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Differential equations
Navier–Stokes differential equations used to simulate airflow around an obstruction.
Scope Natural sciences Engineering Astronomy Physics Chemistry Biology Geology Applied mathematics Continuum mechanics Chaos theory Dynamical systems Social sciences Economics Population dynamics
Classification
Types Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear By variable type Dependent and independent variables Autonomous Complex Coupled / Decoupled Exact Homogeneous / Nonhomogeneous Features Order Operator Notation
Relation to processes Difference (discrete analogue) Stochastic Delay
Solution
General topics Picard–Lindelöf theorem Wronskian Phase portrait Phase space Lyapunov / Asymptotic / Exponential stability Rate of convergence Series / Integral solutions Numerical integration Dirac delta function
Solution methods Inspection Method of characteristics Euler Exponential response formula Finite difference (Crank–Nicolson) Finite element Infinite element Finite volume Galerkin Petrov–Galerkin Integrating factor Integral transforms Perturbation theory Runge–Kutta Separation of variables Undetermined coefficients
People Isaac Newton Gottfried Leibniz Leonhard Euler Émile Picard Józef Maria Hoene-Wroński Ernst Lindelöf Rudolf Lipschitz Augustin-Louis Cauchy John Crank Phyllis Nicolson Carl David Tolmé Runge Martin Kutta
v t e

In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial 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.

Ordinary differential equations (ODE)

Suppose a differential equation can be written in the form

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

which we can write more simply by letting y=f(x){displaystyle y=f(x)}:

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

As long as h(y) ≠ 0, we can rearrange terms to obtain:

dyh(y)=g(x)dx,{displaystyle {dy over h(y)}=g(x),dx,}

so that the two variables x and y have been separated. dx (and dy) 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 of dx as a differential (infinitesimal) is somewhat advanced.

Alternative notation

Those who dislike Leibniz's notation may prefer to write this as

1h(y)dydx=g(x),{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 x{displaystyle x}, we have

∫1h(y)dydxdx=∫g(x)dx,(1){displaystyle int {frac {1}{h(y)}}{frac {dy}{dx}},dx=int g(x),dx,qquad qquad (1)}

or equivalently,

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

because of the substitution 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 the derivative dydx{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 two constants of integration, in equation (1) as in

∫1h(y)dy+C1=∫g(x)dx+C2,{displaystyle int {frac {1}{h(y)}},dy+C_{1}=int g(x),dx+C_{2},}

because a single constant C=C2−C1{displaystyle C=C_{2}-C_{1}} is equivalent.)

Example

Population growth is often modeled by the differential equation

dPdt=kP(1−PK){displaystyle {frac {dP}{dt}}=kPleft(1-{frac {P}{K}}right)}

where P{displaystyle P} is the population with respect to time t{displaystyle t}, k{displaystyle k} is the rate of growth, and K{displaystyle K} is the carrying capacity of the environment.

Separation of variables may be used to solve this differential equation.

dPdt=kP(1−PK)∫dPP(1−PK)=∫kdt{displaystyle {begin{aligned}&{frac {dP}{dt}}=kPleft(1-{frac {P}{K}}right)\[5pt]&int {frac {dP}{Pleft(1-{frac {P}{K}}right)}}=int k,dtend{aligned}}}

To evaluate the integral on the left side, we simplify the fraction

1P(1−PK)=KP(K−P){displaystyle {frac {1}{Pleft(1-{frac {P}{K}}right)}}={frac {K}{Pleft(K-Pright)}}}

and then, we decompose the fraction into partial fractions

KP(K−P)=1P+1K−P{displaystyle {frac {K}{P(K-P)}}={frac {1}{P}}+{frac {1}{K-P}}}

Thus we have

∫(1P+1K−P)dP=∫kdtln⁡|P|−ln⁡|K−P|=kt+Cln⁡|K−P|−ln⁡|P|=−kt−Cln⁡|K−PP|=−kt−C|K−PP|=e−kt−C|K−PP|=e−Ce−ktK−PP=±e−Ce−ktLet A=±e−C.K−PP=Ae−ktKP−1=Ae−ktKP=1+Ae−ktPK=11+Ae−ktP=K1+Ae−kt{displaystyle {begin{aligned}&int left({frac {1}{P}}+{frac {1}{K-P}}right),dP=int k,dt\[6pt]&ln {begin{vmatrix}Pend{vmatrix}}-ln {begin{vmatrix}K-Pend{vmatrix}}=kt+C\[6pt]&ln {begin{vmatrix}K-Pend{vmatrix}}-ln {begin{vmatrix}Pend{vmatrix}}=-kt-C\[6pt]&ln {begin{vmatrix}{cfrac {K-P}{P}}end{vmatrix}}=-kt-C\[6pt]&{begin{vmatrix}{dfrac {K-P}{P}}end{vmatrix}}=e^{-kt-C}\[6pt]&{begin{vmatrix}{dfrac {K-P}{P}}end{vmatrix}}=e^{-C}e^{-kt}\[6pt]&{frac {K-P}{P}}=pm e^{-C}e^{-kt}\[6pt]{text{Let }}&A=pm e^{-C}.\[6pt]&{frac {K-P}{P}}=Ae^{-kt}\[6pt]&{frac {K}{P}}-1=Ae^{-kt}\[6pt]&{frac {K}{P}}=1+Ae^{-kt}\[6pt]&{frac {P}{K}}={frac {1}{1+Ae^{-kt}}}\[6pt]&P={frac {K}{1+Ae^{-kt}}}end{aligned}}}

