First-Order Linear and Quasilinear Partial Differential Equations

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First-Order Linear and Quasilinear Partial Differential Equations

1. General form of first-order quasilinear PDE

A first-order quasilinear partial differential equation with two independent variables has the general form
$∂w ∂w f(x,y,w )∂x- + g(x,y, w)-∂y = h(x,y,w ).$ (1)
Such equations are encountered in various applications (continuum mechanics, gas dynamics, hydrodynamics, heat and mass transfer, wave theory, acoustics, multiphase flows, chemical engineering, etc.).

If the functions $f$ , $g$ , and $h$ are independent of the unknown $w$ , then equation (1) is called linear.

2. Characteristic system. General solution

Suppose that two independent integrals,
$u1 (x, y,w ) = C1, u2 (x, y,w ) = C2,$ (2)
of the characteristic system of ordinary differential equations
$---dx-----= ---dy----= ---dw----- f(x,y, w) g(x,y,w ) h(x,y, w)$ (3)
are known. Then the general solution to equation (1) is given by
$Φ (u1,u2) = 0,$ (4)
where $Φ$ is an arbitrary function of two variables. With equation (4) solved for $u2$ , one often specifies the general solution in the form $u = Ψ (u ) 2 1$ , where $Ψ(u )$ is an arbitrary function of one variable.

Remark. If $h (x,y,w ) ≡ 0$ , then $w = C2$ can be used as the second integral in (2).

Example. Consider the linear partial differential equation $∂w- ∂w- ∂x + a ∂y = bx.$ The associated characteristic system of ordinary differential equations $dx dy dw ---= ---= --- 1 a bx$ has two integrals $y - ax = C1$ , $1 w - --bx2 = C2 2$ . Therefore, the general solution to this PDE can be written as $w - 1bx2 = Ψ (y - ax) 2$ , or $w = 1bx2 + Ψ (y - ax ), 2$ where $Ψ(z)$ is an arbitrary function.

3. Cauchy Problem: Two Formulations. Solving the Cauchy Problem

Generalized Cauchy problem: find a solution $w = w (x, y)$ to equation (1) satisfying the initial conditions
$x = φ1 (ξ), y = φ2(ξ), w = φ3(ξ),$ (5)
where $ξ$ is a parameter $(α ≤ ξ ≤ β)$ and the $φk (ξ)$ are given functions.

Geometric interpretation: find an integral surface of equation (1) passing through the line defined parametrically by equation (5).

Classical Cauchy problem: find a solution $w = w (x, y)$ of equation (1) satisfying the initial condition
$w = φ(y) at x = 0,$ (6)
where $φ(y)$ is a given function.

It is often convenient to represent the classical Cauchy problem as a generalized Cauchy problem by rewriting condition (6) in the parametric form
$x = 0, y = ξ, w = φ (ξ).$ (7)

Existence and uniqueness theorem. If the coefficients $f$ , $g$ , and $h$ of equation (1) and the functions $φk$ in (5) are continuously differentiable with respect to each of their arguments and if the inequalities $′ ′ fφ 2 - g φ1 ⁄= 0$ and $(φ ′1)2 + (φ ′2)2 ⁄= 0$ hold along a line (5), then there is a unique solution to the Cauchy problem (in a neighborhood of the line (5)).

4. Procedure of solving the Cauchy problem

The procedure for solving the Cauchy problem (1), (5) involves several steps. First, two independent integrals (2) of the characteristic system (3) are determined. Then, to find the constants of integration $C1$ and $C2$ , the initial data (5) must be substituted into the integrals (2) to obtain
$u1(φ1(ξ),φ2 (ξ ),φ3 (ξ)) = C1, u2 (φ1 (ξ),φ2(ξ),φ3(ξ)) = C2.$ (8)
Eliminating $C1$ and $C 2$ from (2) and (8) yields
$u1 (x, y,w ) = u1 (φ1(ξ),φ2(ξ),φ3(ξ)), u2 (x, y,w ) = u2 (φ1(ξ),φ2(ξ),φ3(ξ)).$ (9)
Formulas (9) are a parametric form of the solution to the Cauchy problem (1), (5). In some cases, one may succeed in eliminating the parameter $ξ$ from relations (9), thus obtaining the solution in an explicit form.

In the cases where first integrals (2) of the characteristic system (3) cannot be found using analytical methods, one should employ numerical methods to solve the Cauchy problem (1), (5) (or (1), (6)).

Example. Consider the Cauchy problem for Hopf's equation
$∂w-+ w ∂w- = 0 ∂x ∂y$ (10)
subject to the initial condition (6).

First, we rewrite the initial condition (6) in the parametric form (7). Solving the characteristic system $dx- = dy- = dw-, 1 w 0$ we find two independent integrals,
$w = C1, y - wx = C2.$ (11)

Using the initial conditions (7), we find that $C1 = φ(ξ)$ and $C2 = ξ$ . Substituting these expressions into (11) yields the solution of the Cauchy problem (10), (6) in the parametric form
$w = φ(ξ),$ (12)

$y = ξ + φ(ξ)x.$ (13)
The characteristics (13) are straight lines in the $xy$ -plane with slope $φ(ξ)$ which intersect the $y$ -axis at the points $ξ$ . On each characteristic, the function $w$ has the same value equal to $φ(ξ)$ (generally, $w$ takes different values on different characteristics).

For $φ′(ξ) > 0$ , different characteristics do not intersect, and hence, formulas (12) and (13) define a unique solution.

References

A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002.
E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen, II, Partielle Differentialgleichungen Erster Ordnung für eine gesuchte Funktion, Akad. Verlagsgesellschaft Geest & Portig, Leipzig, 1965.