. The solution is only valid over the domain in yellow
$$
(x,y) in lbrace (t + y(y-t)/2, y), 0<t<1, y<2rbrace
$$
Indeed, we start following characteristic curves at some point $(t,t)$ with $0<t<1$. On the one hand, we can decrease $y$ without restriction. But on the other hand, we cannot increase $y$ further than $y=2^-$. At the point $(x,y) = (2,2)$, all characteristic curves coming from $lbrace(t,t), 0<t<1rbrace$ intersect. The classical solution collapses (similar case to this one).
The reason why the domain of validity is restricted to $y<2$ can be compared to the case of the Riccati ODE initial-value problem
$$
x'(t) = -x(t)^2 qquadtext{with}qquad x(0) = -1, .
$$
Indeed, the solution $x(t) = frac{1}{t-1}$ blows up at $t=1$ and cannot be used for larger times (even if the solution function is defined for $t>1$). Note that this simple ODE problem can be linked to the PDE problem by following a derivation by Lax, leading to Eq. (6.10) p. 36 of (1).
(1) P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, 1973 doi:10.1137/1.9781611970562.ch1
$endgroup$
$begingroup$
Why is the solution only valid on the domain in yellow when $f(x,y)$ is defined for all $x$, $y neq 2$ ? It seems to me that the validity of the solution still holds outside although we might lose unicity since the associated system of ODE does not have initial conditions.
$endgroup$
– h3h325
Dec 5 '18 at 15:45
$begingroup$
I understand why the solution is not valid for $y =2$, I also see why the idea behind the method should fail if the characteristic curves intersect. But as I see it for $y>2$ the function $f(x,y)$ satisfies the pde and also the initial condition, so why can we not take it as a solution? Thanks a lot for the answer by the way.
$endgroup$
– h3h325
Dec 5 '18 at 16:12
$begingroup$
I see! So the conclusion is that the solution still holds but we cannot guarantee uniqueness, right?
$endgroup$
– h3h325
Dec 5 '18 at 16:22
$begingroup$
Sorry if im beeing too picky but I still don't get your point. The system of ODEs does have a solution by Picard-Lindelöf (even if it is not unique), since 1, 1 and $f$ are lipschitz, for any $[S, infty)$, for some $S$ corresponding to $y >2$, so why are we losing existence? Also $f(x,y)$ satisfies the PDE everywhere where it is defined, and also the initial conditions. So how is $f$ not a solution? Could you maybe refer me somewhere were this is explained?
$endgroup$
– h3h325
Dec 5 '18 at 16:43
1
$begingroup$
@h3h325 I'm troubled by your statement about Picard Lindelof, please note that it is a local existence and uniqueness theorem
$endgroup$
– qbert
Dec 7 '18 at 18:24
|
show 1 more comment
$begingroup$
This PDE is a non-homogeneous inviscid Burgers' equation. Let us apply the method of characteristics:
$frac{text d y}{text ds} = 1$, letting $y(0) = t$, we know $y = t + s$
$frac{text d f}{text ds} = 1$, letting $f(0) = t/2$, we know $f = t/2 + s$
$frac{text d x}{text ds} = f$, letting $x(0) = t$, we know $x = t + s(t + s)/2$
Therefore, the characteristics are the curves $x = t + y(y-t)/2$ for $0<t<1$, along which $f$ is given by
$$
f(x,y) = y - frac{x-y^2/2}{2-y} , .
$$
One can note that this expression is not defined at $y=2$, where the denominator vanishes. The solution is only valid over the domain in yellow
$$
(x,y) in lbrace (t + y(y-t)/2, y), 0<t<1, y<2rbrace
$$
Indeed, we start following characteristic curves at some point $(t,t)$ with $0<t<1$. On the one hand, we can decrease $y$ without restriction. But on the other hand, we cannot increase $y$ further than $y=2^-$. At the point $(x,y) = (2,2)$, all characteristic curves coming from $lbrace(t,t), 0<t<1rbrace$ intersect. The classical solution collapses (similar case to this one).
The reason why the domain of validity is restricted to $y<2$ can be compared to the case of the Riccati ODE initial-value problem
$$
x'(t) = -x(t)^2 qquadtext{with}qquad x(0) = -1, .
$$
Indeed, the solution $x(t) = frac{1}{t-1}$ blows up at $t=1$ and cannot be used for larger times (even if the solution function is defined for $t>1$). Note that this simple ODE problem can be linked to the PDE problem by following a derivation by Lax, leading to Eq. (6.10) p. 36 of (1).
