Inmathematics,separation of variables (also known as theFourier method) is any of several methods for solvingordinary andpartial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.

A differential equation for the unknown${\displaystyle f(x)}$ is separable if it can be written in the form

${\displaystyle \frac{d}{dx}f(x)=g(x)h(f(x))}$

where${\displaystyle g}$ and${\displaystyle h}$ are given functions. This is perhaps more transparent when written using${\displaystyle y=f(x)}$ as:

${\displaystyle \frac{dy}{dx}=g(x)h(y).}$

So now as long ash(y) ≠ 0, we can rearrange terms to obtain:

${\displaystyle \frac{dy}{h(y)}=g(x)dx,}$

where the two variablesx andy have been separated. Notedx (anddy) can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition ofdx as adifferential (infinitesimal) is somewhat advanced.

Those who dislikeLeibniz's notation may prefer to write this as

${\displaystyle \frac{1}{h(y)}\frac{dy}{dx}=g(x),}$

but that fails to make it quite as obvious why this is called "separation of variables". Integrating both sides of the equation with respect to${\displaystyle x}$, we have

${\displaystyle \int \frac{1}{h(y)}\frac{dy}{dx}dx=\int g(x)dx,}$

A1

or equivalently,

${\displaystyle \int \frac{1}{h(y)}dy=\int g(x)dx}$

because of thesubstitution rule for integrals.

If one can evaluate the two integrals, one can find a solution to the differential equation. Observe that this process effectively allows us to treat thederivative${\displaystyle \frac{dy}{dx}}$ as a fraction which can be separated. This allows us to solve separable differential equations more conveniently, as demonstrated in the example below.

(Note that we do not need to use twoconstants of integration, in equation (A1) as in

${\displaystyle \int \frac{1}{h(y)}dy+C_1=\int g(x)dx+C_2,}$

because a single constant${\displaystyle C=C_2-C_1}$ is equivalent.)

Population growth is often modeled by the "logistic" differential equation

${\displaystyle \frac{dP}{dt}=kP\left(1-\frac{P}{K}\right)}$

where${\displaystyle P}$ is the population with respect to time${\displaystyle t}$,${\displaystyle k}$ is the rate of growth, and${\displaystyle K}$ is thecarrying capacity of the environment.Separation of variables now leads to

which is readily integrated using partial fractions on the left side yielding

${\displaystyle P(t)=\frac{K}{1+A{e}^{-kt}}}$

where A is the constant of integration. We can find${\displaystyle A}$ in terms of${\displaystyle P\left(0\right)=P_0}$ at t=0. Noting${\displaystyle e^0=1}$ we get

${\displaystyle A=\frac{K-P_0}{P_0}.}$

Much like one can speak of a separable first-order ODE, one can speak of a separable second-order, third-order ornth-order ODE. Consider the separable first-order ODE:

${\displaystyle \frac{dy}{dx}=f(y)g(x)}$

The derivative can alternatively be written the following way to underscore that it is an operator working on the unknown function,y:

${\displaystyle \frac{dy}{dx}=\frac{d}{dx}(y)}$

Thus, when one separates variables for first-order equations, one in fact moves thedx denominator of the operator to the side with thex variable, and thed(y) is left on the side with they variable. The second-derivative operator, by analogy, breaks down as follows:

${\displaystyle \frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(\frac{d}{dx}(y)\right)}$

The third-, fourth- andnth-derivative operators break down in the same way. Thus, much like a first-order separable ODE is reducible to the form

${\displaystyle \frac{dy}{dx}=f(y)g(x)}$

a separable second-order ODE is reducible to the form

${\displaystyle \frac{d^{2}y}{d{x}^2}=f\left(y^{\prime }\right)g(x)}$

and an nth-order separable ODE is reducible to

${\displaystyle \frac{d^{n}y}{d{x}^n}=f\left(y^{(n-1)}\right)g(x)}$

Consider the simple nonlinear second-order differential equation:$${\displaystyle y^{″}=(y^{\prime }{)}^2.}$$This equation is an equation only ofy'' andy', meaning it is reducible to the general form described above and is, therefore, separable. Since it is a second-order separable equation, collect allx variables on one side and ally' variables on the other to get:$${\displaystyle \frac{d(y^{\prime })}{(y^{\prime }{)}^2}=dx.}$$Now, integrate the right side with respect tox and the left with respect toy':$${\displaystyle \int \frac{d(y^{\prime })}{(y^{\prime }{)}^2}=\int dx.}$$This gives$${\displaystyle -\frac{1}{y^{\prime }}=x+C_1,}$$which simplifies to:$${\displaystyle y^{\prime }=-\frac{1}{x+C_1}.}$$This is now a simple integral problem that gives the final answer:$${\displaystyle y=C_2-\ln |x+C_1|.}$$

The method of separation of variables is also used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as theheat equation,wave equation,Laplace equation,Helmholtz equation andbiharmonic equation.

