Laplace Equation

The second-order partial differential equation which is commonly used in physics as its solution R also known as harmonic functions takes place in problems related to electric, magnetic and gravitational potentials of state temperatures as well as of hydrodynamics is called Laplace’s equation. The French mathematician and astronomer Pierre-Simon Laplace discovered Laplace’s equation. According to Laplace’s equation the total of second-order partial derivatives of R, the unknown function, as per the Cartesian coordinates, is equal to zero. Laplace’s equation can be depicted as: Some pivotal points to note about Laplace’s equation are:

• The equation does not have a dependence on time rather it has a dependence on the spatial variables such as x and y.
• Laplace’s equation can also be described as a steady-state equation such that it has steady-state temperature distribution, stress distribution and potential distribution.

Laplace’s equation in polar coordinates

After leading to the derivation of Laplace’s equation in polar coordinates, it can be convenient in representing the heat and wave equation in polar coordinates. Laplace’s equation in polar coordinates can be described as:
U_yy + U_xx = U_rr + (1/r)U_r + (1/r²)U_θθ = Zero
For the heat equation, the solution U(x,y,t) is equal to U(r, θ, t), that is,
U_t is equal to K(U_xx +U_yy) which is equal to K[u_rr +(1/r)u_r + (1/r²)u_θθ]  where,
K > zero: diffusivity
On the other hand, for the wave equation, one can see,
U_tt is equal to c²(u_xx +u_yy) is equal to c²[U_rr + (1/r)U_r +(1/r²) U_θθ] where,
C > zero:wave velocity

Conclusion

As we have reached the conclusion of this topic, the concept around the Laplace equation has become crystal clear. It can be understood that one can describe the Laplace equation as of immense significance as per the world of science. One can conveniently explain the Laplace equation as the second-order partial differential equation that is frequently used in physics as its solution R also known as harmonic functions takes place in problems related to electric, magnetic and gravitational potentials of state temperatures as well as of hydrodynamics. The equation was first discovered by French mathematician and astronomer Pierre-Simon Laplace. From the above discussion, it can be observed that the Laplace equation does not have any dependence on time, instead it has a dependence on the spatial variables such as x and y. Last but not the least, it can be concluded that once we have led to the derivation of Laplace’s equation in polar coordinates, it can be convenient in representing the heat and wave equation in polar coordinates