(1) P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, 1973 doi:10.1137/1.9781611970562.ch1
$endgroup$
This PDE is a non-homogeneous inviscid Burgers' equation. Let us apply the method of characteristics:
$frac{text d y}{text ds} = 1$, letting $y(0) = t$, we know $y = t + s$
$frac{text d f}{text ds} = 1$, letting $f(0) = t/2$, we know $f = t/2 + s$
$frac{text d x}{text ds} = f$, letting $x(0) = t$, we know $x = t + s(t + s)/2$
Therefore, the characteristics are the curves $x = t + y(y-t)/2$ for $0<t<1$, along which $f$ is given by
$$
f(x,y) = y - frac{x-y^2/2}{2-y} , .
$$
One can note that this expression is not defined at $y=2$, where the denominator vanishes. The solution is only valid over the domain in yellow
$$
(x,y) in lbrace (t + y(y-t)/2, y), 0<t<1, y<2rbrace
$$
Indeed, we start following characteristic curves at some point $(t,t)$ with $0<t<1$. On the one hand, we can decrease $y$ without restriction. But on the other hand, we cannot increase $y$ further than $y=2^-$. At the point $(x,y) = (2,2)$, all characteristic curves coming from $lbrace(t,t), 0<t<1rbrace$ intersect. The classical solution collapses (similar case to this one).
The reason why the domain of validity is restricted to $y<2$ can be compared to the case of the Riccati ODE initial-value problem
$$
x'(t) = -x(t)^2 qquadtext{with}qquad x(0) = -1, .
$$
Indeed, the solution $x(t) = frac{1}{t-1}$ blows up at $t=1$ and cannot be used for larger times (even if the solution function is defined for $t>1$). Note that this simple ODE problem can be linked to the PDE problem by following a derivation by Lax, leading to Eq. (6.10) p. 36 of (1).
(1) P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, 1973 doi:10.1137/1.9781611970562.ch1
edited Dec 8 '18 at 11:05
answered Dec 5 '18 at 14:09
Harry49Harry49
6,19331132
6,19331132
$begingroup$
Why is the solution only valid on the domain in yellow when $f(x,y)$ is defined for all $x$, $y neq 2$ ? It seems to me that the validity of the solution still holds outside although we might lose unicity since the associated system of ODE does not have initial conditions.
$endgroup$
– h3h325
Dec 5 '18 at 15:45
$begingroup$
I understand why the solution is not valid for $y =2$, I also see why the idea behind the method should fail if the characteristic curves intersect. But as I see it for $y>2$ the function $f(x,y)$ satisfies the pde and also the initial condition, so why can we not take it as a solution? Thanks a lot for the answer by the way.
$endgroup$
– h3h325
Dec 5 '18 at 16:12
$begingroup$
I see! So the conclusion is that the solution still holds but we cannot guarantee uniqueness, right?
$endgroup$
– h3h325
Dec 5 '18 at 16:22
$begingroup$
Sorry if im beeing too picky but I still don't get your point. The system of ODEs does have a solution by Picard-Lindelöf (even if it is not unique), since 1, 1 and $f$ are lipschitz, for any $[S, infty)$, for some $S$ corresponding to $y >2$, so why are we losing existence? Also $f(x,y)$ satisfies the PDE everywhere where it is defined, and also the initial conditions. So how is $f$ not a solution? Could you maybe refer me somewhere were this is explained?
$endgroup$
– h3h325
Dec 5 '18 at 16:43
1
$begingroup$
@h3h325 I'm troubled by your statement about Picard Lindelof, please note that it is a local existence and uniqueness theorem
$endgroup$
– qbert
Dec 7 '18 at 18:24
|
show 1 more comment
$begingroup$
Why is the solution only valid on the domain in yellow when $f(x,y)$ is defined for all $x$, $y neq 2$ ? It seems to me that the validity of the solution still holds outside although we might lose unicity since the associated system of ODE does not have initial conditions.
$endgroup$
– h3h325
Dec 5 '18 at 15:45
$begingroup$
I understand why the solution is not valid for $y =2$, I also see why the idea behind the method should fail if the characteristic curves intersect. But as I see it for $y>2$ the function $f(x,y)$ satisfies the pde and also the initial condition, so why can we not take it as a solution? Thanks a lot for the answer by the way.
$endgroup$
– h3h325
Dec 5 '18 at 16:12
$begingroup$
I see! So the conclusion is that the solution still holds but we cannot guarantee uniqueness, right?
$endgroup$
– h3h325
Dec 5 '18 at 16:22
$begingroup$
Sorry if im beeing too picky but I still don't get your point. The system of ODEs does have a solution by Picard-Lindelöf (even if it is not unique), since 1, 1 and $f$ are lipschitz, for any $[S, infty)$, for some $S$ corresponding to $y >2$, so why are we losing existence? Also $f(x,y)$ satisfies the PDE everywhere where it is defined, and also the initial conditions. So how is $f$ not a solution? Could you maybe refer me somewhere were this is explained?
