Equations {differential equation} can use derivatives to model change, without considering initial or boundary conditions.
purpose
Differential equations solve elasticity, spring, vibration, catenary, pendulum, astronomy, Earth-shape, and tractrix problems.
types
Ordinary differential equations have no partial derivatives. Partial differential equations have at least one partial derivative.
order
Differential equation has highest derivative {order, differential}.
degree
Differential equations have variable power {degree, equation} in highest-order derivative. First-degree differential equations {linear differential equation, degree} model additive phenomena.
Differential equations {ordinary differential equation} with no partial derivatives can model past or future conditions over time.
Integrating differential equation removes derivative and finds solution {general solution, differentiation}. Equations with nth derivative integrate n times. Variable power in highest-order derivative determines integration method.
integration constant
Because derivatives of constants are zero, general solutions are true to within an additive constant. General solutions are solution envelopes.
initial condition
Knowing one function value {boundary value} {initial value} or function derivative {initial condition} allows finding integration constant and so exact solution {particular solution} {singular solution}. If equation has nth derivative, n initial conditions find exact solution. Sum of general solution and particular solution is solution {complete solution}.
partial differential equations
Using variable-separation methods and/or infinite-series methods, to make ordinary differential equations, can solve partial differential equations.
conditions
To model situations that depend on conditions, use same differential equation and add integral equation to account for conditions separately.
existence proofs
To prove solution existence, demonstrate condition {Lipschitz condition}, demonstrate theorem {Cauchy-Lipschitz theorem}, or use iteration to reach solution {successive approximation method} {method of successive approximation} {existence of solutions}.
Terms {company} on non-homogeneous differential-equation right side can be similar to terms on left side.
Slight deviations from conic sections {method of perturbations} {perturbations method} can solve differential equations.
To solve differential equations with derivatives of x^n, sin(a*x), or cos(a*x), reset coefficients to one or zero, solve, and then put back coefficients {method of undetermined coefficients} {undetermined coefficients method}.
Small integral-value changes {method of variation of constants of integration} {variation of constants of integration method} can solve differential equations.
To solve differential equations with derivatives of functions that are not x^n, sin(a*x), or cos(a*x), use parameters to make ordinary differential equation and vary parameters to simplify equation {method of variation of parameters} {parameter variation method} {variation of parameters method}.
To solve differential equations with derivatives of functions that are not x^n, sin(a*x), or cos(a*x), substitute power series, such as Taylor series, for function {power series method}.
Substituting with algebraic equations can solve differential-equation systems {relaxation method, mathematics} {relaxation process}. Over an interval, select number of discrete points equal to number of variables in differential equations. At points, find approximate function values. At points, find partial derivative slope {differential coefficient} with respect to each variable: Df(x(i)) / Dx(i), where D denotes partial derivative, x is variable, and i is point/variable index. Write same number of algebraic equations as number of variables and points, each with a differential coefficient. Solve algebraic-equation system by computer.
iteration
Recognition algorithms can use iteration to move simultaneously toward optimum parameter values. Enhance some frequencies. Correlate with template. Equalize frequency histogram for more contrast. Subtract slowly varying information {background, recognition}. Find edge that has fast intensity change, using templates. Find surface orientations by neighboring reflectances. Find distances. Find velocities by comparing succeeding images. Find discontinuities and continuities.
After integration, solutions need point {boundary value, solution} {initial value, solution} to find integration constant. Problem can have no boundary or initial value {boundary value problem} {initial value problem}. Method of arithmetic means and method of sweeping out can find solutions to ordinary and partial differential-equation systems.
Potential function or harmonic function may or may not exist at boundary {Dirichlet problem} {first boundary-value problem}.
For homogeneous differential equations, equations {indicial equation} {characteristic equation, solution} can find solutions using base e raised to a power. r^n + a1 * r^(n - 1) + a2 * r^(n - 2) + ... + an = 0, where n is equation order, and r is general-solution highest power of e. Indicial equations remove highest-power term from differential equations, reducing equation degree.