Therefore, the solution to the logistic equation is

P(t)=K1+Ae−kt{displaystyle P(t)={frac {K}{1+Ae^{-kt}}}}

To find A{displaystyle A}, let t=0{displaystyle t=0} and P(0)=P0{displaystyle Pleft(0right)=P_{0}}. Then we have

P0=K1+Ae0{displaystyle P_{0}={frac {K}{1+Ae^{0}}}}

Noting that e0=1{displaystyle e^{0}=1}, and solving for A we get

A=K−P0P0.{displaystyle A={frac {K-P_{0}}{P_{0}}}.}

Partial differential equations

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 the heat equation, wave equation, Laplace equation, Helmholtz equation and biharmonic equation.

Example: homogeneous case

Consider the one-dimensional heat equation. The equation is

∂u∂t−α∂2u∂x2=0{displaystyle {frac {partial u}{partial t}}-alpha {frac {partial ^{2}u}{partial x^{2}}}=0}

(1)

The variable u denotes temperature. The boundary condition is homogeneous, that is

u|x=0=u|x=L=0{displaystyle u{big |}_{x=0}=u{big |}_{x=L}=0}

(2)

Let us attempt to find a solution which is not identically zero satisfying the boundary conditions but with the following property: u is a product in which the dependence of u on x, t is separated, that is:

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

(3)

Substituting u back into equation (1) and using the product rule,

T′(t)αT(t)=X″(x)X(x).{displaystyle {frac {T'(t)}{alpha T(t)}}={frac {X''(x)}{X(x)}}.}

(4)

Since the right hand side depends only on x and the left hand side only on t, both sides are equal to some constant value − λ. Thus:

T′(t)=−λαT(t),{displaystyle T'(t)=-lambda alpha T(t),}

(5)

and

X″(x)=−λX(x).{displaystyle X''(x)=-lambda X(x).}

(6)

− λ here is the eigenvalue for both differential operators, and T(t) and X(x) are corresponding eigenfunctions.

We will now show that solutions for X(x) for values of λ ≤ 0 cannot occur:

Suppose that λ < 0. Then there exist real numbers B, C such that

X(x)=Be−λx+Ce−−λx.{displaystyle X(x)=Be^{{sqrt {-lambda }},x}+Ce^{-{sqrt {-lambda }},x}.}

From (2) we get

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

(7)

and therefore B = 0 = C which implies u is identically 0.

Suppose that λ = 0. Then there exist real numbers B, C such that

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

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

Therefore, it must be the case that λ > 0. Then there exist real numbers A, B, C such that

T(t)=Ae−λαt,{displaystyle T(t)=Ae^{-lambda alpha t},}

and

X(x)=Bsin⁡(λx)+Ccos⁡(λx).{displaystyle X(x)=Bsin({sqrt {lambda }},x)+Ccos({sqrt {lambda }},x).}

From (7) we get C = 0 and that for some positive integer n,

λ=nπL.{displaystyle {sqrt {lambda }}=n{frac {pi }{L}}.}

This solves the heat equation in the special case that the dependence of u 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

u(x,t)=∑n=1∞Dnsin⁡nπxLexp⁡(−n2π2αtL2),{displaystyle u(x,t)=sum _{n=1}^{infty }D_{n}sin {frac {npi x}{L}}exp left(-{frac {n^{2}pi ^{2}alpha t}{L^{2}}}right),}

where D_n are coefficients determined by initial condition.

Given the initial condition

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

we can get

f(x)=∑n=1∞Dnsin⁡nπxL.{displaystyle f(x)=sum _{n=1}^{infty }D_{n}sin {frac {npi x}{L}}.}

This is the sine series expansion of f(x). Multiplying both sides with sin⁡nπxL{displaystyle sin {frac {npi x}{L}}} and integrating over [0,L] result in

Dn=2L∫0Lf(x)sin⁡nπxLdx.{displaystyle D_{n}={frac {2}{L}}int _{0}^{L}f(x)sin {frac {npi x}{L}},dx.}

This method requires that the eigenfunctions of x, here {sin⁡nπxL}n=1∞{displaystyle left{sin {frac {npi x}{L}}right}_{n=1}^{infty }}, are orthogonal and complete. In general this is guaranteed by Sturm-Liouville theory.

