Laplace's Equation

Laplace’s equation named after Pierre-Simon Laplace, is a second order partial differential equation and written as;

∇²φ=0

In the above mentioned formula ∇² is the Laplace operator and φ is a scalar function. Both the Laplace’s and Poisson's equations are said to be the simplest examples of elliptic partial differential equations.

Laplace’s equation as potential theory

The common theory of solutions to Laplace's equation is called as potential theory. The solutions of Laplace’s equation are all harmonic functions and are essential in most of the areas of science, remarkably the areas of electromagnetism, astronomy and fluid dynamics. And this is because they can be used to accurately describe the behavior of electric, gravitational and fluid potentials. In the study of heat conduction, the Laplace equation is the constant state heat equation.

The Dirichlet problem for Laplace’s equation comprises of identifying a solution φ on some domain D such that φ on the boundary of D is equal to some given function. Because of the Laplace operator which appears in the heat equation, a physical interpretation of this problem such as fixing the temperature on the boundary of the domain according to the given specification of the boundary condition.

Temperature distribution and boundaries

Permit heat to move until a stationary state is reached that the temperature at each point on the domain doesn’t change anymore. The temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem. The Neumann boundary conditions for Laplace's equation state not the function  itself φ on the boundary of D but its normal derivative. Physically this is in relation to the building of a potential for a vector field whose effect is known at the boundary of D alone.

The solutions of Laplace’s equation are known as harmonic functions which are analytic within the domain where the equation is satisfied. Whether the functions are solutions to Laplace’s equation or their sum is a solution. And this particular property is known as the principle of superposition which is very helpful for instance; solutions to difficult problems can be built by summing easier solutions.

And thus Laplace's equations is widely used in the fields of chemistry, physics and even in mathematics for its solutions are benefitting the electrical, magnetic and gravitational potentials, of steady-state temperatures and of hydrodynamics. And the equation can be solved by the separation of variables in all 11 coordinate systems that the Helmholtz differential equation can perform.

Questions:

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