User:Stewartadcock/Grand challenge equations

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

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

$$P_{n+1} = r p_{n} (1-p_{n})$$

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

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

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

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

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

$$\nabla \cdot \vec D = \rho$$

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

$$\nabla \cdot \vec B = 0$$

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

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

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

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

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