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

Linear, Semilinear, and Nonlinear Second-Order PDEs

Linear second-order PDEs and their properties. Principle of linear superposition

A second-order linear partial differential equation with two independent variables has the form \[\tag{1} a(x,y)\frac{\partial^2w}{\partial x^2}+ 2b(x,y)\frac{\partial^2w}{\partial x\,\partial y}+ c(x,y)\frac{\partial^2w}{\partial y^2}= \alpha(x,y)\frac{\partial w}{\partial x}+ \beta(x,y)\frac{\partial w}{\partial y}+ \gamma(x, y)w+\delta(x,y). \]

If \(\delta\equiv 0\), equation (1) is a homogeneous linear equation, and if \(\delta\not\equiv 0\), it is a nonhomogeneous linear equation. The functions \(a(x,y)\), \(b(x,y)\), ..., \(\gamma(x,y)\), \(\delta(x,y)\) are called coefficients of equation (1).

Some properties of a homogeneous linear equation (with \(\delta\equiv 0\)):

1. A homogeneous linear equation has a particular solution \(w=0\ .\)
2. The principle of linear superposition holds; namely, if \(w_1(x,y)\), \(w_2(x,y)\), ..., \(w_n(x,y)\) are particular solutions to homogeneous linear equation, then the function \(A_1w_1(x,y)+A_2w_2(x,y)+\cdots+A_nw_n(x,y),\) where \(A_1\), \(A_2\), ..., \(A_n\) are arbitrary numbers is also an exact solution to that equation.
3. Suppose equation (1) has a particular solution \(\tilde w=\tilde w(x,y;\mu)\) that depends on a parameter \(\mu\), and the coefficients of the linear differential equation are independent of \(\mu\) (but can depend on \(x\) and \(y\)). Then, by differentiating \(\tilde w\) with respect to \(\mu\), one obtains other solutions to the equation, \( \frac{\partial\tilde w}{\partial\mu},\quad \frac{\partial^2\tilde w}{\partial\mu^2},\quad \ldots,\quad \frac{\partial^k\tilde w}{\partial\mu^k},\quad \ldots \)
4. Let \(\tilde w=\tilde w(x,y;\mu)\) be a particular solution as described in property 3. Multiplying \(\tilde w\) by an arbitrary function \(\varphi(\mu)\) and integrating the resulting expression with respect to \(\mu\) over some interval \([\mu_1,\mu_2]\), one obtains a new function \( \int_{\mu_1}^{\mu_2}\tilde w(x,y;\mu)\varphi(\mu)\,d\mu, \) which is also a solution to the original homogeneous linear equation.
5. Suppose the coefficients of the homogeneous linear equation (1) are independent of \(x\ .\) Then: (i) there is a particular solution of the form \(w=e^{\lambda x}u(y)\), where \(\lambda\) is an arbitrary number and \(u(y)\) is determined by a linear second-order ordinary differential equation, and (ii) differentiating any particular solution with respect to \(x\) also results in a particular solution to equation (1).

Properties 2–5 are widely used for constructing solutions to problems governed by linear PDEs.

Examples of particular solutions to linear PDEs can be found in the subsections Heat equation and Laplace equation below.

Semilinear and nonlinear second-order PDEs

A second-order semilinear partial differential equation with two independent variables has the form \[\tag{2} a(x,y)\frac{\partial^2w}{\partial x^2}+ 2b(x,y)\frac{\partial^2w}{\partial x\,\partial y}+ c(x, y)\frac{\partial^2w}{\partial y^2}= F\biggl(x,y,w,\frac{\partial w}{\partial x},\frac{\partial w}{\partial x}\biggr). \]

In the general case, a second-order nonlinear partial differential equation with two independent variables has the form \[ F\biggl(x,y,w,\frac{\partial w}{\partial x},\frac{\partial w}{\partial y}, \frac{\partial^2w}{\partial x^2},\frac{\partial^2w}{\partial x\,\partial y}, \frac{\partial^2w}{\partial y^2}\biggr)=0. \]

The classification and the procedure for reducing linear and semilinear equations of the form (1) and (2) to a canonical form are only determined by the left-hand side of the equations (see below for details).

Some Linear Equations Encountered in Applications

Three basic types of linear partial differential equations are distinguished—parabolic, hyperbolic, and elliptic (for details, see below). The solutions of the equations pertaining to each of the types have their own characteristic qualitative differences.

