# Verify that the formula provides a solution of the pde

• MHB
• evinda
In summary, you are trying to find the partial derivatives of $z$ with respect to $x$ and $t$, but you are not getting the desired result because the relation $x - z = ta(\phi(z))$ is different than the relation you are using. You need to go back to the original relation and apply the implicit function theorem.
evinda
Gold Member
MHB
Hello! (Wave)

I want to check by direct differentation that the formula $u(x,t)=\phi(z)$, where $z$ is given implicitly by $x-z=t a(\phi(z))$, does indeed provide a solution of the pde $u_t+a(u) u_x=0$.I have tried the following, but we do not get the desired result. Have I done something wrong?$x-z=ta\phi (z)\Rightarrow \phi (z)=\frac{x-z}{ta}$

$u(x,t)=\phi (z)=\frac{x-z}{ta}$

$u_t=-\frac{x-z}{at^2}$

$u_x=\frac{1}{ta}$

$u_t+a(u)u_x=-\frac{x-z}{at^2}+a\frac{1}{ta}=-\frac{x-z}{at^2}+\frac{at}{t^2a}=\frac{-x+z+at}{at^2}$.

evinda said:
Hello! (Wave)

I want to check by direct differentation that the formula $u(x,t)=\phi(z)$, where $z$ is given implicitly by $x-z=t a(\phi(z))$, does indeed provide a solution of the pde $u_t+a(u) u_x=0$.I have tried the following, but we do not get the desired result. Have I done something wrong?$x-z=ta\phi (z)\Rightarrow \phi (z)=\frac{x-z}{ta}$

You previously had the defining relation $x - z = ta(\phi(z))$, which is different. I think you need to go back to that relation and apply the implicit function theorem.

evinda said:
$u(x,t)=\phi (z)=\frac{x-z}{ta}$

$u_t=-\frac{x-z}{at^2}$

$u_x=\frac{1}{ta}$

$u_t+a(u)u_x=-\frac{x-z}{at^2}+a\frac{1}{ta}=-\frac{x-z}{at^2}+\frac{at}{t^2a}=\frac{-x+z+at}{at^2}$.

Hold on, I was going to explain.

So, I assume that $\phi$ is any smooth function of a real variable.

Define: $F : \mathbb{R^3} \to \mathbb{R}$ by
$F(x, t, z) := x - z - t a(\phi(z)).$
You want to use the relation $F(x, t, z) = 0$ to define $z$ as a function of $x$ and $t$. By the implicit function theorem, this can be done in a sufficiently small neighbourhood of any point $(x_0, t_0, z_0)$ for which it holds that $F(x_0, t_0, z_0) = 0$ and the derivative $D_zF(x_0, t_0, z_0) \neq 0$. So, near such a point we have $F(x, t, z(x, t)) = 0$, i.e.
$x - z(x,t) - t a(\phi(z(x,t))) = 0.$
Moreover, by the same theorem, you can find the derivatives of $z$ by implicit differentiation. Can you implicitly differentiate the above equality with respect to $x$ and $t$ to find the partial derivatives of $z$?

Once you have done this, note that $u_x(x,t) = \phi'(z(x,t)) D_xz(x,t)$ and $u_t(x,t) = \phi'(z(x,t)) D_tz(x,t)$, where the partials of $z$ that you just calculated appear again. Substitute for these and see if you can prove that $u$ indeed satisfies the PDE.

## 1. What is a PDE?

A PDE, or partial differential equation, is a mathematical equation that involves multiple variables and their partial derivatives. It is used to describe how a system changes over time and space.

## 2. How do you verify that a formula provides a solution of a PDE?

To verify that a formula provides a solution of a PDE, you must substitute the formula into the PDE and check if it satisfies the equation. This involves taking the partial derivatives of the formula and comparing it to the given PDE.

## 3. What are the steps for verifying a solution to a PDE?

The steps for verifying a solution to a PDE are: 1) Substitute the formula into the PDE, 2) Take the partial derivatives of the formula, 3) Substitute the derivatives into the PDE, and 4) Check if the resulting equation is true. If it is, then the formula is a solution of the PDE.

## 4. Can a PDE have multiple solutions?

Yes, a PDE can have multiple solutions. In fact, most PDEs have an infinite number of solutions. It is important to note that not all solutions may be physically meaningful or relevant to the problem being studied.

## 5. How do you know if a solution to a PDE is physically meaningful?

To determine if a solution to a PDE is physically meaningful, you must consider the initial and boundary conditions of the problem. The solution must satisfy these conditions in order to accurately describe the behavior of the system being studied.

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