Factors {integrating factor} can multiply an equation to make equation homogeneous.
To solve homogeneous differential equations, isolate variables {method of separation of variables} {separation of variables method}. Roots are e^(q*x) * (a + b*x + c * x^2 + ...), where q is coefficient, x is independent variable, and a b c are coefficients.
For first-order partial differential equations with n variables, variable separation can make ordinary differential equations with n parameters {method of characteristics} {characteristics method}. Characteristic curves and integrals are envelopes.
For first-order partial differential equations, variable separation can result in ordinary differential equations with parameters {Lagrange method}.
Power series with convergence domain can solve partial-differential-equation systems {method of majorant functions} {majorant function method}.
Green's theorem and Green's function can solve partial differential equations {method of singularities} {singularities method}.
Differential-equation solution can have infinite value {singularity, solution} at point {singular point, differential equation}. Singular point has at least one discontinuous differential coefficient. Singular point can be stable at focal point, where all curves through the point are convex. Singular point can be stable at center. Singular point can be unstable at point {node, intersection} where paths meet and end.
Theories {Fuchsian theory} can smooth singularities in linear differential equations.
Groups {monodromy group} can explain singularities in linear differential equations.
Singular points {saddle point}| can be unstable where convex and concave curves are orthogonal.
Non-homogeneous nth-order differential equations {adjoint equation} can have non-constant coefficients.
dy/dx = p(x) * y + q(x) * y^a, where a is not zero and a is not one {Bernoulli equation}.
y = x * y' + f(y') {Clairaut's equation} {Clairaut equation}.
Y(n,z) = sum from n = 0 to n = infinity of ((-1)^r * (z/2)^(n + 2*r) / (r! * (n + r)!)) * (2 * log(z/2) + 2*c - (sum from m = 1 to m = n + r of (1/m)) - (sum from m = 1 to m = r of (1/m))) - (sum from r = 0 to r = n - 1 of (z/2)^(2*r - n) * (n - r - 1)! / r!) {factorial equation}.
Special functions {Hermite function} can solve ordinary differential equations over infinite or semi-infinite intervals.
(p^2 - h^2) * (r^2 - k^2) * ((d^2)E(r) / (dr)^2) + r * (2 * p^2 - h^2 - k^2) * (dE(r) / dr) + ((h^2 + k^2) * p - n * (n + 1) * r^2) * E(r) = 0, where (d^2) is second derivative, r is radius, p is vertical dimension, n is parameter, and (k,h) is point {Lamé's differential equation} {Lamé differential equation}. Solution functions {Lamé function} are elliptical harmonics of first or second kind.
Linear equation has variables raised only to first power. Differential equation has derivatives {linear differential equation, calculus}. Second-order differential equation has second derivatives. Homogeneous equation has function equal to zero.
homogeneous
Second-order first-degree linear homogeneous differential general equation is a * (d^2)x + b * dx + c = 0, where (d^2) is second differential, d is first differential, x is independent variable, and a, b, and c are coefficients. General solution is c1 * e^(r1 * x) + c2 * e^(r2 * x), where e is base of natural logarithms, c1 and c2 are constants, r1 and r2 are roots, and x is independent variable.
non-homogeneous
Second-order, first-degree linear non-homogeneous differential general equation is a * (d^2)x + b * dx = -c, with same general solution.
Functions can have multiple values. Partial-differential-equation systems can model multiple-value functions {theory of multiple-valued functions} {multiple-valued function theory}.
To solve non-linear differential equations {non-linear differential equation}, look for stable point using qualitative theory or find characteristic equation, using theorems {Poincaré-Bendixson theorem} and operations {Painleve transcendent operations}.
dy/dx = a0(x) + a1(x) * y + a2(x) * y^2 {Riccati equation}.
Second-order ordinary differential equation can expand into infinite series of eigenfunctions {Sturm-Liouville theory, differential equation}.