Heat equation (a parabolic equation)

1. The simplest example of a parabolic equation is the heat equation \[\tag{3} \frac{\partial w}{\partial t}-\frac{\partial^2w}{\partial x^2}=0, \]

where the variables \(t\) and \(x\) play the role of time and a spatial coordinate, respectively. Note that equation (3) contains only one highest derivative term.

Equation (3) is often encountered in the theory of heat and mass transfer. It describes one-dimensional unsteady thermal processes in quiescent media or solids with constant thermal diffusivity. A similar equation is used in studying corresponding one-dimensional unsteady mass-exchange processes with constant diffusivity.

2. The heat equation (3) has infinitely many particular solutions (this fact is common to many PDEs); in particular, it admits solutions \[ \begin{align*} w(x, t)&=A(x^2+2t)+B,\\ w(x, t)&=A\exp(\mu^2t \pm \mu x)+B,\\ w(x, t)&=A\frac 1{\sqrt t}\exp\biggl(-\frac{x^2}{4t}\biggr)+B,\\ w(x, t)&=A\exp(-\mu^2t)\cos(\mu x+B)+C,\\ w(x, t)&=A\exp(-\mu x)\cos(\mu x-2\mu^2t+B)+C, \end{align*} \] where \(A\), \(B\), \(C\), and \(\mu\) are arbitrary constants.

See also Linear heat equations from EqWorld and Heat equation from Wikipedia.

Wave equation (a hyperbolic equation)

1. The simplest example of a hyperbolic equation is the wave equation \[\tag{4} \frac{\partial^2w}{\partial t^2}-\frac{\partial^2w}{\partial x^2}=0, \]

where the variables \(t\) and \(x\) play the role of time and the spatial coordinate, respectively. Note that the highest derivative terms in equation (4) differ in sign.

This equation is also known as the equation of vibration of a string. It is often encountered in elasticity, aerodynamics, acoustics, and electrodynamics.

2. The general solution of the wave equation (4) is \[\tag{5} w=\varphi(x+t)+\psi(x-t), \]

where \(\varphi(x)\) and \(\psi(x)\) are arbitrary twice continuously differentiable functions. This solution has the physical interpretation of two traveling waves of arbitrary shape that propagate to the right and to the left along the \(x\)-axis with a constant speed equal to 1.

See also Wave equation from Wikipedia and Linear hyperbolic equations from EqWorld.

Laplace equation (an elliptic equation)

1. The simplest example of an elliptic equation is the Laplace equation \[\tag{6} \frac{\partial^2w}{\partial x^2}+\frac{\partial^2w}{\partial y^2}=0, \]

where \(x\) and \(y\) play the role of the spatial coordinates. Note that the highest derivative terms in equation (6) have like signs. The Laplace equation is often written briefly as \(\Delta w=0\), where \(\Delta\) is the Laplace operator.

The Laplace equation is often encountered in heat and mass transfer theory, fluid mechanics, elasticity, electrostatics, and other areas of mechanics and physics. For example, in heat and mass transfer theory, this equation describes steady-state temperature distribution in the absence of heat sources and sinks in the domain under study.

A solution to the Laplace equation (6) is called a harmonic function.

2. Note some particular solutions of the Laplace equation (6): \[ \begin{array}{rcl} w(x,y)&=&Ax+By+C,\\ w(x,y)&=&A(x^2-y^2)+Bxy,\\ w(x,y)&=&\displaystyle\frac{Ax+By}{x^2+y^2}+C,\\ w(x,y)&=&(A\sinh\mu x+B\cosh \mu x)(C\cos\mu y+D\sin\mu y),\\ w(x,y)&=&(A\cos \mu x+B\sin \mu x)(C\sinh\mu y+D\cosh\mu y), \end{array} \] where \(A\), \(B\), \(C\), \(D\), and \(\mu\) are arbitrary constants.

A fairly general method for constructing solutions to the Laplace equation (6) involves the following. Let \(f(z)=u(x, y)+iv(x, y)\) be any analytic function of the complex variable \(z=x+iy\) (\(u\) and \(v\) are real functions of the real variables \(x\) and \(y\ ;\) \(i^2=-1\)). Then the real and imaginary parts of \(f\) both satisfy the Laplace equation, \[ \Delta u=0,\qquad \Delta v=0. \] Thus, by specifying analytic functions \(f(z)\) and taking their real and imaginary parts, one obtains various solutions of the Laplace equation (6).

Classification of Second-Order Partial Differential Equations

Types of equations

Any semilinear partial differential equation of the second-order with two independent variables (2) can be reduced, by appropriate manipulations, to a simpler equation that has one of the three highest derivative combinations specified above in examples (3), (4), and (6).