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– h3h325
Dec 5 '18 at 16:43
1
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@h3h325 I'm troubled by your statement about Picard Lindelof, please note that it is a local existence and uniqueness theorem
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– qbert
Dec 7 '18 at 18:24
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Why is the solution only valid on the domain in yellow when $f(x,y)$ is defined for all $x$, $y neq 2$ ? It seems to me that the validity of the solution still holds outside although we might lose unicity since the associated system of ODE does not have initial conditions.
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– h3h325
Dec 5 '18 at 15:45
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Why is the solution only valid on the domain in yellow when $f(x,y)$ is defined for all $x$, $y neq 2$ ? It seems to me that the validity of the solution still holds outside although we might lose unicity since the associated system of ODE does not have initial conditions.
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– h3h325
Dec 5 '18 at 15:45
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I understand why the solution is not valid for $y =2$, I also see why the idea behind the method should fail if the characteristic curves intersect. But as I see it for $y>2$ the function $f(x,y)$ satisfies the pde and also the initial condition, so why can we not take it as a solution? Thanks a lot for the answer by the way.
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– h3h325
Dec 5 '18 at 16:12
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I understand why the solution is not valid for $y =2$, I also see why the idea behind the method should fail if the characteristic curves intersect. But as I see it for $y>2$ the function $f(x,y)$ satisfies the pde and also the initial condition, so why can we not take it as a solution? Thanks a lot for the answer by the way.
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– h3h325
Dec 5 '18 at 16:12
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I see! So the conclusion is that the solution still holds but we cannot guarantee uniqueness, right?
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– h3h325
Dec 5 '18 at 16:22
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I see! So the conclusion is that the solution still holds but we cannot guarantee uniqueness, right?
$endgroup$
– h3h325
Dec 5 '18 at 16:22
$begingroup$
Sorry if im beeing too picky but I still don't get your point. The system of ODEs does have a solution by Picard-Lindelöf (even if it is not unique), since 1, 1 and $f$ are lipschitz, for any $[S, infty)$, for some $S$ corresponding to $y >2$, so why are we losing existence? Also $f(x,y)$ satisfies the PDE everywhere where it is defined, and also the initial conditions. So how is $f$ not a solution? Could you maybe refer me somewhere were this is explained?
$endgroup$
– h3h325
Dec 5 '18 at 16:43
$begingroup$
Sorry if im beeing too picky but I still don't get your point. The system of ODEs does have a solution by Picard-Lindelöf (even if it is not unique), since 1, 1 and $f$ are lipschitz, for any $[S, infty)$, for some $S$ corresponding to $y >2$, so why are we losing existence? Also $f(x,y)$ satisfies the PDE everywhere where it is defined, and also the initial conditions. So how is $f$ not a solution? Could you maybe refer me somewhere were this is explained?
$endgroup$
– h3h325
Dec 5 '18 at 16:43
1
1
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@h3h325 I'm troubled by your statement about Picard Lindelof, please note that it is a local existence and uniqueness theorem
$endgroup$
– qbert
Dec 7 '18 at 18:24
$begingroup$
@h3h325 I'm troubled by your statement about Picard Lindelof, please note that it is a local existence and uniqueness theorem
$endgroup$
– qbert
Dec 7 '18 at 18:24
|
show 1 more comment
$begingroup$
$$f(x,y)frac{partial f}{partial x} + frac{partial f}{partial y}= 1 tag 1$$
Charpit-Lagrange equations :
$$frac{dx}{f}=frac{dy}{1}=frac{df}{1}$$
First family of characteristic curves, from $frac{dy}{1}=frac{df}{1}$ :
$$f-y=c_1$$
Second family of characteristic curves, from $frac{dx}{f}=frac{df}{1}$
$$frac12 f^2-x=c_2$$
General solution of the PDE expressed on the form of implicit equation :
$$frac12 f^2-x=Phi(f-y) tag 2$$
Where $Phi$ is an arbitrary function, to be determined according to boundary condition.
Condition : $f(x,x)=frac{x}{2}$
$frac12 (frac{x}{2})^2-x=Phi(frac{x}{2}-x)quad;quad frac{x^2}{8}-x=Phi(-frac{x}{2})$
Let $X=-frac{x}{2}quad;quad x=-2X$
$ frac{(-2X)^2}{8}-(-2X)=Phi(X)$
$$Phi(X)=frac12 X^2+2X$$
So, the function $Phi$ is determined. We put it into the general solution Eq.$(2)$ where $X=f-y$ :
$$frac12 f^2-x=frac12 (f-y)^2+2(f-y)$$
After simplification :
$$f(x,y)=frac{y^2-4y+2x}{2y-4}qquad yneq 2.$$
This is the solution of the PDE which satisfies the boundary condition.
One can check that this solution is valid in putting it into the PDE.
