# 1st order PDE, seems easy but still confusing

• Irid
In summary, the conversation discusses solving a PDE with two types of initial conditions using the Method of Characteristics. Two tentative solutions are presented, but the speaker realizes they do not satisfy the required initial conditions. Another speaker suggests separating the equation into two ordinary ones and solving them, resulting in a general solution with an arbitrary function. The final solution for the given initial condition is given.
Irid
Hello,
I am doing some physics and I end up with this PDE:

$$\frac{\partial q(x,y,t)}{\partial t} = -(x^2 + y^2)q(x,y,t) + ax\frac{\partial q(x,y,t)}{\partial y}$$
where q(x,y,t) is the scalar field to determine and a is a parameter. I need to consider two types of initial conditions: q(x,y,t=0) = 1; and q(x,y,t=0) = delta(x,y).

I have found two tentative solutions:

$$q(x,y,t) = C\exp \left(-\lambda s + \frac{y}{ax}(x^2 + y^2/3 - \lambda) \right)$$
where lambda is any (?) number. Another solution is

$$q(x,y,t) = C\exp \left(-sx^2 + -y^3/3ax\right)$$

They both seem to satisfy the PDE, but I can't make them satisfy the required initial condition (either 1 or delta function). Any ideas or experience with this kind of equations?

I don't see a 't' in either solution. Did you substitute it with s?

jaytech said:
I don't see a 't' in either solution. Did you substitute it with s?

Oops, sorry, I did.
Actually I think I figured out how to find the solution. It's by the Method of Characteristics. No more help needed...

Substitute $q(x,y,t)=X(x)Y(y)T(t)$ and treat x and X(x) as constants and then try to separate the equation to two ordinary ones.Then solve those two ODEs.The general answer is the linear combination of the answers to the ODEs but the coefficients will depend on x.

The general solution to your PDE is

$q(x,y,t) =C(x,axt+y)\exp(\frac{xy}{a}+\frac{y^3}{3ax})$,

where $C(x,y)$ is an arbitrary function.

So

$q(x,y,t) =δ(x,axt+y)\exp(-\frac{t(a^2t^2x^2+3atxy+3x^2+3y^2)}{3})$

is the solution with

$q(x,y,0) =δ(x,y)$.

## 1. What is a first-order PDE?

A first-order PDE (partial differential equation) is a mathematical equation that involves partial derivatives of a function with respect to one or more independent variables. These equations are used to describe physical phenomena in fields such as physics, engineering, and finance.

## 2. How do I solve a first-order PDE?

Solving a first-order PDE involves finding a function that satisfies the given equation and any given boundary conditions. This can be done using various techniques such as the method of characteristics, separation of variables, or using numerical methods.

## 3. What are some common types of first-order PDEs?

Some common types of first-order PDEs include linear and nonlinear equations, homogeneous and non-homogeneous equations, and hyperbolic, parabolic, and elliptic equations. These different types have distinct properties and require different solution methods.

## 4. What are some applications of first-order PDEs?

First-order PDEs have many applications in fields such as fluid dynamics, heat transfer, electromagnetism, and quantum mechanics. They are also used in mathematical modeling and data analysis in various industries.

## 5. Why do first-order PDEs seem easy but can still be confusing?

First-order PDEs can seem easy because they only involve partial derivatives with respect to one independent variable. However, they can still be confusing because they can have complex solutions and require advanced mathematical techniques to solve. Additionally, real-world problems often involve more than one PDE, making the overall problem more challenging.

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