Given a point \((x,y)\), equation (2) is said to be \[ \begin{array}{ll} \mbox{parabolic}& \mbox{if} \ \ b^2-ac=0,\\ \mbox{hyperbolic}&\mbox{if} \ \ b^2-ac>0,\\ \mbox{elliptic}& \mbox{if} \ \ b^2-ac<0 \end{array} \] at this point.

Characteristic equations

In order to reduce equation (2) to a canonical form, one should first write out the characteristic equation \[ a\,(dy)^2-2b\,dx\,dy+c\,(dx)^2=0, \] which with \(a\not\equiv 0\) splits into two equations \[\tag{7} a\,dy-\bigl(b+\sqrt{b^2-ac}\,\bigr)\,dx=0 \]

and \[\tag{8} a\,dy-\bigl(b-\sqrt{b^2-ac}\,\bigr)\,dx=0, \]

and then find their general integrals.

Remark. If \(a\equiv 0\), the simpler equations \[ \begin{array}{rcl} dx&=&0,\\ 2b\,dy-c\,dx&=&0 \end{array} \] should be used instead of (7) and (8). The first equation has the obvious general solution \(x=C\ .\)

Canonical form of parabolic equations (case \(b^2-ac=0\))

In this case, equations (7) and (8) coincide and have a common general integral, \[ u(x, y)=C. \]

By passing from \(x\), \(y\) to new independent variables \(\xi\), \(\eta\) in accordance with the relations \[ \xi=u(x, y),\qquad \eta=\eta(x, y), \] where \(\eta=\eta(x, y)\) is any twice differentiable function that satisfies the condition of nondegeneracy of the Jacobian \(\frac {D(\xi, \eta)}{D(x, y)}\) in the given domain, one reduces equation (2) to the canonical form \[\tag{9} \frac{\partial^2w}{\partial\eta^2}= F_1\biggl(\xi, \eta, w, \frac{\partial w}{\partial \xi}, \frac{\partial w}{\partial \eta}\biggr). \]

As \(\eta\), one can take \(\eta=x\) or \(\eta=y\ .\)

It is apparent that the transformed equation (9) has only one highest-derivative term, just as the heat equation (3).

Two canonical forms of hyperbolic equations (case \(b^2-ac>0\))

1. The general integrals \[ u_1(x, y)=C_1,\qquad u_2(x, y)=C_2 \] of equations (7) and (8) are real and different. These integrals determine two different families of real characteristics.

By passing from \(x\), \(y\) to new independent variables \(\xi\), \(\eta\) in accordance with the relations \[ \xi=u_1(x, y),\qquad \eta=u_2(x, y), \] one reduces equation (2) to \[ \frac{\partial^2w}{\partial\xi\,\partial\eta}= F_2\biggl(\xi, \eta, w, \frac{\partial w}{\partial \xi}, \frac{\partial w}{\partial \eta}\biggr). \] This is the so-called first canonical form of a hyperbolic equation.

2. The transformation \[ \xi=t+z,\qquad \eta=t-z \] brings the above equation to another canonical form, \[ \frac{\partial^2w}{\partial t^2}-\frac{\partial^2w}{\partial z^2}= F_3\biggl(t, z, w, \frac{\partial w}{\partial t}, \frac{\partial w}{\partial z}\biggr), \] where \(F_3=4F_2\ .\) This is the so-called second canonical form of a hyperbolic equation. Apart from notation, the left-hand side of the last equation coincides with that of the wave equation (4).

Canonical form of elliptic equations (case \(b^2-ac<0\))

In this case the general integrals of equations (7) and (8) are complex conjugates; these determine two families of complex characteristics.

Let the general integral of equation (7) have the form \[ u_1(x, y)+iu_2(x, y)=C,\qquad i^2=-1, \] where \(u_1(x, y)\) and \(u_2(x, y)\) are real-valued functions.

By passing from \(x\), \(y\) to new independent variables \(\xi\), \(\eta\) in accordance with the relations \[ \xi=u_1(x, y),\qquad \eta=u_2(x, y), \] one reduces equation (2) to the canonical form \[ \frac{\partial^2w}{\partial\xi^2}+ \frac{\partial^2w}{\partial\eta^2}=F_4\biggl(\xi, \eta, w, \frac{\partial w}{\partial \xi}, \frac{\partial w}{\partial \eta}\biggr). \] Apart from notation, the left-hand side of the last equation coincides with that of the Laplace equation (6).

