![]() Furthermore, a change of variables t = cos θ transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial P ℓ m(cos θ). Imposing this regularity in the solution Θ of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter λ to be of the form λ = ℓ ( ℓ + 1) for some non-negative integer with ℓ ≥ | m| this is also explained below in terms of the orbital angular momentum. The solution function Y( θ, φ) is regular at the poles of the sphere, where θ = 0, π. A priori, m is a complex constant, but because Φ must be a periodic function whose period evenly divides 2 π, m is necessarily an integer and Φ is a linear combination of the complex exponentials e ± imφ. Λ sin 2 θ + sin θ Θ d d θ ( sin θ d Θ d θ ) = m 2 įor some number m. Laplace's equation in two independent variables in rectangular coordinates has the form For example, solutions to complex problems can be constructed by summing simple solutions. This property, called the principle of superposition, is very useful. If any two functions are solutions to Laplace's equation (or any linear homogeneous differential equation), their sum (or any linear combination) is also a solution. Solutions of Laplace's equation are called harmonic functions they are all analytic within the domain where the equation is satisfied. In particular, at an adiabatic boundary, the normal derivative of φ is zero. For the example of the heat equation it amounts to prescribing the heat flux through the boundary. Physically, this corresponds to the construction of a potential for a vector field whose effect is known at the boundary of D alone. The Neumann boundary conditions for Laplace's equation specify not the function φ itself on the boundary of D but its normal derivative. The temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem. Allow heat to flow until a stationary state is reached in which the temperature at each point on the domain doesn't change anymore. Since the Laplace operator appears in the heat equation, one physical interpretation of this problem is as follows: fix the temperature on the boundary of the domain according to the given specification of the boundary condition. The Dirichlet problem for Laplace's equation consists of finding a solution φ on some domain D such that φ on the boundary of D is equal to some given function.
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