Example: nonhomogeneous case

Suppose the equation is nonhomogeneous,

∂u∂t−α∂2u∂x2=h(x,t){displaystyle {frac {partial u}{partial t}}-alpha {frac {partial ^{2}u}{partial x^{2}}}=h(x,t)}

(8)

with the boundary condition the same as (2).

Expand h(x,t), u(x,t) and f(x) into

h(x,t)=∑n=1∞hn(t)sin⁡nπxL,{displaystyle h(x,t)=sum _{n=1}^{infty }h_{n}(t)sin {frac {npi x}{L}},}

(9)

u(x,t)=∑n=1∞un(t)sin⁡nπxL,{displaystyle u(x,t)=sum _{n=1}^{infty }u_{n}(t)sin {frac {npi x}{L}},}

(10)

f(x)=∑n=1∞bnsin⁡nπxL,{displaystyle f(x)=sum _{n=1}^{infty }b_{n}sin {frac {npi x}{L}},}

(11)

where h_n(t) and b_n can be calculated by integration, while u_n(t) is to be determined.

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

un′(t)+αn2π2L2un(t)=hn(t),{displaystyle u'_{n}(t)+alpha {frac {n^{2}pi ^{2}}{L^{2}}}u_{n}(t)=h_{n}(t),}

which are a sequence of linear differential equations that can be readily solved with, for instance, Laplace transform, or Integrating factor. Finally, we can get

un(t)=e−αn2π2L2t(bn+∫0thn(s)eαn2π2L2sds).{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},dsright).}

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

Example: mixed derivatives

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-dimensional biharmonic equation

∂4u∂x4+2∂4u∂x2∂y2+∂4u∂y4=0.{displaystyle {frac {partial ^{4}u}{partial x^{4}}}+2{frac {partial ^{4}u}{partial x^{2}partial y^{2}}}+{frac {partial ^{4}u}{partial y^{4}}}=0.}

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

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

and we obtain the equation

X(4)(x)X(x)+2X″(x)X(x)Y″(y)Y(y)+Y(4)(y)Y(y)=0.{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

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

we see that the derivative with respect to x and y eliminates the first and last terms, so that

F′(x)G′(y)=0,{displaystyle F'(x)G'(y)=0,}

i.e. either F(x) or G(y) must be a constant, say -λ. This further implies that either −E(x)=F(x)G(y)+H(y){displaystyle -E(x)=F(x)G(y)+H(y)} or −H(y)=E(x)+F(x)G(y){displaystyle -H(y)=E(x)+F(x)G(y)} are constant. Returning to the equation for X and Y, we have two cases

X″(x)=−λ1X(x)X(4)(x)=μ1X(x)Y(4)(y)−2λ1Y″(y)=−μ1Y(y){displaystyle {begin{aligned}X''(x)&=-lambda _{1}X(x)\X^{(4)}(x)&=mu _{1}X(x)\Y^{(4)}(y)-2lambda _{1}Y''(y)&=-mu _{1}Y(y)end{aligned}}}

and

Y″(y)=−λ2Y(y)Y(4)(y)=μ2Y(y)X(4)(x)−2λ2X″(x)=−μ2X(x){displaystyle {begin{aligned}Y''(y)&=-lambda _{2}Y(y)\Y^{(4)}(y)&=mu _{2}Y(y)\X^{(4)}(x)-2lambda _{2}X''(x)&=-mu _{2}X(x)end{aligned}}}

which can each be solved by considering the separate cases for λi<0,λi=0,λi>0{displaystyle lambda _{i}<0,lambda _{i}=0,lambda _{i}>0} and noting that μi=λi2{displaystyle mu _{i}=lambda _{i}^{2}}.

Curvilinear coordinates

In orthogonal 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. See spherical harmonics for example.

Matrices

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

As an example we consider the 2D discrete Laplacian on a regular grid:

L=Dxx⊕Dyy=Dxx⊗I+I⊗Dyy,{displaystyle L=mathbf {D_{xx}} oplus mathbf {D_{yy}} =mathbf {D_{xx}} otimes mathbf {I} +mathbf {I} otimes mathbf {D_{yy}} ,,}

where Dxx{displaystyle mathbf {D_{xx}} } and Dyy{displaystyle mathbf {D_{yy}} } are 1D discrete Laplacians in the x- and y-directions, correspondingly, and I{displaystyle mathbf {I} } are the identities of appropriate sizes. See the main article Kronecker sum of discrete Laplacians for details.

See also

• Inseparable differential equation

References

• 
  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..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  


• 
  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. Retrieved 2011-03-29.
  


• 
  Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Graduate Studies in Mathematics. 140. Providence, RI: American Mathematical Society. ISBN 978-0-8218-8328-0.
  

External links

• 
  Hazewinkel, Michiel, ed. (2001) [1994], "Fourier method", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  


• John Renze, Eric W. Weisstein, Separation of variables (Differential Equation) at MathWorld.


• 
  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"