Basic Problems for PDEs of Mathematical Physics

Most PDEs of mathematical physics govern infinitely many qualitatively similar phenomena or processes. This follows from the fact that differential equations have, as a rule, infinitely many particular solutions. The specific solution that describes the physical phenomenon under study is separated from the set of particular solutions of the given differential equation by means of the initial and boundary conditions.

For simplicity and clarity of illustration, the basic problems of mathematical physics will be presented for the simplest linear equations (3), (4), and (6) only.

Cauchy problem and boundary value problems for parabolic equations

Cauchy problem (\(t\ge 0\), \(-\infty<x<\infty\)). Find a function \(w\) that satisfies heat equation (3) for \(t>0\) and the initial condition \[\tag{10} w=\varphi(x)\quad\hbox{at}\quad t=0. \]

The solution of the Cauchy problem (3), (10) is \[ w(x, t)=\int^{\infty}_{-\infty}\varphi(\xi)E(x, \xi, t)\,d\xi, \] where \(E(x, \xi, t)\) is the fundamental solution of the Cauchy problem, \[ E(x, \xi, t)=\frac 1{2\sqrt{\pi at}} \exp\biggl[-\frac{(x-\xi)^2}{4at}\biggr]. \]

In all boundary value problems (or initial-boundary value problems) below, it will be required to find a function \(w\), in a domain \(t\ge 0\ ,\) \(x_1\le x\le x_2\) (\(-\infty<x_1<x_2<\infty\)), that satisfies the heat equation (3) for \(t>0\) and the initial condition (10). In addition, all problems will be supplemented with some boundary conditions as given below.

First boundary value problem. The function \(w(x,t)\) takes prescribed values on the boundary: \[\tag{11} \begin{array}{lll} w=\psi_1(t)& \hbox{at}& x=x_1,\\ w=\psi_2(t)& \hbox{at}& x=x_2. \end{array} \]

In particular, the solution to the first boundary value problem (3), (10), (11) with \(\psi_1(t)=\psi_2(t)\equiv 0\), \(x_1=0\), and \(x_2=l\) is expressed as \[ w(x,t)=\int^l_0\varphi(\xi)G(x,\xi,t)\,d\xi, \] where the Green's function \(G(x,\xi,t)\) is defined by the formulas \[ \begin{align*} G(x, \xi, t)&= \frac 2l\sum^{\infty}_{n=1}\sin\biggl(\frac{n\pi x}l\biggr) \sin\biggl(\frac{n\pi\xi}l\biggr)\exp\biggl(-\frac{an^2\pi^2t}{l^2}\biggr)\\ &=\frac 1{2\sqrt{\pi at}} \sum^{\infty}_{n=-\infty}\biggl\{\exp\biggl[-\frac{(x-\xi+2nl)^2}{4at}\biggr]- \exp\biggl[-\frac{(x+\xi+2nl)^2}{4at}\biggr]\biggr\}. \end{align*} \] The first series converges rapidly at large \(t\) and the second series at small \(t\ .\)

Second boundary value problem. The derivatives of the function \(w(x,t)\) are prescribed on the boundary: \[\tag{12} \begin{aligned} \frac{\partial w}{\partial x}=\psi_1(t)&\quad \hbox{at}\quad x=x_1,\\ \frac{\partial w}{\partial x}=\psi_2(t)&\quad \hbox{at}\quad x=x_2. \end{aligned} \]

Third boundary value problem. A linear relationship between the unknown function and its derivatives are prescribed on the boundary: \[\tag{13} \begin{aligned} \frac{\partial w}{\partial x}-k_1w=\psi_1(t)&\quad \hbox{at}\quad x=x_1,\\ \frac{\partial w}{\partial x}+k_2w=\psi_2(t)&\quad \hbox{at}\quad x=x_2. \end{aligned} \]

Mixed boundary value problems. Conditions of different type, listed above, are set on the boundary of the domain in question, for example, \[\tag{14} \begin{array}{rll} x=\psi_1(t)& \hbox{at}& x=x_1,\\ \displaystyle\frac{\partial w}{\partial x}=\psi_2(t)& \hbox{at}& x=x_2. \end{array} \]

The boundary conditions (11)–(14) are called homogeneous if \(\psi_1(t)=\psi_2(t)\equiv 0\ .\)

Solutions to the above initial-boundary value problems for the heat equation can be obtained by separation of variables (Fourier method) in the form of infinite series or by the method of integral transforms using the Laplace transform.

For other linear heat equations, their exact solutions, and solutions to associated Cauchy problems and boundary value problems, see Linear heat equations at EqWorld.

