Linear equations of second order

 

Classification[edit]

Notation[edit]

When writing PDEs, it is common to denote partial derivatives using subscripts. For example:

$${\displaystyle {�}_{�}=\frac{\mathrm{\partial }�}{\mathrm{\partial }�},{�}_{��}=\frac{{\mathrm{\partial }}^2�}{\mathrm{\partial }{�}^2},{�}_{��}=\frac{{\mathrm{\partial }}^2�}{\mathrm{\partial }�\mathrm{\partial }�}=\frac{\mathrm{\partial }}{\mathrm{\partial }�}\left(\frac{\mathrm{\partial }�}{\mathrm{\partial }�}\right).}$$

In the general situation that u is a function of n variables, then u_i denotes the first partial derivative relative to the i-th input, u_ij denotes the second partial derivative relative to the i-th and j-th inputs, and so on.

The Greek letter Δ denotes the Laplace operator; if u is a function of n variables, then

$${\displaystyle \mathrm{\Delta }�={�}_{11}+{�}_{22}+\cdots +{�}_{��}.}$$

In the physics literature, the Laplace operator is often denoted by ∇²; in the mathematics literature, ∇²u may also denote the Hessian matrix of u.

Equations of first order[edit]

Main article: First-order partial differential equation

Linear and nonlinear equations[edit]

Linear equations[edit]

A PDE is called linear if it is linear in the unknown and its derivatives. For example, for a function u of x and y, a second order linear PDE is of the form

$${\displaystyle {�}_1(�,�){�}_{��}+{�}_2(�,�){�}_{��}+{�}_3(�,�){�}_{��}+{�}_4(�,�){�}_{��}+{�}_5(�,�){�}_{�}+{�}_6(�,�){�}_{�}+{�}_7(�,�)�=�(�,�)}$$

where a_i and f are functions of the independent variables only. (Often the mixed-partial derivatives u_xy and u_yx will be equated, but this is not required for the discussion of linearity.) If the a_i are constants (independent of x and y) then the PDE is called linear with constant coefficients. If f is zero everywhere then the linear PDE is homogeneous, otherwise it is inhomogeneous. (This is separate from asymptotic homogenization, which studies the effects of high-frequency oscillations in the coefficients upon solutions to PDEs.)

Nonlinear equations[edit]

Three main types of nonlinear PDEs are semilinear PDEs, quasilinear PDEs, and fully nonlinear PDEs.

Nearest to linear PDEs are semilinear PDEs, where only the highest order derivatives appear as linear terms, with coefficients that are functions of the independent variables. The lower order derivatives and the unknown function may appear arbitrarily. For example, a general second order semilinear PDE in two variables is

$${\displaystyle {�}_1(�,�){�}_{��}+{�}_2(�,�){�}_{��}+{�}_3(�,�){�}_{��}+{�}_4(�,�){�}_{��}+�({�}_{�},{�}_{�},�,�,�)=0}$$

In a quasilinear PDE the highest order derivatives likewise appear only as linear terms, but with coefficients possibly functions of the unknown and lower-order derivatives:

$${\displaystyle {�}_1({�}_{�},{�}_{�},�,�,�){�}_{��}+{�}_2({�}_{�},{�}_{�},�,�,�){�}_{��}+{�}_3({�}_{�},{�}_{�},�,�,�){�}_{��}+{�}_4({�}_{�},{�}_{�},�,�,�){�}_{��}+�({�}_{�},{�}_{�},�,�,�)=0}$$

Many of the fundamental PDEs in physics are quasilinear, such as the Einstein equations of general relativity and the Navier–Stokes equations describing fluid motion.

A PDE without any linearity properties is called fully nonlinear, and possesses nonlinearities on one or more of the highest-order derivatives. An example is the Monge–Ampère equation, which arises in differential geometry.^[3]

Linear equations of second order[edit]

Elliptic, parabolic, and hyperbolic partial differential equations of order two have been widely studied since the beginning of the twentieth century. However, there are many other important types of PDE, including the Korteweg–de Vries equation. There are also hybrids such as the Euler–Tricomi equation, which vary from elliptic to hyperbolic for different regions of the domain. There are also important extensions of these basic types to higher-order PDE, but such knowledge is more specialized.

