Separation of variables
Encyclopedia
In mathematics
, separation of variables is any of several methods for solving ordinary and partial differential equation
s, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.
which we can write more simply by letting
:
As long as h(y) ≠ 0, we can rearrange terms to obtain:
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.
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
, we have
or equivalently,
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
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 (2) as in
because a single constant
is equivalent.)
may be written as
If we let
and 
, we can write the differential equation in the form of equation (1) above. Thus, the differential equation is separable.
As shown above, we can treat
and 
as separate values, so that both sides of the equation may be multiplied by 
. Subsequently dividing both sides by 
, we have
At this point we have separated the variables x and y from each other, since x appears only on the right side of the equation and y only on the left.
Integrating both sides, we get
which, via partial fraction
s, becomes
and then
where C is the constant of integration
. A bit of algebra
gives a solution for y:
One may check our solution by taking the derivative with respect to x of the function we found, where B is an arbitrary constant. The result should be equal to our original problem. (One must be careful with the absolute values when solving the equation above. It turns out that the different signs of the absolute value contribute the positive and negative values for B, respectively. And the B = 0 case is contributed by the case that y = 1, as discussed below.)
Note that since we divided by
and 
we must check to see whether the solutions 
and 
solve the differential equation
(in this case they are both solutions). See also: singular solution
s.
where
is the population with respect to time 
, 
is the rate of growth, and 
is the carrying capacity
of the environment.
Separation of variables may be used to solve this differential equation.
To evaluate the integral on the left side, we simplify the fraction
and then, we decompose the fraction into partial fractions
Thus we have
Therefore, the solution to the logistic equation is
To find
, let 
and 
. Then we have
Noting that
, and solving for A we get
, wave equation
, Laplace equation and Helmholtz equation
.
.The equation is
The boundary condition is homogeneous, that is
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:
Substituting u back into equation,
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:
and
− λ here is the eigenvalue for both differential operators, and T(t) and X(x) are corresponding eigenfunction
s.
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
From we get
and therefore B = 0 = C which implies u is identically 0.
Suppose that λ = 0. Then there exist real numbers B, C such that
From 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
and
From we get C = 0 and that for some positive integer n,
This solves the heat equation in the special case that the dependence of u has the special form of .
In general, the sum of solutions to which satisfy the boundary conditions also satisfies and . Hence a complete solution can be given as
where Dn are coefficients determined by initial condition.
Given the initial condition
we can get
This is the sine series expansion of f(x). Multiplying both sides with
and integrating over [0,L] result in
This method requires that the eigenfunctions of x, here
, are orthogonal and complete. In general this is guaranteed by Sturm-Liouville theory
.
with the boundary condition the same as .
Expand h(x,t) ,u(x,t) and f(x,t) into
where hn(t) and bn can be calculated by integration, while un(t) is to be determined.
Substitute and back to and considering the orthogonality of sine functions we get
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
If the boundary condition is nonhomogeneous, then the expansion of and 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.
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.
As an example we consider the 2D discrete Laplacian
on a regular grid
:
where
and 
are 1D discrete Laplacians in the x- and y-directions, correspondingly, and 
are the identities of appropriate sizes. See the main article Kronecker sum of discrete Laplacians
for details.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, separation of variables is any of several methods for solving ordinary and partial differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s, 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 formwhich we can write more simply by letting
:As long as h(y) ≠ 0, we can rearrange terms to obtain:
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
Some who dislike Leibniz's notation may prefer to write this asbut that fails to make it quite as obvious why this is called "separation of variables".
Integrating both sides of the equation with respect to
, we haveor equivalently,
because of the substitution rule for integrals
Integration by substitution
In calculus, integration by substitution is a method for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool for mathematicians...
.
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
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
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
Arbitrary constant of integration
In calculus, the indefinite integral of a given function is only defined up to an additive constant, the constant of integration. This constant expresses an ambiguity inherent in the construction of antiderivatives...
, in equation (2) as in
because a single constant
is equivalent.)Example (I)
The ordinary differential equationmay be written as
If we let
and , we can write the differential equation in the form of equation (1) above. Thus, the differential equation is separable.As shown above, we can treat
and as separate values, so that both sides of the equation may be multiplied by . Subsequently dividing both sides by , we haveAt this point we have separated the variables x and y from each other, since x appears only on the right side of the equation and y only on the left.