The analytical method of separation of variables for solving partial differential equations has also been generalized into a computational method of decomposition in invariant structures that can be used to solve systems of partial differential equations.^[1]

Consider the one-dimensionalheat equation. The equation is

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}-\alpha \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}=0}$

1

The variableu denotes temperature. The boundary condition is homogeneous, that is

${\displaystyle u{|}_{x=0}=u{|}_{x=L}=0}$

2

Let us attempt to find anontrivial solution satisfying the boundary conditions but with the following property:u is a product in which the dependence ofu onx,t is separated, that is:

${\displaystyle u(x,t)=X(x)T(t).}$

3

Substitutingu back into equation (1) and using theproduct rule,

${\displaystyle \frac{T^{\prime }(t)}{\alpha T(t)}=\frac{X^{″}(x)}{X(x)}=-\lambda ,}$

4

whereλ must be constant since the right hand side depends only onx and the left hand side only ont. Thus:

${\displaystyle T^{\prime }(t)=-\lambda \alpha T(t),}$

5

and

${\displaystyle X^{″}(x)=-\lambda X(x).}$

6

−λ here is theeigenvalue for both differential operators, andT(t) andX(x) are correspondingeigenfunctions.

We will now show that solutions forX(x) for values ofλ ≤ 0 cannot occur:

Suppose thatλ < 0. Then there exist real numbersB,C such that

${\displaystyle X(x)=B{e}^{\sqrt{-\lambda }x}+C{e}^{-\sqrt{-\lambda }x}.}$

From (2) we get

${\displaystyle X(0)=0=X(L),}$

7

and thereforeB = 0 =C which impliesu is identically 0.

Suppose thatλ = 0. Then there exist real numbersB,C such that

${\displaystyle X(x)=Bx+C.}$

From (7) we conclude in the same manner as in 1 thatu is identically 0.

Therefore, it must be the case thatλ > 0. Then there exist real numbersA,B,C such that

${\displaystyle T(t)=A{e}^{-\lambda \alpha t},}$

and

${\displaystyle X(x)=B\sin (\sqrt{\lambda }x)+C\cos (\sqrt{\lambda }x).}$

From (7) we getC = 0 and that for some positive integern,

${\displaystyle \sqrt{\lambda }=n\frac{\pi }{L}.}$

This solves the heat equation in the special case that the dependence ofu has the special form of (3).

In general, the sum of solutions to (1) which satisfy the boundary conditions (2) also satisfies (1) and (3). Hence a complete solution can be given as

${\displaystyle u(x,t)=\sum _{n=1}^{\mathrm{\infty }}{D}_n\sin \frac{n\pi x}{L}\exp \left(-\frac{n^2{\pi }^2\alpha t}{L^2}\right),}$

whereD_n are coefficients determined by initial condition.

Given the initial condition

${\displaystyle u{|}_{t=0}=f(x),}$

8

we can get

${\displaystyle f(x)=\sum _{n=1}^{\mathrm{\infty }}{D}_n\sin \frac{n\pi x}{L}.}$

This is theFourier sine series expansion off(x) which is amenable toFourier analysis. Multiplying both sides with$\sin \frac{n\pi x}{L}$ and integrating over[0,L] results in

${\displaystyle D_n=\frac{2}{L}{\int }_0^{L}f(x)\sin \frac{n\pi x}{L}dx.}$

This method requires that the eigenfunctionsX, here${\left\{\sin \frac{n\pi x}{L}\right\}}_{n=1}^{\mathrm{\infty }}$, areorthogonal andcomplete. In general this is guaranteed bySturm–Liouville theory.

Suppose the equation is nonhomogeneous,

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}-\alpha \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}=h(x,t)}$

8

with the boundary condition the same as (2) and initial condition same as (8).

