First-Order Nonlinear Partial Differential Equations and the Method of Characteristics

Partial differential equations govern the universe, and most of these equations are nonlinear. This poses a challenge to engineers because nonlinear equations are difficult to solve. In many cases, there is no explicit formula for a solution and it often takes several pages of mathematical arguments to prove whether or not a solution even exists. However, with some clever trickery, we can deduce the general behavior of the solutions to some nonlinear equations in certain regions of their domains.  Today, we will look at the general first-order nonlinear PDE of the form 

F(∇u, u, x)  = 0,

over an open subset Ω of ℝⁿ. We will also assume a boundary condition 

u = g on Γ,

where Γ is a subset of ∂Ω. For now, both F and g are assumed to be smooth. We will seek to solve this equation by finding parametric equations of curves in Ω along which the value of u and ∇u is known. Our overall strategy is as follows: First, choose x^* ∈ Ω. Then, we will connect x^* with some point x⁰ ∈ Γ via a curve given parametrically by x(s) = (x₁(s), …, x_n(s)), where x(0) = x⁰. This curve will be selected so that we can compute u and ∇u along x(s). Since u(x(0)) = g(x⁰) is known, we will then be able to deduce the value of u at x^* by computing u(x(s)) for some value of s that makes x(s) = x^*. This special curve x(s) will be called the projected characteristic curve, and this entire method will be called the method of characteristics.

The fundamental question that remains is how we will determine the curve defined by x(s). For the remainder of this blog, we will address this problem by deriving a system of ordinary differential equations that will characterize these special curves. To begin, let’s define 

z(s) = u(x(s)),

and 

p(s) = ∇u(x(s)).

The scalar function z will record the value of u along the curve and the vector p(s) = (p₁(s), …, p_n(s)) will record the value of ∇u along the curve. If x(s) is the right curve, then we will need to be able to find z(s) and p(s). Therefore, we will choose x(s) carefully so that z(s) and p(s) can be readily derived. To find the correct equations for x(s), let’s differentiate the identity 

p(s) = ∇u(x(s)) 

with respect to s to obtain 

dp_i/ds = ∑ _j (∂_xj p_i) dx_j/ds = ∑ _j (∂_xj ∂_xi u) dx_j/ds,

for 1 ≤ i ≤ n. Since our original equation did not involve any second derivatives, we seek to eliminate ∂_xj ∂_xi u from the differential equation above. To do this, we take the total derivative of the PDE 

F(p,z,x) = 0

with respect to x_i to get 

dF/dx_i = ∑ _j (∂_pj F) dp_j/dx_i + (∂_z F) p_i + ∂_xi F = 0. 

But dp_j/dx_i = ∂_xj ∂_xi u, so we can close the problem if we let 

∑ _j (∂_xj ∂_xi u) ∂_pj F = ∑ _j (∂_xj ∂_xi u) dx_j/ds.

This is the same as having dx_j/ds = ∂_pj F, so we will require 

dx/ds = ∇_p F 

to be satisfied, where ∇_p denotes the gradient with respect to the p variables. With this condition, we can now see that 

dp_i/ds = ∑ _j (∂_xj ∂_xi u) dx_j/ds = -(∂_z F) p_i – ∂_xi F.

So our condition on p will be that 

dp/ds = -p ∂_z F – ∇_x F.

It remains to find an ODE satisfied by z. To this end, we can differentiate the identity 

z(s) = u(x(s)) 

with respect to s to find that 

dz/ds = ∑ _j (∂_xj u) dx_j/ds =  ∑ _j p_j (∂_pj F) = p • ∇_p F. 

In summary, we have derived

dx/ds = ∇_p F(p,z,x)

dp/ds = -p ∂_z F(p,z,x) – ∇_x F(p,z,x)

dz/ds = p • ∇_p F(p,z,x)

These equations together are called the characteristic ODEs of our partial differential equation, and their solution will define the curve x(s), the value of u along x(s), and the value of ∇u along x(s). The initial condition for x(s) is just 

x(0) = x⁰, 

and the initial condition for z(s) is just 

z(0) = z⁰ = g(x⁰). 

The initial condition for p(s), however, is not immediately obvious. Since u = g on Γ, we can infer any directional derivative of u along Γ by taking a derivative of g along Γ. What we still don’t know is the value of the normal derivative ∂_ν u at x⁰, where ν is the outward unit normal vector to Γ. To get over this hurdle, we can leverage the Implicit Function Theorem. Specifically, this theorem tells us that we can uniquely solve for ∂_ν u in terms of x, u, and the tangential derivatives of u in a local neighborhood of x⁰ provided that 

∇_p F(p⁰,z⁰,x⁰) • ν(x⁰) ≠ 0. 

This condition is called the non-characteristic condition, and it is instrumental in allowing us to use the characteristic ODEs to construct a local solution to the PDE. In the next blog post, I hope to shed some light on how we can precisely use the characteristic ODEs to define such a local solution. I also hope to cover some example PDEs, such as Burgers’ equation and the Hamilton-Jacobi equation. These examples will illustrate how in some cases, we can extend local smooth solutions to the whole domain, but in other cases, we cannot. Until then, please take care. 

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