Linear First-Order PDEs

PDEs · Axiom Academy

Solving linear first-order partial differential equations using the method of characteristics 1 General Form of Linear First-Order PDEs A linear first-order PDE in two independent variables has the general form: where a(x,y) , b(x,y) , and c(x,y,u) are given functions, and u(x,y) is the unknown function we're solving for. The equation is linear in u and its derivatives u x and u y . Notice: The derivatives appear to the first power only No products like u · u x or u x · u y The coefficients a and b can depend on x and y , but not on u This equation describes how a quantity u (like temperature or concentration) changes along a direction in the xy -plane. The coefficients a and b determine the direction of propagation. The simplest case occurs when a and b are constants and c = 0 : where a and b are constants (at least one non-zero). Key Insight: Directional Derivative This equation says that the directional derivative of u in the direction of the vector v = (a, b) is zero: This means u is constant along lines parallel to (a, b)! Consider the equation u x + 2u y = 0 . The solution is constant along lines in direction (1, 2), meaning along lines y = 2x + C . Therefore, u(x,y) = f(2x - y) for any differentiable function f . 3 Method of Characteristics: Finding the Solution For the general linear equation, we use characteristic curves to transform the PDE into an ODE. For the PDE au x + bu y = c , the characteristic curves satisfy:

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