$endgroup$
add a comment |
$begingroup$
$$f(x,y)frac{partial f}{partial x} + frac{partial f}{partial y}= 1 tag 1$$
Charpit-Lagrange equations :
$$frac{dx}{f}=frac{dy}{1}=frac{df}{1}$$
First family of characteristic curves, from $frac{dy}{1}=frac{df}{1}$ :
$$f-y=c_1$$
Second family of characteristic curves, from $frac{dx}{f}=frac{df}{1}$
$$frac12 f^2-x=c_2$$
General solution of the PDE expressed on the form of implicit equation :
$$frac12 f^2-x=Phi(f-y) tag 2$$
Where $Phi$ is an arbitrary function, to be determined according to boundary condition.
Condition : $f(x,x)=frac{x}{2}$
$frac12 (frac{x}{2})^2-x=Phi(frac{x}{2}-x)quad;quad frac{x^2}{8}-x=Phi(-frac{x}{2})$
Let $X=-frac{x}{2}quad;quad x=-2X$
$ frac{(-2X)^2}{8}-(-2X)=Phi(X)$
$$Phi(X)=frac12 X^2+2X$$
So, the function $Phi$ is determined. We put it into the general solution Eq.$(2)$ where $X=f-y$ :
$$frac12 f^2-x=frac12 (f-y)^2+2(f-y)$$
After simplification :
$$f(x,y)=frac{y^2-4y+2x}{2y-4}qquad yneq 2.$$
This is the solution of the PDE which satisfies the boundary condition.
One can check that this solution is valid in putting it into the PDE.
$endgroup$
add a comment |
$begingroup$
$$f(x,y)frac{partial f}{partial x} + frac{partial f}{partial y}= 1 tag 1$$
Charpit-Lagrange equations :
$$frac{dx}{f}=frac{dy}{1}=frac{df}{1}$$
First family of characteristic curves, from $frac{dy}{1}=frac{df}{1}$ :
$$f-y=c_1$$
Second family of characteristic curves, from $frac{dx}{f}=frac{df}{1}$
$$frac12 f^2-x=c_2$$
General solution of the PDE expressed on the form of implicit equation :
$$frac12 f^2-x=Phi(f-y) tag 2$$
Where $Phi$ is an arbitrary function, to be determined according to boundary condition.
Condition : $f(x,x)=frac{x}{2}$
$frac12 (frac{x}{2})^2-x=Phi(frac{x}{2}-x)quad;quad frac{x^2}{8}-x=Phi(-frac{x}{2})$
Let $X=-frac{x}{2}quad;quad x=-2X$
$ frac{(-2X)^2}{8}-(-2X)=Phi(X)$
$$Phi(X)=frac12 X^2+2X$$
So, the function $Phi$ is determined. We put it into the general solution Eq.$(2)$ where $X=f-y$ :
$$frac12 f^2-x=frac12 (f-y)^2+2(f-y)$$
After simplification :
$$f(x,y)=frac{y^2-4y+2x}{2y-4}qquad yneq 2.$$
This is the solution of the PDE which satisfies the boundary condition.
One can check that this solution is valid in putting it into the PDE.
$endgroup$
$$f(x,y)frac{partial f}{partial x} + frac{partial f}{partial y}= 1 tag 1$$
Charpit-Lagrange equations :
$$frac{dx}{f}=frac{dy}{1}=frac{df}{1}$$
First family of characteristic curves, from $frac{dy}{1}=frac{df}{1}$ :
$$f-y=c_1$$
Second family of characteristic curves, from $frac{dx}{f}=frac{df}{1}$
$$frac12 f^2-x=c_2$$
General solution of the PDE expressed on the form of implicit equation :
$$frac12 f^2-x=Phi(f-y) tag 2$$
Where $Phi$ is an arbitrary function, to be determined according to boundary condition.
Condition : $f(x,x)=frac{x}{2}$
$frac12 (frac{x}{2})^2-x=Phi(frac{x}{2}-x)quad;quad frac{x^2}{8}-x=Phi(-frac{x}{2})$
Let $X=-frac{x}{2}quad;quad x=-2X$
$ frac{(-2X)^2}{8}-(-2X)=Phi(X)$
$$Phi(X)=frac12 X^2+2X$$
So, the function $Phi$ is determined. We put it into the general solution Eq.$(2)$ where $X=f-y$ :
$$frac12 f^2-x=frac12 (f-y)^2+2(f-y)$$
After simplification :
$$f(x,y)=frac{y^2-4y+2x}{2y-4}qquad yneq 2.$$
This is the solution of the PDE which satisfies the boundary condition.
One can check that this solution is valid in putting it into the PDE.
answered Dec 8 '18 at 8:11
JJacquelinJJacquelin
43.5k21853
43.5k21853
add a comment |
add a comment |
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