Expandh(x,t),u(x,t) andf(x) into

${\displaystyle h(x,t)=\sum _{n=1}^{\mathrm{\infty }}{h}_n(t)\sin \frac{n\pi x}{L},}$

9

${\displaystyle u(x,t)=\sum _{n=1}^{\mathrm{\infty }}{u}_n(t)\sin \frac{n\pi x}{L},}$

10

${\displaystyle f(x)=\sum _{n=1}^{\mathrm{\infty }}{b}_n\sin \frac{n\pi x}{L},}$

11

whereh_n(t) andb_n can be calculated by integration, whileu_n(t) is to be determined.

Substitute (9) and (10) back to (8) and considering the orthogonality of sine functions we get

${\displaystyle u_n^{\prime }(t)+\alpha \frac{n^2{\pi }^2}{L^2}{u}_n(t)=h_n(t),}$

which are a sequence oflinear differential equations that can be readily solved with, for instance,Laplace transform, orIntegrating factor. Finally, we can get

${\displaystyle u_n(t)=e^{-\alpha \frac{n^2{\pi }^2}{L^2}t}\left(b_n+{\int }_0^t{h}_n(s)e^{\alpha \frac{n^2{\pi }^2}{L^2}s}ds\right).}$

If the boundary condition is nonhomogeneous, then the expansion of (9) and (10) is no longer valid. One has to find a functionv that satisfies the boundary condition only, and subtract it fromu. The functionu-v then satisfies homogeneous boundary condition, and can be solved with the above method.

For some equations involving mixed derivatives, the equation does not separate as easily as the heat equation did in the first example above, but nonetheless separation of variables may still be applied. Consider the two-dimensionalbiharmonic equation

${\displaystyle \frac{{\mathrm{\partial }}^{4}u}{\mathrm{\partial }{x}^4}+2\frac{{\mathrm{\partial }}^{4}u}{\mathrm{\partial }{x}^2\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{4}u}{\mathrm{\partial }{y}^4}=0.}$

Proceeding in the usual manner, we look for solutions of the form

${\displaystyle u(x,y)=X(x)Y(y)}$

and we obtain the equation

${\displaystyle \frac{X^{(4)}(x)}{X(x)}+2\frac{X^{″}(x)}{X(x)}\frac{Y^{″}(y)}{Y(y)}+\frac{Y^{(4)}(y)}{Y(y)}=0.}$

Writing this equation in the form

${\displaystyle E(x)+F(x)G(y)+H(y)=0,}$

Taking the derivative of this expression with respect to${\displaystyle x}$ gives${\displaystyle E^{\prime }(x)+F^{\prime }(x)G(y)=0}$ which means${\displaystyle G(y)=const.}$ or${\displaystyle F^{\prime }(x)=0}$ and likewise, taking derivative with respect to${\displaystyle y}$ leads to${\displaystyle F(x)G^{\prime }(y)+H^{\prime }(y)=0}$ and thus${\displaystyle F(x)=const.}$ or${\displaystyle G^{\prime }(y)=0}$, hence eitherF(x) orG(y) must be a constant, say −λ. This further implies that either${\displaystyle -E(x)=F(x)G(y)+H(y)}$ or${\displaystyle -H(y)=E(x)+F(x)G(y)}$ are constant. Returning to the equation forX andY, we have two cases

and

which can each be solved by considering the separate cases for and noting that${\displaystyle {\mu }_i={\lambda }_i^2}$.

Inorthogonal curvilinear coordinates, separation of variables can still be used, but in some details different from that in Cartesian coordinates. For instance, regularity or periodic condition may determine the eigenvalues in place of boundary conditions. Seespherical harmonics for example.

For many PDEs, such as the wave equation, Helmholtz equation and Schrödinger equation, the applicability of separation of variables is a result of thespectral theorem. In some cases, separation of variables may not be possible. Separation of variables may be possible in some coordinate systems but not others,^[2] and which coordinate systems allow for separation depends on the symmetry properties of the equation.^[3] Below is an outline of an argument demonstrating the applicability of the method to certain linear equations, although the precise method may differ in individual cases (for instance in the biharmonic equation above).

Consider an initial boundary value problem for a function${\displaystyle u(x,t)}$ on${\displaystyle D=\{(x,t):x\in [0,l],t\ge 0\}}$ in two variables:

${\displaystyle (Tu)(x,t)=(Su)(x,t)}$

where${\displaystyle T}$ is a differential operator with respect to${\displaystyle x}$ and${\displaystyle S}$ is a differential operator with respect to${\displaystyle t}$ with boundary data:

${\displaystyle (Tu)(0,t)=(Tu)(l,t)=0}$ for${\displaystyle t\ge 0}$

${\displaystyle (Su)(x,0)=h(x)}$ for${\displaystyle 0\le x\le l}$

where${\displaystyle h}$ is a known function.

