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.
3-Calculus-Differential Equation-Kinds
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Date Modified: 2022.0225