User:Stewartadcock/Grand challenge equations

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The Grand Challenge Equations.

Arguably, these are the most important fundamental equations in science.

Discovery of efficient/exact methods for solving any of these equations would be expected to revolutionise their respective fields and the modern computational sciences.

  • Newton's Equations
\vec F = m \vec a
Newton's equations describe the motion of bodies and are the basis of classical mechanics.
  • Schroedinger Equation (Time dependent)
- \frac{\hbar^{2}}{2m} \nabla^{2} \Psi (r,t) + V \Psi (r,t) = - \frac{\hbar}{i} \frac{\partial \Psi (r,t)}{\partial t}
Describe the quantum mechanical wavefunction, the basis of quantum chemistry.
  • Navier-Stokes Equation
  • Poisson Equation
  • Heat Equation
  • Helmhltz Equation
  • Discrete Fourier Transform
  • Maxwell's Equations
  • Partition Function
  • Population Dynamics
  • Combined First and Second Laws of Thermodynamics
  • Radiosity
  • Rational B-Spline

P(t)=\frac{\sum_{i} W_i B_{i}(t) P_i}{\sum_{i} W_{i} B_{i}(t)}

Pn + 1 = rpn(1 − pn)

B_{i}A_{i} = E_{i}A_{i} + \rho_{i} \sum_{j} B_{j}A_{j}F_{ji} \frac{}{}

Z = \sum_{j} g_{j} e^{ \frac{-E_{j}}{kT} }

f = - \nabla^2 u + \lambda u

\nabla^2 u = \frac{\partial u}{\partial t}

\nabla \times \vec E = - \frac{\partial \vec B}{\partial t}

\nabla \cdot \vec D = \rho

\nabla \times \vec H = \frac{\partial \vec D}{\partial t} + \vec J

\nabla \cdot \vec B = 0

\frac{\partial \vec u}{\partial t} + \left( \vec u \cdot \nabla \right) = - \frac{1}{\rho} \nabla p + \gamma \nabla^2 \vec u + \frac{1}{\rho} \vec F

dU = \left(\frac{\partial \vec U}{\partial S}\right)_{V} dS + \left(\frac{\partial \vec U}{\partial V}\right)_{S} dV

f = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}

F_{j} = \sum_{k=0}^{N-1} f_{k} e^{\frac{2 \pi ijk}{N}}

\frac{\partial \vec u}{\partial t} + \left(\vec u \cdot \nabla \right) \vec u = - \frac{1}{\rho} \nabla + \gamma \nabla^2 \vec u + \frac{1}{\rho} \vec F