The elliptic/parabolic/hyperbolic classification provides a guide to appropriate initial and boundary conditions and to the smoothness of the solutions. Assuming u_xy = u_yx, the general linear second-order PDE in two independent variables has the form

$${\displaystyle �{�}_{��}+2�{�}_{��}+�{�}_{��}+\cdots \text{(lower order terms)}=0,}$$

where the coefficients A, B, C... may depend upon x and y. If A² + B² + C² > 0 over a region of the xy-plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:

$${\displaystyle �{�}^2+2���+�{�}^2+\cdots =0.}$$

More precisely, replacing ∂_x by X, and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.

Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant B² − 4AC, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by B² − AC due to the convention of the xy term being 2B rather than B; formally, the discriminant (of the associated quadratic form) is (2B)² − 4AC = 4(B² − AC), with the factor of 4 dropped for simplicity.

1. B² − AC < 0 (elliptic partial differential equation): Solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where x < 0.
2. B² − AC = 0 (parabolic partial differential equation): Equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where x = 0.
3. B² − AC > 0 (hyperbolic partial differential equation): hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where x > 0.

If there are n independent variables x₁, x₂ , …, x_n, a general linear partial differential equation of second order has the form

$${\displaystyle ��=\sum _{�=1}^{�}\sum _{�=1}^{�}{�}_{�,�}\frac{{\mathrm{\partial }}^2�}{\mathrm{\partial }{�}_{�}\mathrm{\partial }{�}_{�}}+\text{lower-order terms}=0.}$$

The classification depends upon the signature of the eigenvalues of the coefficient matrix a_i,j.

1. Elliptic: the eigenvalues are all positive or all negative.
2. Parabolic: the eigenvalues are all positive or all negative, except one that is zero.
3. Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
4. Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues.^[4]

The theory of elliptic, parabolic, and hyperbolic equations have been studied for centuries, largely centered around or based upon the standard examples of the Laplace equation, the heat equation, and the wave equation.[edit]

Notation[edit]

When writing PDEs, it is common to denote partial derivatives using subscripts. For example:

$${\displaystyle {�}_{�}=\frac{\mathrm{\partial }�}{\mathrm{\partial }�},{�}_{��}=\frac{{\mathrm{\partial }}^2�}{\mathrm{\partial }{�}^2},{�}_{��}=\frac{{\mathrm{\partial }}^2�}{\mathrm{\partial }�\mathrm{\partial }�}=\frac{\mathrm{\partial }}{\mathrm{\partial }�}\left(\frac{\mathrm{\partial }�}{\mathrm{\partial }�}\right).}$$

In the general situation that u is a function of n variables, then u_i denotes the first partial derivative relative to the i-th input, u_ij denotes the second partial derivative relative to the i-th and j-th inputs, and so on.

The Greek letter Δ denotes the Laplace operator; if u is a function of n variables, then

$${\displaystyle \mathrm{\Delta }�={�}_{11}+{�}_{22}+\cdots +{�}_{��}.}$$

In the physics literature, the Laplace operator is often denoted by ∇²; in the mathematics literature, ∇²u may also denote the Hessian matrix of u.

Equations of first order[edit]

Main article: First-order partial differential equation

Linear and nonlinear equations[edit]

Linear equations[edit]

A PDE is called linear if it is linear in the unknown and its derivatives. For example, for a function u of x and y, a second order linear PDE is of the form

$${\displaystyle {�}_1(�,�){�}_{��}+{�}_2(�,�){�}_{��}+{�}_3(�,�){�}_{��}+{�}_4(�,�){�}_{��}+{�}_5(�,�){�}_{�}+{�}_6(�,�){�}_{�}+{�}_7(�,�)�=�(�,�)}$$

where a_i and f are functions of the independent variables only. (Often the mixed-partial derivatives u_xy and u_yx will be equated, but this is not required for the discussion of linearity.) If the a_i are constants (independent of x and y) then the PDE is called linear with constant coefficients. If f is zero everywhere then the linear PDE is homogeneous, otherwise it is inhomogeneous. (This is separate from asymptotic homogenization, which studies the effects of high-frequency oscillations in the coefficients upon solutions to PDEs.)