No exact model exists for three mutually gravitationally interacting bodies {three-body problem}. Approximate solutions model mass-center straight-line motions and use energy and momentum conservation laws.
two dimensions
Perhaps, physical problems in three dimensions can reduce to problems in two dimensions using information concepts. Information is on surface, instead of in volume. Projection onto surface from volume has same information about positions, momenta, and transition probabilities. For special cases, reducing to two dimensions can solve the three-body problem.
Functions {Mathieu function} {Weber function}, in mutually orthogonal curvilinear coordinates, can solve the potential equation.
Circular, elliptic, hyperbolic, and other analytic functions {automorphic function} can generalize to find higher properties.
invariance
Automorphic functions are invariant if z' = (a*z + b) / (c*z + d) where a*d - b*c = 1, z is complex number, and z' is complex conjugate.
theta
theta(z) = sum from i = 0 to i = infinity of (c(i) * z + d(i))^(-2 * m) * H(z(i)), where m > 1 and H is rational function. Automorphic-function groups can be discrete or discontinuous groups of infinite order {theory of automorphic functions, discontinuous} {automorphic function theory, discontinuous}.
(d^2)u / (dr)^2 + du / (r * dr) + a^2 - b^2 / r^2 = 0 {Bessel equation}. x^2 * y'' + x * y' + (x^2 - n^2) * y = 0, where (d^2) is second derivative and x and n are complex, has two solutions. J(n,x) = (1 / (2 * pi)) * (integral from u = 0 to u = 2*pi of (cos(n*u) - x * sin(u)) * du). x * J(n+1,x) - 2 * n * J(n,x) + x * J(n-1,x) = 0.
Partial differential equation {partial differential equation} can have order greater than one, with second or higher derivatives. Partial differential equations of order greater than one are equivalent to first-order partial-differential-equation systems {system of partial differential equations}. For example, homogeneous, linear, second-order partial differential equation can be two first-order partial differential equations. c1 * (D^2)x + c2 * Dx + c3 = 0, where (D^2) is second derivative, D is first derivative, and c1, c2, and c3 are constants. c11 * Dx + c12 = 0 and d21 * Dx + d22 = 0, where D is first derivative and c11, c12, c21, and c22 are constants.
conditions
Partial differential equations can use boundary values and initial values.
Methods {arithmetic means method} {method of arithmetic means} {sweeping out method} {method of sweeping out} similar to ordinary-differential-equation methods can find partial-differential equation-system solutions.
Partial differential equation {heat-flow equation} {heat equation} can represent heat flow. Second derivatives of heat with respect to distance equal constant squared times first partial derivative of heat with respect to time: (D^2)T / Dx + (D^2)T / Dy + (D^2)T / Dz = (k^2) * (DT / Dt), where T is heat, (D^2) is second partial derivative, D is partial derivative, k is constant, and x, y, z, and t are coordinates.
Variable separation on partial differential equations can result in ordinary differential equations that use parameters {eigenfunction}| that have value sequences {eigenvalue, mathematics}. Ordinary differential equation solutions use eigenvalues. Second-order ordinary differential equations can expand into infinite series of eigenfunctions {Sturm-Liouville theory, eigenfunction}.
For homogeneous functions u with n variables, n*u = x * (Du/Dx) + y * (Du/Dy) + ..., where D are partial differentials {Euler's theorem} {Euler theorem}.
First-order partial differential equations {Navier-Stokes equation} describe fluid dynamics, using velocity, pressure, density, and viscosity. Examples are fluid motions and viscous-media object motions.
Partial differential equations {Plateau's problem} {Plateau problem} can represent surfaces of least area under closed boundaries. Example is soap film in loop.
Partial differential equations {total differential equation} can be P*dx + Q*dy + R*dz = 0.
Partial differential equations {excess function} {E-function} can represent energy function.