Cauchy problem and boundary value problems for hyperbolic equations

Cauchy problem (\(t\ge 0\), \(-\infty<x<\infty\)). Find a function \(w\) that satisfies the wave equation (4) for \(t>0\) and two initial conditions \[\tag{15} \begin{array}{rll} w=\varphi_0(x)& \hbox{at}& t=0,\\ \displaystyle\frac{\partial w}{\partial t}=\varphi_1(x)& \hbox{at}& t=0. \end{array} \]

The solution of the Cauchy problem (4), (15) is given by D'Alembert's formula: \[ w(x, t)=\frac 12[\varphi_0(x+at)+\varphi_0(x-at)]+\frac 1{2a}\int^{x+at}_{x-at}\varphi_1(\xi)\,d\xi. \]

Boundary value problems. In all boundary value problems, it is required to find a function \(w\), in a domain \(t\ge 0\), \(x_1\le x\le x_2\) (\(-\infty<x_1<x_2<\infty\)), that satisfies the wave equation (4) for \(t>0\) and the initial conditions (15). In addition, appropriate boundary conditions, (11), (12), (13), or (14), are imposed.

Solutions to these boundary value problems for the wave equation can be obtained by separation of variables (Fourier method) in the form of infinite series. In particular, the solution to the first boundary value problem (4), (11), (15) with homogeneous boundary conditions, \(\psi_1(t)=\psi_2(t)\equiv 0\) at \(x_1=0\) and \(x_2=l\), is expressed as \[\tag{16} w(x,t)=\frac{\partial}{\partial t}\int^l_0\varphi_0(\xi)G(x,\xi,t)\,d\xi +\int^l_0\varphi_1(\xi)G(x,\xi,t)\,d\xi, \]

where \[ G(x, \xi, t) =\frac 2{a\pi}\sum^{\infty}_{n=1}\frac 1n\sin\Bigl(\frac{n\pi x}l\Bigr) \sin\Bigl(\frac{n\pi\xi}l\Bigr)\sin\Bigl(\frac{n\pi at}l\Bigr). \]

Goursat problem. On the characteristics of the wave equation (4), values of the unknown function \(w\) are prescribed: \[\tag{17} \begin{array}{llll} w=\varphi(x)& \hbox{for}& x-t=0& (0\le x\le a),\\ w=\psi(x)& \hbox{for}& x+t=0& (0\le x\le b), \end{array} \]

with the consistency condition \(\varphi(0)=\psi(0)\) implied to hold.

Substituting the values set on the characteristics (17) into the general solution of the wave equation (5), one arrives at a system of linear algebraic equations for \(\varphi(x)\) and \(\psi(x)\ .\) As a result, the solution to the Goursat problem (4), (17) is obtained in the form \[ w(x,t)=\varphi\biggl(\frac{x+t}2\biggr)+\psi\biggl(\frac{x-t}2\biggr)-\varphi(0). \] The solution propagation domain is the parallelogram bounded by the four lines \[ x-t=0,\quad x+t=0,\quad x-t=2b,\quad x+t=2a. \]

For other linear wave equations, their exact solutions, and solutions to associated Cauchy problems and boundary value problems, see Linear hyperbolic equations at EqWorld.

Boundary value problems for elliptic equations

Setting boundary conditions for the first, second, and third boundary value problems for the Laplace equation (6) means prescribing values of the unknown function, its first derivative, and a linear combination of the unknown function and its derivative, respectively.

For example, the first boundary value problem in a rectangular domain \(0\le x\le a\), \(0\le y\le b\) is characterized by the boundary conditions \[\tag{18} \begin{array}{llllll} w=\varphi_1(y)& \hbox{at}& x=0,\quad& w=\varphi_2(y)& \hbox{at}& x=a,\\ w=\varphi_3(x)& \hbox{at}& y=0,\quad& w=\varphi_4(x)& \hbox{at}& y=b. \end{array} \]

The solution to problem (6), (18) with \(\varphi_3(x)=\varphi_4(x)\equiv 0\) is given by \[ w(x, y)=\sum^{\infty}_{n=1} A_n\sinh\biggl[\frac{n\pi}b(a-x)\biggr]\sin\biggl(\frac{n\pi}by\biggr) +\sum^{\infty}_{n=1} B_n\sinh\biggl(\frac{n\pi}bx\biggr)\sin\biggl(\frac{n\pi}by\biggr), \] where the coefficients \(A_n\) and \(B_n\) are expressed as \[ A_n=\frac{2}{\lambda_n}\int^b_0\varphi_1(\xi)\sin\biggl(\frac{n\pi\xi}b\biggr)d\xi,\quad B_n=\frac{2}{\lambda_n}\int^b_0\varphi_2(\xi)\sin\biggl(\frac{n\pi\xi}b\biggr)d\xi,\quad \lambda_n=b\sinh\biggl(\frac{n\pi a}b\biggr). \]

Remark. For elliptic equations, the first boundary value problem is often called the Dirichlet problem, and the second boundary value problem is called the Neumann problem.

