First-Order Partial Differential Equations

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< Partial differential equation
Andrei D. Polyanin et al. (2009), Scholarpedia, 4(1):4605. revision #55858 [link to/cite this article]

Curator: Dr. Andrei D. Polyanin, Institute for Problems in Mechanics, Moscow, Russia
Curator: Dr. William E. Schiesser, Lehigh University and University of Pennsylvania, USA
Curator: Dr. Alexei I. Zhurov, Cardiff University, UK, and Institute for Problems in Mechanics, Moscow, Russia.

Contents

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First-Order Partial Differential Equations

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General Form of First-Order Partial Differential Equation

A first-order partial differential equation with nindependent 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.

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Quasilinear Equations. Characteristic System. General Solution

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General form of first-order quasilinear PDE

A first-order quasilinear partial differential equation with two independent variables has the general form

(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.

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Characteristic system. General solution

The system of ordinary differential equations

(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

(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

(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-x=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-x),

where \Psi(z) is an arbitrary function.

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Cauchy Problem: Two Formulations. Solving the Cauchy Problem

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Generalized Cauchy problem

Generalized Cauchy problem: find a solution w=w(x,y) to equation (1) satisfying the initial conditions

(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).

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Classical Cauchy problem

Classical Cauchy problem: find a solution w=w(x,y) of equation (1) satisfying the initial condition

(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).
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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)).

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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

(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

(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)).

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Second-Order Partial Differential Equations

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Higher-Order Partial Differential Equations

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Approximate and Numerical Methods

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References

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External links


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Action editor: Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
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