# Mixed Derivative in Differential Equations: Analytical and Numerical Solutions

• Gonzolo
In summary, the conversation is discussing a differential equation with mixed derivatives and the possibility of solving it analytically or numerically. The equation is shown with constants and known variables, and there is a suggestion to simplify the problem by converting the mixed derivatives to non-mixed derivatives. The conversation ends with a request for a method to solve the equation.
Gonzolo
Hi, has anyone here ever seen anything like this?

f = f(x,y,t)
g = g(x,y,t)

$$\frac{ \partial^{2}{f} }{ \partial {y}\partial {t} } } + \frac{ \partial^{2}{f} }{ \partial {x}\partial {t} } }+ \frac{\partial{f}}{\partial{t}}+ \frac{\partial{f}}{\partial{y}}+ \frac{\partial{f}}{\partial{x}}+f = g$$

Personnally, my blood pressure has begun to drop dangerously. You can drop in a real constant by each term. Any info or sites on DE's with mixed derivatives would be helpful. Can any DE with mixed derivatives at all be solved analytically? A good direction for numerical solving would be helpful too.

If it's as satanic as it first seems, I'll probably do approximations to simplify my model.

Thanks.

Actually, the units don't work without the constants, so it should be :

$$C_1\frac{ \partial^{2}{f} }{ \partial {y}\partial {t} } } + C_2\frac{ \partial^{2}{f} }{ \partial {x}\partial {t} } }+C_3\frac{\partial{f}}{\partial{t}}+C_4\frac{\partial{f}}{\partial{y}}+C_5\frac{\partial{f}}{\partial{x}}+C_6f = g$$

g(x,y,t) is known
Finding f(x,y,t) is the problem.

It is always possible, by rotating the coordinate system, to convert mixed derivatives to 'non-mixed' derivatives. One way is to set up the second derivative coefficients as a matrix and find the eigen-vectors. Those should be the new axes.

Yea, Gonzolo, I'd like to see that one solved too. However, I think a good approach is to first analyze it in just 2-D. You know, look first at f(x,y) and g(x,y):

$$\frac{\partial^2f}{\partial x\partial y}+\frac{\partial f}{\partial x}+f=g$$

I mean, just any solution, any initial condition, any boundary conditions. Can anyone here propose a method for solving this one?

## 1. What is a mixed derivative in differential equations?

A mixed derivative in differential equations refers to a second-order partial derivative where the variables in the equation are of different types. In other words, one variable is being differentiated with respect to another variable that is also being differentiated with respect to a third variable.

## 2. How is a mixed derivative represented in mathematical notation?

A mixed derivative is typically represented using a combination of notations, such as f''(x,y) or fxy(x,y), depending on the specific equation and context. The variables x and y indicate which variables are being differentiated with respect to, and the number of primes or subscripts indicates the order of the derivative.

## 3. What is the difference between analytical and numerical solutions for mixed derivatives?

An analytical solution for a mixed derivative involves finding an exact mathematical expression for the derivative, while a numerical solution involves using numerical methods, such as finite differences or Euler's method, to approximate the value of the derivative at specific points.

## 4. Can mixed derivatives be solved using both analytical and numerical methods?

Yes, mixed derivatives can be solved using both analytical and numerical methods. However, the approach and level of complexity may vary depending on the specific equation and the desired level of accuracy.

## 5. Why are mixed derivatives important in differential equations?

Mixed derivatives are important in differential equations because they allow us to study the relationship between multiple variables in a single equation. They also play a crucial role in many physical and scientific phenomena, such as fluid flow, heat transfer, and elasticity.

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