For other linear elliptic equations, their exact solutions, and solutions to associated boundary value problems, see Linear elliptic equations at EqWorld.

Some Nonlinear Equations Encountered in Applications

Nonlinear heat equation:

\[\tag{19} \frac{\partial w}{\partial t}=\frac{\partial}{\partial x} \biggl[f(w)\frac{\partial w}{\partial x}\biggr]. \]

This equation describes one-dimensional unsteady thermal processes in quiescent media or solids in the case where the thermal diffusivity is temperature dependent, \(f(w)>0\ .\) In the special case \(f(w)\equiv 1\), the nonlinear equation (19) becomes the linear heat equation (3).

In general, the nonlinear heat equation (19) admits exact solutions of the form \[ \begin{array}{ll} w=W(kx-\lambda t)& (\hbox{traveling-wave solution}),\\ w=U(x/\!\sqrt t\,)& (\hbox{self-similar solution}), \end{array} \] where \(W=W(z)\) and \(U=U(r)\) are determined by ordinary differential equations, and \(k\) and \(\lambda\) are arbitrary constants.

Kolmogorov–Petrovskii–Piskunov equation:

\[\tag{20} \frac{\partial w}{\partial t}=a\frac{\partial^2w}{\partial x^2}+f(w),\qquad a>0. \]

Equations of this form are often encountered in various problems of mass and heat transfer (with \(f\) being the rate of a volume chemical reaction), combustion theory, biology, and ecology.

In the special case of \(f(w)\equiv 0\) and \(a=1\), the nonlinear equation (20) becomes the linear heat equation (3).

Remark. Equation (20) is also called a heat equation with a nonlinear source.

Burgers equation:

\[\tag{21} \frac{\partial w}{\partial t}+w\frac{\partial w}{\partial x}=\frac{\partial^2w}{\partial x^2}. \]

This equation is used for describing wave processes in gas dynamics, hydrodynamics, and acoustics.

1. Exact solutions to the Burgers equation can be obtained using the following formula (Hopf–Cole transformation): \[ w(x,t)=-\frac 2u\frac{\partial u}{\partial x}, \] where \(u=u(x,t)\) is a solution to the linear heat equation \(u_t=u_{xx}\) (see above for details).

2. The solution to the Cauchy problem for the Burgers equation with the initial condition \[ w=f(x)\quad {\rm at}\quad t=0 \qquad (-\infty<x<\infty) \] has the form \[ w(x,t)=-2\frac {\partial}{\partial x}\ln F(x,t), \] where \[ F(x, t)=\frac 1{\sqrt{4\pi t}}\int^{\infty}_{-\infty} \exp\biggl[-\frac{(x-\xi)^2}{4t}+\frac12\int^{\xi}_0f(\xi')\,d\xi'\biggr]d\xi. \]

Nonlinear wave equation:

\[\tag{22} \frac{\partial^2w}{\partial t^2}=\frac{\partial}{\partial x} \biggl[f(w)\frac{\partial w}{\partial x}\biggr]. \]

This equation is encountered in wave and gas dynamics, \(f(w)>0\ .\) In the special case \(f(w)\equiv 1\), the nonlinear equation (22) becomes the linear wave equation (4).

Equation (22) admits exact solutions in implicit form: \[ \begin{array}{rcl} x+t\sqrt{f(w)}&=&\varphi(w),\\ x-t\sqrt{f(w)}&=&\psi(w), \end{array} \] where \(\varphi(w)\) and \(\psi(w)\) are arbitrary functions.

Equation (22) can be reduced to a linear equation (see Polyanin and Zaitsev, 2004).

Nonlinear Klein–Gordon equation:

\[\tag{23} \frac{\partial^2w}{\partial t^2}=a\frac{\partial^2w}{\partial x^2}+f(w), \qquad a>0. \]

Equations of this form arise in differential geometry and various areas of physics (superconductivity, dislocations in crystals, waves in ferromagnetic materials, laser pulses in two-phase media, and others). For \(f(w)\equiv 0\) and \(a=1\), equation (23) coincides with the linear wave equation (4).