Energy or force equations can minimize quantities {least constraint principle} {principle of least constraint}. For example, sum of kinetic-energy-to-potential-energy changes over time {action} can be minimum: integral of (kinetic energy - potential energy) * dt.
Partial differential equations {Hamilton-Jacobi equation} can represent potential energy plus kinetic energy equals total energy. Sum of second partial derivatives of potential with respect to each coordinate and partial derivative of potential with respect to time equals zero: (D^2)V / Dx + (D^2)V / Dy + (D^2)V / Dz - DV / Dt = 0, where V is potential, (D^2) is second partial derivative, D is partial derivative, and x, y, z, and t are coordinates.
Operators {Laplace operator} {Laplace's operator}, on vector fields or potentials {del squared of f}, can be second derivatives, describe field-variation smoothness, be vectors, and be non-linear.
potential
Partial differential equations {potential equation} {Laplace's equation} can represent potentials. Potential V depends on distance r from mass or charge center: r = (x^2 + y^2 + z^2)^0.5.
Second partial derivative of potential V with respect to distance along x-axis plus second partial derivative of potential V with respect to distance along y-axis plus second partial derivative of potential V with respect to distance along z-axis equals zero: (D^2)V / Dx + (D^2)V / Dy + (D^2)V / Dz = 0, where (D^2) is second partial derivative, D is partial derivative, and V is constant times distance from center, because dx^2 / dx = 2 * x and d(2*x) / dx = 0.
solution
Spherical functions or Legendre polynomials can solve potential equation.
(1 - x^2) * y'' - 2 * x * y' + n * (n + 1) * y = 0, where n is parameter {Legendre differential equation}. Solutions are polynomials {Legendre polynomial}, potential equation spherical coordinates derived by variable separation, or spherical harmonics of second kind.
For boundaries with potential change zero, calculations can find potential change normal to region {Neumann problem} {second fundamental problem}.
If potential-equation right side equals -4 * pi * (energy density), rather than zero, equation describes gravitation and electrostatics {Poisson's equation} {Poisson equation}. Energy density is pressure.
Functions {periodic function} can solve partial differential equations (D^2)y / Dt = (a^2) * ((D^2)y / Dx), where (D^2) is second partial derivative, D is partial derivative, a is constant, t is time, x is distance, and y is function of time and distance. Representing functions by infinite trigonometric series can solve periodic equations. Parameters can analyze function, so y(t,x) = h(t) * g(x). Parameters set equation eigenfunction and eigenvalues.
First-order partial differential equation {electromagnetic wave equation} describes electromagnetic-wave energy oscillations.
Waves {cylindrical wave} can have partial differential equations. Second partial derivative of velocity with respect to time, times 1/c^2, equals three times partial derivative of velocity with respect to distance along pipe length, times 1/z, plus second partial derivative of velocity with respect to distance: ((D^2)v / Dt) * (1 / c^2) = 3 * (Dv / Dz) * (1/z) + (D^2)v / Dz, where (D^2) is second partial derivative, D is partial derivative, v is velocity, z is distance, t is time, and c is constant.
Waves {spherical wave} can have partial differential equations. Second partial derivative of radial velocity with respect to time, times 1/c^2, equals four times partial derivative of radial velocity with respect to radius, times 1/V, plus second partial derivative of radial velocity with respect to radius: ((D^2)s / Dt) * (1/c^2) = 4 * (Ds / DV) * (1/V) + (D^2)v / DV, where (D^2) is second partial derivative, D is partial derivative, v is radial velocity (ds/dt), c is constant, and radius V = (x^2 + y^2 + z^2)^0.5.
Vibrators with fixed endpoints can have stationary waves. Wave equations {stationary wave equation} can model steady-state waves. Wavefunction del operator, potential energy change, plus constant times wavefunction, kinetic energy change, equals zero {reduced wave equation} {Helmholtz equation}: Dw + (k^2) * w = 0, where w is wavefunction, D is delta function, and k is constant. The solution is an exponential function with complex exponents.
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225