Such continuum problems must often be discretized in order to obtain a numerical solution.

The Theory of Consistent Approximations provides conditions under which solutions to a series of increasingly accurate discretized optimal control problem converge to the solution of the original, continuous-time problem.

The solution of the discretized problem converges to the solution of the continuous problem as the grid size goes to zero, and the speed of convergence is one of the factors of the efficiency of the method.

Complete solid solution alloys give single solid phase microstructure, while partial solutions give two or more phases that may or may not be homogeneous in distribution, depending on thermal ( heat treatment ) history.

When the partial pressure of is reduced, for example when a can of soda is opened, the equilibrium for each of the forms of carbonate ( carbonate, bicarbonate, carbon dioxide, and carbonic acid ) shifts until the concentration of in the solution is equal to the solubility of CO < sub > 2 </ sub > at that temperature and pressure.

One partial solution to this problem has been to double pump the bus.

A partial solution to this is some programs ' ability to view the composite-order of elements ( such as images, effects, or other attributes ) with a visual diagram called a flowchart to nest compositions, or " comps ," directly into other compositions, thereby adding complexity to the render-order by first compositing layers in the beginning composition, then combining that resultant image with the layered images from the proceeding composition, and so on.

Stereophonic sound provided a partial solution to the problem of creating some semblance of the illusion of live orchestral performers by creating a phantom middle channel when the listener sits exactly in the middle of the two front loudspeakers.

Making the arm longer to reduce this angle is a partial solution, but less than ideal.

While he is best known for the Kolmogorov – Arnold – Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory, topology, algebraic geometry, classical mechanics and singularity theory, including posing the ADE classification problem, since his first main result — the partial solution of Hilbert's thirteenth problem in 1957 at the age of 19.

A partial solution to these problems is the robots exclusion protocol, also known as the robots. txt protocol that is a standard for administrators to indicate which parts of their Web servers should not be accessed by crawlers.

The one dimensional wave equation is unusual for a partial differential equation in that a relatively simple general solution may be found.

: p < sub > i </ sub > is the partial pressure of the component i in the mixture ( in the solution )

An important application of finite differences is in numerical analysis, especially in numerical differential equations, which aim at the numerical solution of ordinary and partial differential equations respectively.

In order to prevent cycling and encourage greater movement through the solution space, a tabu list is maintained of partial or complete solutions.

* In partial differential equations, when the solution of the equation for the right-hand side f can be written as Tf above, the kernel becomes the Green's function.

At a given temperature, the composition of a pure carbonic acid solution ( or of a pure CO < sub > 2 </ sub > solution ) is completely determined by the partial pressure of carbon dioxide above the solution.

A topological soliton, also called a topological defect, is any solution of a set of partial differential equations that is stable against decay to the " trivial solution.

The Cauchy – Kowalevski theorem states that the Cauchy problem for any partial differential equation whose coefficients are analytic in the unknown function and its derivatives, has a locally unique analytic solution.

Even if the solution of a partial differential equation exists and is unique, it may nevertheless have undesirable properties.

where the index i iterates the components, N < sub > i </ sub > is the mole fraction of the i < sup > th </ sup > component in the solution, P is the pressure, the index T refers to constant temperature, V < sub > i, aq </ sub > is the partial molar volume of the i < sup > th </ sup > component in the solution, V < sub > i, cr </ sub > is the partial molar volume of the i < sup > th </ sup > component in the dissolving solid, and R is the universal gas constant.

DCTs are important to numerous applications in science and engineering, from lossy compression of audio ( e. g. MP3 ) and images ( e. g. JPEG ) ( where small high-frequency components can be discarded ), to spectral methods for the numerical solution of partial differential equations.

Its importance in physics is the result of its being the solution to the differential equation describing forced resonance.

If the differential equations are nonlinear and have a known solution, it may be possible to linearize the nonlinear differential equations at that solution.

The transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant solution of the nonlinear differential equations describing the system.

The following examples show how to solve differential equations in a few simple cases when an exact solution exists.

It is easy to confirm that this is a solution by plugging it into the original differential equation:

The evolution function Φ < sup > t </ sup > is often the solution of a differential equation of motion

Critical for the solution of certain differential equations, these functions are used throughout both classical and quantum physics.

Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.

The potential field is given by a function V: R < sup > 3 </ sup > → R and the trajectory is a solution of the differential equation

* Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations.

The general solution comes from the study of ordinary differential equations and can be solved by the use of a Green's function.

Oliver Heaviside FRS ( ( 18 May 1850 – 3 February 1925 ) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations ( later found to be equivalent to Laplace transforms ), reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis.

The general solution of the above differential equation is

Solving the differential equation above, a solution which is a sinusoidal function is obtained.

Such a vector field serves to define a generalized ordinary differential equation on a manifold: a solution to such a differential equation is a differentiable curve on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field.

If any two functions are solutions to Laplace's equation ( or any linear homogeneous differential equation ), their sum ( or any linear combination ) is also a solution.

The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.

His most original contribution is the elementary solution he provided for the Riemann – Hilbert problem f < sub >+</ sub > = g f < sub >−</ sub > about the existence of a differential equation with given monodromy group.

As an example, the field of real numbers is not algebraically closed, because the polynomial equation x < sup > 2 </ sup > + 1 = 0 has no solution in real numbers, even though all its coefficients ( 1 and 0 ) are real.

Assuming that the solution of this equation can be written as

The equation on the left is the Bessel equation which has the general solution

The equation on the right has the general solution

Then the solution of the acoustic wave equation is

Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution ; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.

In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation.

Adding concentrated acid reverses the equation, and the copper ions go back into an aqueous solution.

Assuming that all the particles start from the origin at the initial time t = 0, the diffusion equation has the solution

Bessel functions are also known as cylinder functions or cylindrical harmonics because they are found in the solution to Laplace's equation in cylindrical coordinates.

Y < sub > α </ sub >( x ) is necessary as the second linearly independent solution of the Bessel's equation when α is an integer.

* Conic bundle, an algebraic variety that appears as a solution of a Cartesian equation

Swiss physicist Felix Bloch provided a wave function solution to the Schrödinger equation with a periodic potential, called the Bloch wave.

As these methods are pushed to the limit, they approach the exact solution of the non-relativistic Schrödinger equation.

This problem leads to an equation of the eighth degree, of which one solution, the Earth's orbit, is known.

Often, ray tracing methods are utilized to approximate the solution to the rendering equation by applying Monte Carlo methods to it.

* is a constant connected with the solution to the Lane-Emden equation.

In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane.

Given r, the rate of rotation is easy to infer from the constant angular momentum L, so a 2D solution can be easily reconstructed from a 1D solution of this equation.

Since the separation of variables in this case involves dividing by y, we must check if the constant function y = 0 is a solution of the original equation.

Trivially, if y = 0 then y '= 0, so y = 0 is actually a solution of the original equation.