1. In general, the nonlinear Klein–Gordon equation (23) admits exact solutions of the form \[ \begin{array}{ll} w=W(z),& z=kx-\lambda t,\\ w=U(\xi),& \xi=(\sqrt a\,t+C_1)^2-(x+C_2)^2, \end{array} \] where \(W=W(z)\) and \(U=U(\xi)\) are determined by ordinary differential equations, while \(k\), \(\lambda\), \(C_1\), and \(C_2\) are arbitrary constants.

2. In the special case \[ f(w)=be^{\beta w}, \] the general solution of equation (23) is expressed as \[ w(x,t)=\frac 1{\beta}\bigl[\varphi(z)+\psi(y)\bigr]- \frac 2{\beta}\ln\biggl|k\int \exp\bigl[\varphi(z)\bigr]\,dz -\frac{b\beta}{8ak}\int\exp\bigl[\psi(y)\bigr]\,dy\biggr|, \] \[ z=x-\sqrt a\,t,\qquad y=x+\sqrt a\,t, \] where \(\varphi=\varphi(z)\) and \(\psi=\psi(y)\) are arbitrary functions and \(k\) is an arbitrary constant.

Remark. In the special cases \(f(w)=b\sin(\beta w)\) and \(f(w)=b\sinh(\beta w)\), equation (23) is called the sine-Gordon equation and the sinh-Gordon equation, respectively.

Nonlinear Laplace equation:

\[\tag{24} \frac{\partial^2w}{\partial x^2}+\frac{\partial^2w}{\partial y^2}=f(w). \]

This equation is also called a stationary heat equation with a nonlinear source.

1. In general, the nonlinear heat equation (24) admits exact solutions of the form \[ \begin{array}{ll} w=W(z),& z=k_1x+k_2y,\\ w=U(r),& r=\sqrt{(x+C_1)^2+(y+C_2)^2}, \end{array} \] where \(W=W(z)\) and \(U=U(r)\) are determined by ordinary differential equations, while \(k_1\), \(k_2\), \(C_1\), and \(C_2\) are arbitrary constants.

2. In the special case \[ f(w)=ae^{\beta w}, \] the general solution of equation (24) is expressed as \[ w(x,y)=-\frac 2\beta\ln\frac{\bigl|1-2a\beta\Phi(z)\overline{\Phi(z)}\,\bigr|}{4|\Phi'_z(z)|}, \] where \(\Phi=\Phi(z)\) is an arbitrary analytic function of the complex variable \(z=x+iy\) with nonzero derivative, and the bar over a symbol denotes the complex conjugate.

Monge–Ampere equation:

\[ \biggl(\frac{\partial^2w}{\partial x\,\partial y}\biggr)^{\!2}- \frac{\partial^2w}{\partial x^2} \frac{\partial^2w}{\partial y^2}=f(x,y). \] The equation is encountered in differential geometry, gas dynamics, and meteorology.

Below are solutions to the homogeneous Monge–Ampere equation for the special case \(f(x,y)\equiv 0\ .\)

1. Exact solutions involving one arbitrary function: \[ w(x,y)=\varphi(C_1x+C_2 y)+C_3x+C_4y+C_5, \] \[ w(x,y)=(C_1x+C_2y)\,\varphi\biggl(\frac yx\biggr)+C_3x+C_4y+C_5, \] \[ w(x,y)=(C_1x+C_2y+C_3)\,\varphi\biggl(\frac{C_4x+C_5y+C_6}{C_1x+C_2y+C_3}\biggr) +C_7x+C_8y+C_9, \] where \(C_1\), ..., \(C_{9}\) are arbitrary constants and \(\varphi=\varphi(z)\) is an arbitrary function.

2. General solution in parametric form: \[ w=t x+\varphi(t)y+\psi(t), \] \[ x+\varphi'(t)y+\psi'(t)=0, \] where \(t\) is the parameter, and \(\varphi=\varphi(t)\) and \(\psi=\psi(t)\) are arbitrary functions.

Simplest Types of Exact Solutions of Nonlinear PDEs

Preliminary remarks

The following classes of solutions are usually regarded as exact solutions to nonlinear partial differential equations of mathematical physics:

1. Solutions expressible in terms of elementary functions.
2. Solutions expressed by quadrature.
3. Solutions described by ordinary differential equations (or systems of ordinary differential equations).
4. Solutions expressible in terms of solutions to linear partial differential equations (and/or solutions to linear integral equations).

The simplest types of exact solutions to nonlinear PDEs are traveling-wave solutions and self-similar solutions. They often occur in various applications.

In what follows, it is assumed that the unknown \(w\) depends on two variables, \(x\) and \(t\), where \(t\) plays the role of time and \(x\) is a spatial coordinate.