We look for solutions of the form${\displaystyle u(x,t)=f(x)g(t)}$. Dividing the PDE through by${\displaystyle f(x)g(t)}$ gives

${\displaystyle \frac{Tf}{f}=\frac{Sg}{g}}$

The right hand side depends only on${\displaystyle x}$ and the left hand side only on${\displaystyle t}$ so both must be equal to a constant${\displaystyle K}$, which gives two ordinary differential equations

${\displaystyle Tf=Kf,Sg=Kg}$

which we can recognize as eigenvalue problems for the operators for${\displaystyle T}$ and${\displaystyle S}$. If${\displaystyle T}$ is a compact, self-adjoint operator on the space${\displaystyle L^2[0,l]}$ along with the relevant boundary conditions, then by the Spectral theorem there exists a basis for${\displaystyle L^2[0,l]}$ consisting of eigenfunctions for${\displaystyle T}$. Let the spectrum of${\displaystyle T}$ be${\displaystyle E}$ and let${\displaystyle f_{\lambda }}$ be an eigenfunction with eigenvalue${\displaystyle \lambda \in E}$. Then for any function which at each time${\displaystyle t}$ is square-integrable with respect to${\displaystyle x}$, we can write this function as a linear combination of the${\displaystyle f_{\lambda }}$. In particular, we know the solution${\displaystyle u}$ can be written as

${\displaystyle u(x,t)=\sum _{\lambda \in E}{c}_{\lambda }(t)f_{\lambda }(x)}$

For some functions${\displaystyle c_{\lambda }(t)}$. In the separation of variables, these functions are given by solutions to${\displaystyle Sg=Kg}$

Hence, the spectral theorem ensures that the separation of variables will (when it is possible) find all the solutions.

For many differential operators, such as${\displaystyle \frac{d^2}{d{x}^2}}$, we can show that they are self-adjoint by integration by parts. While these operators may not be compact, their inverses (when they exist) may be, as in the case of the wave equation, and these inverses have the same eigenfunctions and eigenvalues as the original operator (with the possible exception of zero).^[4]

The matrix form of the separation of variables is theKronecker sum.

As an example we consider the 2Ddiscrete Laplacian on aregular grid:

${\displaystyle L={\mathbf{D}}_{\mathbf{x}\mathbf{x}}\oplus {\mathbf{D}}_{\mathbf{y}\mathbf{y}}={\mathbf{D}}_{\mathbf{x}\mathbf{x}}\otimes \mathbf{I}+\mathbf{I}\otimes {\mathbf{D}}_{\mathbf{y}\mathbf{y}},}$

where${\displaystyle {\mathbf{D}}_{\mathbf{x}\mathbf{x}}}$ and${\displaystyle {\mathbf{D}}_{\mathbf{y}\mathbf{y}}}$ are 1D discrete Laplacians in thex- andy-directions, correspondingly, and${\displaystyle \mathbf{I}}$ are the identities of appropriate sizes. See the main articleKronecker sum of discrete Laplacians for details.

Some mathematicalprograms are able to do separation of variables:Xcas^[5] among others.

• Inseparable differential equation

• Polyanin, Andrei D. (2001-11-28).Handbook of Linear Partial Differential Equations for Engineers and Scientists. Boca Raton, FL:Chapman & Hall/CRC.ISBN 1-58488-299-9.
• Myint-U, Tyn;Debnath, Lokenath (2007).Linear Partial Differential Equations for Scientists and Engineers. Boston, MA:Birkhäuser Boston.doi:10.1007/978-0-8176-4560-1.ISBN 978-0-8176-4393-5.
• Teschl, Gerald (2012).Ordinary Differential Equations and Dynamical Systems.Graduate Studies in Mathematics. Vol. 140. Providence, RI:American Mathematical Society.ISBN 978-0-8218-8328-0.

• "Fourier method",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• John Renze,Eric W. Weisstein, "Separation of variables" ("Differential Equation") atMathWorld.
• Methods of Generalized and Functional Separation of Variables at EqWorld: The World of Mathematical Equations
• Examples of separating variables to solve PDEs
• "A Short Justification of Separation of Variables"

• Ordinary differential equations
• Partial differential equations

• Articles with short description
• Short description is different from Wikidata