User:Leo Trottier/PDE/First-order partial differential equations

A first-order partial differential equation with \(n\)independent variables has the general form \[ F\biggl(x_1,x_2,\dots, x_n,w,\frac{\partial w}{\partial x_1}, \frac{\partial w}{\partial x_2},\dots,\frac{\partial w}{\partial x_n}\biggl)=0, \] where \(w=w(x_1,x_2,\dots, x_n)\) is the unknown function and \(F(\dots)\) is a given function.

Quasilinear Equations. Characteristic System. General Solution

General form of first-order quasilinear PDE

A first-order quasilinear partial differential equation with two independent variables has the general form \[\tag{1} f(x,y,w)\frac{\partial w}{\partial x}+g(x,y,w)\frac{\partial w}{\partial y}=h(x,y,w). \]

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.

Characteristic system. General solution

The system of ordinary differential equations \[\tag{2} \frac{dx}{f(x,y,w)}=\frac{dy}{g(x,y,w)}=\frac{dw}{h(x,y,w)} \]

is known as the characteristic system of equation (1). Suppose that two independent particular solutions of this system have been found in the form \[\tag{3} u_{1}(x, y, w)=C_{1},\qquad u_{2}(x,y,w)=C_{2}, \] where \(C_1\) and \(C_2\) are arbitrary constants; such particular solutions are known as integrals of system (2). Then the general solution to equation (1) can be written as \[\tag{4} \Phi(u_{1},u_{2})=0, \] where \(\Phi\) is an arbitrary function of two variables. With equation (4) solved for \(u_2\), one often specifies the general solution in the form \(u_2=\Psi(u_1)\), where \(\Psi(u)\) is an arbitrary function of one variable.

Remark. If \(h(x,y,w)\equiv 0\), then \(w=C_2\) can be used as the second integral in (3).

Example. Consider the linear equation \[ \frac{\partial w}{\partial x}+a\frac{\partial w}{\partial y}=b. \] The associated characteristic system of ordinary differential equations \[ \frac{dx}{1}=\frac{dy}{a}=\frac{dw}{b} \] has two integrals \[ y-ax=C_1,\quad \ w-bx=C_2. \] Therefore, the general solution to this PDE can be written as \(w-bx=\Psi(y-ax)\), or \[ w=bx+\Psi(y-ax), \] where \(\Psi(z)\) is an arbitrary function.

Cauchy Problem: Two Formulations. Solving the Cauchy Problem

Generalized Cauchy problem

Generalized Cauchy problem: find a solution \(w=w(x,y)\) to equation (1) satisfying the initial conditions \[\tag{5} x=\varphi_1(\xi ),\quad y=\varphi_2(\xi ),\quad w=\varphi_3(\xi ), \] where \(\xi\) is a parameter \((\alpha\le \xi \le \beta)\) and the \(\varphi_k(\xi)\) are given functions.

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

Classical Cauchy problem

Classical Cauchy problem: find a solution \(w=w(x,y)\) of equation (1) satisfying the initial condition \[\tag{6} w=\varphi(y)\quad \hbox{at}\quad x=0, \] where \(\varphi(y)\) 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 \[ x=0,\quad y=\xi,\quad w=\varphi(\xi). \]

Existence and uniqueness theorem

If the coefficients \(f\), \(g\), and \(h\) of equation (1) and the functions \(\varphi_k\) in (5) are continuously differentiable with respect to each of their arguments and if the inequalities \(f\varphi'_2-g\varphi'_1\not=0\) and \((\varphi'_1)^2+(\varphi'_2)^2\not=0\) hold along the curve (5), then there is a unique solution to the Cauchy problem (in a neighborhood of the curve (5)).

Procedure of solving the Cauchy problem

The procedure for solving the Cauchy problem (1), (5) involves several steps. First, two independent integrals (3) of the characteristic system (2) are determined. Then, to find the constants of integration \(C_1\) and \(C_2\), the initial data (5) must be substituted into the integrals (3) to obtain \[\tag{7} u_{1}\bigl(\varphi_1(\xi),\varphi_2(\xi),\varphi_3(\xi)\bigr)=C_{1},\qquad u_{2}\bigl(\varphi_1(\xi),\varphi_2(\xi),\varphi_3(\xi)\bigr)=C_{2}. \]

Eliminating \(C_1\) and \(C_2\) from (3) and (7) yields \[\tag{8} \begin{array}{rcl} &&u_{1}(x,y,w)=u_{1}\bigl(\varphi_1(\xi),\varphi_2(\xi),\varphi_3(\xi)\bigr),\\ &&u_{2}(x,y,w)=u_{2}\bigl(\varphi_1(\xi),\varphi_2(\xi),\varphi_3(\xi)\bigr). \end{array} \]

Formulas (8) are a parametric form of the solution to the Cauchy problem (1), (5). In some cases, one may succeed in eliminating the parameter \(\xi\) from relations (8), thus obtaining the solution in an explicit form.

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