Traveling-wave solutions

Traveling-wave solutions, by definition, are of the form \[\tag{25} w(x,t)=W(z),\quad \ z=kx-\lambda t, \]

where \(\lambda/k\) plays the role of the wave propagation velocity (the value \(\lambda =0\) corresponds to a stationary solution, and the value \(k=0\) corresponds to a space-homogeneous solution). Traveling-wave solutions are characterized by the fact that the profiles of these solutions at different time instants are obtained from one another by appropriate shifts (translations) along the \(x\)-axis. Consequently, a Cartesian coordinate system moving with a constant speed can be introduced in which the profile of the desired quantity is stationary. For \(k>0\) and \(\lambda>0\), the wave (25) travels along the \(x\)-axis to the right (in the direction of increasing \(x\)).

Traveling-wave solutions occur for equations that do not explicitly involve independent variables, \[\tag{26} F\biggl(w, \frac{\partial w}{\partial x}, \frac{\partial w}{\partial t}, \frac{\partial^2w}{\partial x^2}, \frac{\partial^2w}{\partial x\,\partial t}, \frac{\partial^2w}{\partial t^2},\ldots\biggr)=0. \]

Substituting (25) into (26), one obtains an autonomous ordinary differential equation for the function \(W(z)\ :\) \[ F(W,kW',-\lambda W',k^2W'',-k\lambda W'',\lambda ^2W'',\ldots)=0, \] where \(k\) and \(\lambda \) are arbitrary constants, and the prime denotes a derivative with respect to \(z\ .\)

Remark. The term traveling-wave solution is also used in the cases where the variable \(t\) plays the role of a spatial coordinate, \(t=y\ .\)

All nonlinear equations considered above, (19)–(24) and (25) with \(f(x,y)=0\), admit traveling-wave solutions.

Self-similar solutions

By definition, a self-similar solution is a solution of the form \[\tag{27} w(x,t)=t^{\alpha}U(\zeta),\quad \ \zeta=xt^\beta. \]

The profiles of these solutions at different time instants are obtained from one another by a similarity transformation (like scaling).

Self-similar solutions exist if the scaling of the independent and dependent variables, \[\tag{28} t=C\bar t,\quad x=C^k\bar x,\quad w=C^m\bar w,\qquad \mbox{where}\ C\not=0\ \mbox{is an arbitrary constant}, \]

for some \(k\) and \(m\) such that \(|k|+|m|\not=0\), is equivalent to the identical transformation.

It can be shown that the parameters in solution (27) and transformation (28) are linked by the simple relations \[\tag{29} \alpha=m, \quad \ \beta=-k. \]

In practice, the above existence criterion is checked and if a pair of \(k\) and \(m\) in (28) has been found, then a self-similar solution is defined by formulas (27) with parameters (29).

Example. Consider the heat equation with a nonlinear power-law source term \[\tag{30} \frac{\partial w}{\partial t}=a\frac{\partial^2w}{\partial x^2}+bw^n. \]

The scaling transformation (28) converts equation (30) into \[\tag{31} C^{m-1}\frac{\partial \bar w}{\partial\bar t}= aC^{m-2k}\frac{\partial^2\bar w}{\partial \bar x^2}+bC^{mn}\bar w^n. \]

In order that equation (31) coincides with (30), one must require that the powers of \(C\) are the same, which yields the following system of linear algebraic equations for the constants \(k\) and \(m\ :\) \[ m-1=m-2k=mn. \] This system admits a unique solution\[\,k=\frac 12\ ,\] \(m=\frac 1{1-n}\ .\) Using this solution together with relations (27) and (29), one obtains self-similar variables in the form \[ w=t^{1/(1-n)}U(\zeta),\quad \ \zeta=xt^{-1/2}. \] Inserting these into (30), one arrives at the following ordinary differential equation for \(U(\zeta)\ :\) \[ aU''_{\zeta\zeta}+\frac12\zeta U'_\zeta+\frac 1{n-1}U+bU^n=0. \]

Cauchy Problem and Boundary Value Problems for Nonlinear Equations

The Cauchy problem and boundary value problems for nonlinear equations are stated in exactly the same way as for linear equations (see Basic Problems for PDEs of Mathematical Physics).

Examples. The Cauchy problem for a nonlinear heat equation is stated as follows: find a solution to equation (19) subject to the initial condition (10).

The first boundary value problem for a nonlinear wave equation as follows: find a solution to equation (24) subject to the initial conditions (10) and the boundary conditions (11).

Problems for nonlinear PDEs are normally solved using numerical methods.