A Partial Differential Equation is a Differential Equation which has more than one independent variable: u(x,y), u(t,x,y), … For instance:

Key Intuition PDEs may have no solutions (unlike Uniqueness and Existance for ODEs) yet, usually, there are too many solutions—so… how do you describe all solutions? usually, there are no explicit formulas Laplacian of u(x,y) Laplacian of u(x,y) Examples Heat Equation See Heat Equation Wave Equation see Wave Equation Transport Equation \begin{equation} \pdv{u}{t} = \pdv{u}{x} \end{equation} generally any u = w(x+t) should solve this Schrodinger Equation We have some:

and its a complex-valued function:

which results in a superposition in linear equations Nonlinear Example \begin{equation} \pdv{u}{t} = \pdv[2]{u}{x} + u(1-u) \end{equation} this is a PDE variant of the logistic equation: this is non-linear Monge-Ampere Equations \begin{equation} u(x,y) \end{equation} Hessian \begin{equation} Hess(u) = \mqty(\pdv[2]{u}{x} & \frac{\partial^{2} u}{\partial x \partial y} \ \frac{\partial^{2} u}{\partial x \partial y} & \pdv[2]{u}{y}) \end{equation} If we take its determinant, we obtain:

Traveling Wave For two-variable PDEs, it is called a Traveling Wave if solutions to u takes on the form:

for some constant c, and where w(x) is a function which depends on only one of the two variables. Bell Curves See also Bell Curves