partial differential equations
While searching for a quantitative description of physical phenomena, the engineer establishes a system of ordinary or partial differential equations — valid in a certain region (or domain) — and imposes on this system suitable boundary and initial conditions. A partial differential equation (PDE) is an equation, involving an unknown function of two or more variables and certain of its partial derivatives. A PDE can be solved numerically with various methods, such as finite difference method, finite volume method, finite element method, spectral method, meshfree method, domain...
The Black-Scholes equation is a parabolic partial differential equation (PDE) for option prices discovered by Black and Scholes in 1973. This discovery revolutionized financial markets in 1990-s. In 1997 Merton and Scholes were awarded the Nobel prize in economics. For any stock price s,
\(0<s<\infty\)
and time t,
\(0<t<T\)
the price u for an option expiring at time T satisfies the following Black-Scholes PDE:
\(\frac{\partial u}{\partial t} + \frac{1}{2} s^2 \sigma^2(s,t)\frac{\partial^2 u}{\partial s^2} + s\mu\frac{\partial u}{\...
Partial differential equations (PDEs) are used widely for the modeling of various physical phenomena. In simple case one can find symbolic solutions to some PDEs. The solution can then be described by means of either additive or multiplicative separable solutions. For the symbolic calculus needed, SymPy is being used - a python module for symbolic mathematics.
Ιn the first example we are going to consider additive separable solutions of the PDE. Consider an equation of two independent variables x, y, and a dependent variable w.
w(x,y,z) = X(x) + u(y,z...
