## Which method is used for solving heat equation?

The Heat equation is a partial differential equation that describes the variation of temperature in a given region over a period of time. Traditionally, the heat equations are often solved by classic methods such as Separation of variables and Fourier series methods.

## What is method of characteristics and why it is needed?

The method of characteristics is a technique for solving hyperbolic partial differential equa- tions (PDE). Typically the method applies to first-order equations, although it is valid for any 3 Page 4 hyperbolic-type PDEs.

**How do you find the characteristic equation in PDE?**

We can use ODE theory to solve the characteristic equations, then piece together these characteristic curves to form a surface. Such a surface will provide us with a solution to our PDE. x(s) = as + c1 t(s) = s + c2 z(s) = c3.

### When can you use method of characteristics?

In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation.

### Which is the correct description of the heat equation?

The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.

**How are diffusive processes governed by the heat equation?**

The heat equation governs heat diffusion, as well as other diffusive processes, such as particle diffusion or the propagation of action potential in nerve cells. Although they are not diffusive in nature, some quantum mechanics problems are also governed by a mathematical analog of the heat equation (see below).

#### How is the evolution of temperature predicted by the heat equation?

Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. The height and redness indicate the temperature at each point. The initial state has a uniformly hot hoof-shaped region (red) surrounded by uniformly cold region (yellow). As time passes the heat diffuses into the cold region.

#### How is the heat equation related to probability theory?

In probability theory, the heat equation is connected with the study of random walks and Brownian motion via the Fokker–Planck equation. The Black–Scholes equation of financial mathematics is a small variant of the heat equation, and the Schrödinger equation of quantum mechanics can be regarded as a heat equation in imaginary time.