Cauchy Problem and d'Alembert Formula

In mathematics, the partial differential equation is one of the important topics in calculus. To find the solution of the partial differential equation (PDE) when defined for a particular surface or wave, we have to apply different techniques and maths formulas. Among these, the Cauchy problem and d’Alembert’s formula are the main tools. In this article, you will learn about the Cauchy problem in PDE and how to find the solution to this problem using d’Alembert’s formula, along with the derivation of d’Alembert’s formula in detail.

Cauchy Problem Definition

In mathematics, a Cauchy problem asks for a PDE (partial differential equation) solution that meets particular conditions given on a hypersurface in the domain field. The Cauchy problem is named after the French mathematician Augustin-Louis Cauchy. A Cauchy problem can also be an initial value problem or a boundary value problem.

Cauchy Problem Example

Let us take the Euler equation as:

Au_xx + Bu_xy + Cu_yy = F (x, y, u, u_x, u_y) ……..(i)

Here, A, B, and C are functions of variables x and y.

Let (x₀, y₀) be any point on L₀, a smooth curve in the xy-plane.

Also, let x₀ = x₀(λ) and y₀ = y₀(λ) be the parametric equations of the above-defined curve L₀.

Here, λ is a parameter.

Suppose two functions, f(λ) and g(λ), are defined along L₀.

Now, the Cauchy problem is one of the determined solutions u(x, y) of equation (i) in the neighbourhood of the curve, i.e. L₀ satisfies the Cauchy conditions given by:

u = f (λ)…….(ii)

∂u/∂n = g (λ)…..(iii) on the curve L₀.

Here,

n = the direction of the normal to L₀,

Besides, n lies to the left of L₀ in the counterclockwise movement of expanding arc length. However, the functions f(λ) and g(λ) are called the Cauchy data.

Cauchy Problem Solution

Consider the Cauchy problem defined above. Now, for every point on the curve L₀, the value of u is determined by the equation u = f(λ).

Thus, the curve L₀ illustrated by equations x₀ = x₀(λ) and y₀ = y₀(λ) with the condition u = f(λ) produces a twisting curve L in the space (x, y, u) whose projection on xy-plane is the curve L₀.

Therefore, the Cauchy problem’s solution is a surface called an integral surface (surface integral) in the space (x, y, u) that passes through the obtained curve L and satisfies the condition ∂u/∂n = g(λ).

Here, ∂u/∂n = g(λ) depicts a tangent plane to the surface integral along the curve L.

d’Alembert’s Formula

In the field of partial differential equations, d’Alembert’s formula is the solution to a one-dimensional wave equation. Suppose u_tt(x, t) = c² u_xx(x, t) is a one-dimensional wave equation with initial conditions at t = 0: u(x, 0) and u_t(x, 0) such that the solution of this Cauchy problem of wave equation is given by:

d’Alembert’s Wave Equation Derivation

Consider a wave equation:

u_tt − c² u_xx = 0, x ∈ R and t > 0

with initial conditions

u(x, 0) = f(x), x ∈ R

and

u_t(x, 0) = g(x), x ∈ R

The characteristic equations of the above partial differential equation are x ± ct = constant.

Now, μ = x + ct and η = x – ct are used to convert the given PDE to the form u_μη = 0.

The general solution of this PDE is given by:

u(μ, η) = F(μ) + G(η)

Here, F and G are C¹ functions.

Thus, u(x, t) = F(x + ct) + G(x – ct)

Also, we can say u is C² if F and G are C².

Here, the solution u(x, t) can be derived as two waves with constant velocity c that move in opposite directions along the x-axis.

Let us consider this solution with Cauchy data as:

u(x, 0) = g(x)

ut(x, 0) = h(x)

Consider u(x, 0) = g(x) then g(x) = F(x) + G(x) …….(i)

Consider ut(x, 0) = h(x) then h(x) = cF’(x) = cG’(x) ……(ii)

By integrating equation (ii), we get;

cF(x) – cG(x) = ∫^x_-∞ h(ξ) dξ + c1…..(iii)

Solving equations (i) and (iii), we get;

F(x) = (-1/2c) [-cg(x) – { ∫^x_-∞ h(ξ) dξ + c1}]

G(x) = (-1/2c) [-cg(x) + { ∫^x_-∞ h(ξ) dξ + c1}]

Now, consider the solution equation:

u(x, t) = F(x + ct) + G(x – ct)

Therefore, u(x, t) = (½) [g(x – ct) + g(x +ct) + (1/2c) ∫^x+ct_x-ct h(ξ) dξ

This is called d’Alembert’s formula.

d’Alembert’s Solution of Wave Equation

We generally use d’Alembert’s formula to find the solution to a wave equation, such as a homogeneous wave equation, non-homogeneous wave equation, and so on.

Learn how to find the solution of a non-homogeneous wave equation using d’Alembert’s formula here.