Potential theory and partial differential equations

Abstract

Order, linearity, homogeneity. Ordinary differential equations contain only functions of one independent variable. By contrast, one speaks of a partial differential equation if the unknown function u = u(x₁, x₂,…,x _n ) depends on several independent variables x₁, x₂,…,x _n and the equation contains partial derivatives ${{\partial u} \over {\partial {x_i}}},\,\,{{{\partial ^2}u} \over {\partial {x_i}\,\partial {x_j}}}$ etc. for i, j = 1, 2,…, n. The order of the highest derivative that appears in the equation determines the order of the equation. The differential equation is called linear if the unknown function and its derivatives occur linearly and are not multiplied together. A linear partial differential equation is called homogeneous if it contains no term free from the unknown function and its derivatives, otherwise inhomogeneous. For linear partial differential equations, as for ordinary ones, the principle of superposition holds: if u₁ and u₂ are solutions, then every linear combination u= C₁u{n1} + C₂u₂, where C₁ and C₂ are constants, is also a solution.