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
${\displaystyle {\vec {F}}=m{\vec {a}}}$
Newton's equations describe the motion of bodies and are the basis of classical mechanics.
• Schroedinger Equation (Time dependent)
${\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi (r,t)+V\Psi (r,t)=-{\frac {\hbar }{i}}{\frac {\partial \Psi (r,t)}{\partial t}}}$
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
• Rational B-Spline

${\displaystyle P(t)={\frac {\sum _{i}W_{i}B_{i}(t)P_{i}}{\sum _{i}W_{i}B_{i}(t)}}}$

${\displaystyle P_{n+1}=rp_{n}(1-p_{n})}$

${\displaystyle B_{i}A_{i}=E_{i}A_{i}+\rho _{i}\sum _{j}B_{j}A_{j}F_{ji}{\frac {}{}}}$

${\displaystyle Z=\sum _{j}g_{j}e^{\frac {-E_{j}}{kT}}}$

${\displaystyle f=-\nabla ^{2}u+\lambda u}$

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

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

${\displaystyle \nabla \cdot {\vec {D}}=\rho }$

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

${\displaystyle \nabla \cdot {\vec {B}}=0}$

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

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

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

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

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