Navier–Stokes differential equations used to simulate airflow around an obstruction.
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.
Mathematics includes the study of such topics as quantity, structure, space, and change.
In mathematics, a partial differential equation (PDE) is a differential equation that contains beforehand unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.
Suppose a differential equation can be written in the form
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.
The term differential is used in calculus to refer to an infinitesimal change in some varying quantity. For example, if x is a variable, then a change in the value of x is often denoted Δx. The differential dx represents an infinitely small change in the variable x. The idea of an infinitely small or infinitely slow change is extremely useful intuitively, and there are a number of ways to make the notion mathematically precise.
Those who dislike Leibniz's notation may prefer to write this as
In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively.
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
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.
The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
(Note that we do not need to use two constants of integration, in equation (1) as in
because a single constant is equivalent.)
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 capacity of the environment.
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. In population biology, carrying capacity is defined as the environment's maximal load, which is different from the concept of population equilibrium. Its effect on population dynamics may be approximated in a logistic model, although this simplification ignores the possibility of overshoot which real systems may exhibit.
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
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.
Consider the one-dimensional heat equation. The equation is
The variable u denotes temperature. 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 ( 1 ) and using the product rule,
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:
− 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
From ( 2 ) 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 ( 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
From ( 7 ) 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 ( 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
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.
Suppose the equation is nonhomogeneous,
with the boundary condition the same as ( 2 ).
Expand h(x,t), u(x,t) and f(x) into
where hn(t) and bn can be calculated by integration, while un(t) is to be determined.
Substitute ( 9 ) and ( 10 ) back to ( 8 ) 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 ( 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.
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
Proceeding in the usual manner, we look for solutions of the form
and we obtain the equation
Writing this equation in the form
we see that the derivative with respect to x and y eliminates the first and last terms, so that
i.e. either F(x) or G(y) must be a constant, say -λ. This further implies that either or are constant. Returning to the equation for X and Y, we have two cases
which can each be solved by considering the separate cases for and noting that .
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.
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:
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.
Differential equations arise in many problems in physics, engineering, and other sciences. The following examples show how to solve differential equations in a few simple cases when an exact solution exists.
In probability theory and statistics, the exponential distribution is the probability distribution that describes the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.
In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.
The heat equation is a parabolic partial differential equation that describes the distribution of heat in a given region over time.
In physics, Langevin equation is a stochastic differential equation describing the time evolution of a subset of the degrees of freedom. These degrees of freedom typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation.
Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.
In mathematics, the Hodge star operator or Hodge star is a linear map introduced by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. The result when applied to an element of the algebra is called the element's Hodge dual.
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials.
The theory of functions of several complex variables is the branch of mathematics dealing with complex valued functions
In mathematics and its applications, a classical Sturm–Liouville theory, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is the theory of a real second-order linear differential equation of the form
In mathematics, a Cauchy-Euler equation is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of the particularly simple equidimensional structure the differential equation can be solved explicitly.
In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two state variables, say, or etc.. It is a two-dimensional case of the general n-dimensional phase space.
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T(v) is a scalar multiple of v. This condition can be written as the equation
Differential entropy is a concept in information theory that began as an attempt by Shannon to extend the idea of (Shannon) entropy, a measure of average surprisal of a random variable, to continuous probability distributions. Unfortunately, Shannon did not derive this formula, and rather just assumed it was the correct continuous analogue of discrete entropy, but it is not. The actual continuous version of discrete entropy is the limiting density of discrete points (LDDP). Differential entropy is commonly encountered in the literature, but it is a limiting case of the LDDP, and one that loses its fundamental association with discrete entropy.
A differential equation can be homogeneous in either of two respects.
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.
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 of its derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to their derivatives.
In mathematics, the method of steepest descent or stationary-phase method or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point, in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals.
The Fokas method, or unified transform, is an algorithmic procedure for analysing boundary value problems for linear partial differential equations and for an important class of nonlinear PDEs belonging to the so-called integrable systems. It is named after Greek mathematician Athanassios S. Fokas.
In mathematics, the Thomas–Fermi equation for the neutral atom is a second order non-linear ordinary differential equation, named after Llewellyn Thomas and Enrico Fermi, which can be derived by applying the Thomas–Fermi model to atoms. The equation reads