Integrating both sides, we get
which, via partial fraction
Partial fraction
In algebra, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function ....
s, becomes
and then
where C is the constant of integration
Arbitrary constant of integration
In calculus, the indefinite integral of a given function is only defined up to an additive constant, the constant of integration. This constant expresses an ambiguity inherent in the construction of antiderivatives...
. A bit of algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
gives a solution for y:
One may check our solution by taking the derivative with respect to x of the function we found, where B is an arbitrary constant. The result should be equal to our original problem. (One must be careful with the absolute values when solving the equation above. It turns out that the different signs of the absolute value contribute the positive and negative values for B, respectively. And the B = 0 case is contributed by the case that y = 1, as discussed below.)
Note that since we divided by
and we must check to see whether the solutions and solve the differential equationDifferential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
(in this case they are both solutions). See also: singular solution
Singular solution
A singular solution ys of an ordinary differential equation is a solution that is singular or one for which the initial value problem fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full...
s.
Example (II)
Population growth is often modeled by the differential equation
where
is the population with respect to time , is the rate of growth, and is the carrying capacityCarrying capacity
The carrying capacity of a biological species in an environment is the maximum population size of the species that the environment can sustain indefinitely, given the food, habitat, water and other necessities available in the environment...
of the environment.
Separation of variables may be used to solve this differential equation.
To evaluate the integral on the left side, we simplify the fraction
and then, we decompose the fraction into partial fractions
Thus we have
-
-
-
-
-
-
-
-
Let
. -
-
-
-
-
Therefore, the solution to the logistic equation is
To find
, let and . Then we have
Noting that
, and solving for A we get
Partial differential equations
The method of separation of variables are also used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as heat equationHeat equation
The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...
, wave equation
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
, Laplace equation and Helmholtz equation
Helmholtz equation
The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation\nabla^2 A + k^2 A = 0where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...
.
Homogeneous case
Consider the one-dimensional heat equationHeat equation
The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...
.The equation is
The boundary condition is homogeneous, that is
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:
Substituting u back into equation,
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:
and
− λ here is the eigenvalue for both differential operators, and T(t) and X(x) are corresponding eigenfunction
Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
s.
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
From we get
and therefore B = 0 = C which implies u is identically 0.
Suppose that λ = 0. Then there exist real numbers B, C such that
From 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
and
From we get C = 0 and that for some positive integer n,
This solves the heat equation in the special case that the dependence of u has the special form of .
In general, the sum of solutions to which satisfy the boundary conditions also satisfies and . Hence a complete solution can be given as
where Dn are coefficients determined by initial condition.
Given the initial condition
we can get
This is the sine series expansion of f(x). Multiplying both sides with
and integrating over [0,L] result inThis method requires that the eigenfunctions of x, here
, are orthogonal and complete. In general this is guaranteed by Sturm-Liouville theorySturm-Liouville theory
In mathematics and its applications, a classical Sturm–Liouville equation, named after Jacques Charles François Sturm and Joseph Liouville , is a real second-order linear differential equation of the form...
.
Nonhomogeneous case
Suppose the equation is nonhomogeneous,with the boundary condition the same as .
Expand h(x,t) ,u(x,t) and f(x,t) into
where hn(t) and bn can be calculated by integration, while un(t) is to be determined.
Substitute and back to and considering the orthogonality of sine functions we get
which are a sequence of linear differential equations that can be readily solved with, for instance, Laplace transform,or Integrating factor
Integrating factor
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus, in this case often multiplying through by an...
. Finally, we can get
If the boundary condition is nonhomogeneous, then the expansion of and 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.
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
Discrete Laplace operator
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid...
on a regular grid
Regular grid
A regular grid is a tessellation of n-dimensional Euclidean space by congruent parallelotopes . Grids of this type appear on graph paper and may be used in finite element analysis as well as finite volume methods and finite difference methods...
:
where
and are 1D discrete Laplacians in the x- and y-directions, correspondingly, and are the identities of appropriate sizes. See the main article Kronecker sum of discrete LaplaciansKronecker sum of discrete Laplacians
In mathematics, the Kronecker sum of discrete Laplacians, named after Leopold Kronecker, is a discrete version of the separation of variables for the continuous Laplacian in a rectangular cuboid domain.-General form of the Kronecker sum of discrete Laplacians:...
for details.
External links
- Methods of Generalized and Functional Separation of Variables at EqWorld: The World of Mathematical Equations.
- Examples of separating variables to solve PDEs.