Special Functions

Consider a straight rod along which there is a uniform flowof heat.

• Let $u(x, t)$ denote the temperature of the rod at time $t$ and location $x$.
• Let $q(x, t)$ denote the rate of heat flow.

The expression $\partial q/\partial x$ denotes the rate at which the rate of heat flow changes per unit length and therefore measures the rate at which heat is accumulating at a given point $x$ at time $t$. If heat is accumulating, the temperature at that point is rising, and the rate is denoted by $\partial u/\partial t$.

1	The principle of conservation of energy leads to $\partial q/\partial x=k\partial u/\partial t$, where k is the specific heat of the rod. This means that the rate at which heat is accumulating at a point is proportional to the rate at which the temperature is increasing.

2

A second relationship between $q$ and $u$ is obtained from Newton's law of cooling, which states that $q = K(\partial u/\partial x)$. Elimination of $q$ between these equations leads to $$\frac{\partial^2 u}{\partial x^2}=(k/K)\frac{\partial u}{\partial t}$$ the partial differential equation for one-dimensional heat flow. The partial differential equation for heat flow in three dimensions takes the form $$\boxed{\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial yx^2}=(k/K)\frac{\partial u}{\partial t}}$$ Often written as $$\boxed{\nabla^2 u=(k/K)\frac{\partial u}{\partial t}}$$ where the symbol $\nabla$, called del or nabla, is known as the Laplace operator.

• Another example to PDEs for dealing with wave propagation problem:

  $$\nabla^2 u=(1/c^2)\frac{\partial^2 u}{\partial t^2}$$

  where c is the speed at which the wave propagates.
• PDEs are harder to solve than ordinary differential equations (ODEs).
• However, the PDEs associated with wave propagation and heat flow can be reduced to a system of ODEs through a process known as separation of variables.
• These ODEs depend on the choice of coordinate system, which in turn is influenced by the physical configuration of the problem.
• The solutions of these ODEs form the majority of the special functions of mathematical physics.

• In the broad sense, a set of several classes of functions that arise in the solution of both theoretical and applied problems in various branches.
• In the narrow sense, the special functions of mathematical physics, which arise when solving PDEs by the method of separation of variables.
• Special functions can be defined by means of power series, generating functions, infinite products, repeated differentiation, integral representations, differential, difference, integral, and functional equations, trigonometric series, or other series in orthogonal functions.
• For example, in solving the equations of heat flow or wave propagation in cylindrical coordinates, the method of separation of variables leads to Bessel's differential equation, a solution of which is the Bessel function, denoted by $J_n(x)$.

• A polynomial ("many terms") is defined as an expression that consist of variables, coefficients and exponents.
• A polynomial can have:
  • variables (like x and y)
  • constants/coefficients (like 6, -10, or $^{3/2}$)
  • exponents (like the 2 in $y^2$)
  • that can be combined using addition, subtraction, multiplication and division
  • but not division by a variable (so something like 2/x is not correct)
  • a monomial is the product of non-negative powers of variables and will only have one term. 13, 3x, 4y$^2$, ...
  • a binomial is the sum of two monomials. 3x + 1, 2x + y, ...
  • a trinomial is the sum of three monomials. x$^2$ + 2x + 1, 2x + 3y + 2, ...
  • can have one or more terms, but not an infinite number of terms.
• The standard form of a polynomial refers to writing a polynomial in the descending power of the variable.
  $2x^3 - 4x^2 + 7x - 4$