Nonlinear equations[edit]

Three main types of nonlinear PDEs are semilinear PDEs, quasilinear PDEs, and fully nonlinear PDEs.

Nearest to linear PDEs are semilinear PDEs, where only the highest order derivatives appear as linear terms, with coefficients that are functions of the independent variables. The lower order derivatives and the unknown function may appear arbitrarily. For example, a general second order semilinear PDE in two variables is

$${\displaystyle {�}_1(�,�){�}_{��}+{�}_2(�,�){�}_{��}+{�}_3(�,�){�}_{��}+{�}_4(�,�){�}_{��}+�({�}_{�},{�}_{�},�,�,�)=0}$$

In a quasilinear PDE the highest order derivatives likewise appear only as linear terms, but with coefficients possibly functions of the unknown and lower-order derivatives:

$${\displaystyle {�}_1({�}_{�},{�}_{�},�,�,�){�}_{��}+{�}_2({�}_{�},{�}_{�},�,�,�){�}_{��}+{�}_3({�}_{�},{�}_{�},�,�,�){�}_{��}+{�}_4({�}_{�},{�}_{�},�,�,�){�}_{��}+�({�}_{�},{�}_{�},�,�,�)=0}$$

Many of the fundamental PDEs in physics are quasilinear, such as the Einstein equations of general relativity and the Navier–Stokes equations describing fluid motion.

A PDE without any linearity properties is called fully nonlinear, and possesses nonlinearities on one or more of the highest-order derivatives. An example is the Monge–Ampère equation, which arises in differential geometry.^[3]

Linear equations of second order[edit]

Elliptic, parabolic, and hyperbolic partial differential equations of order two have been widely studied since the beginning of the twentieth century. However, there are many other important types of PDE, including the Korteweg–de Vries equation. There are also hybrids such as the Euler–Tricomi equation, which vary from elliptic to hyperbolic for different regions of the domain. There are also important extensions of these basic types to higher-order PDE, but such knowledge is more specialized.

The elliptic/parabolic/hyperbolic classification provides a guide to appropriate initial and boundary conditions and to the smoothness of the solutions. Assuming u_xy = u_yx, the general linear second-order PDE in two independent variables has the form

$${\displaystyle �{�}_{��}+2�{�}_{��}+�{�}_{��}+\cdots \text{(lower order terms)}=0,}$$

where the coefficients A, B, C... may depend upon x and y. If A² + B² + C² > 0 over a region of the xy-plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:

$${\displaystyle �{�}^2+2���+�{�}^2+\cdots =0.}$$

More precisely, replacing ∂_x by X, and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.

Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant B² − 4AC, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by B² − AC due to the convention of the xy term being 2B rather than B; formally, the discriminant (of the associated quadratic form) is (2B)² − 4AC = 4(B² − AC), with the factor of 4 dropped for simplicity.

1. B² − AC < 0 (elliptic partial differential equation): Solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where x < 0.
2. B² − AC = 0 (parabolic partial differential equation): Equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where x = 0.
3. B² − AC > 0 (hyperbolic partial differential equation): hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where x > 0.

If there are n independent variables x₁, x₂ , …, x_n, a general linear partial differential equation of second order has the form

$${\displaystyle ��=\sum _{�=1}^{�}\sum _{�=1}^{�}{�}_{�,�}\frac{{\mathrm{\partial }}^2�}{\mathrm{\partial }{�}_{�}\mathrm{\partial }{�}_{�}}+\text{lower-order terms}=0.}$$

The classification depends upon the signature of the eigenvalues of the coefficient matrix a_i,j.

1. Elliptic: the eigenvalues are all positive or all negative.
2. Parabolic: the eigenvalues are all positive or all negative, except one that is zero.
3. Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
4. Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues.^[4]

The theory of elliptic, parabolic, and hyperbolic equations have been studied for centuries, largely centered around or based upon the standard examples of the Laplace equation, the heat equation, and the wave equation.