PARTIAL DIFFERENTIAL EQUATIONS (MT02EC09) MODULE IV – BOUNDARY VALUE PROBLEMS (B. V. Ps) - Pramada Ramachandran Asst. Prof. Department of Mathematics St. Paul’s College, Kalamassery

BOUNDARY VALUE PROBLEMS(B. V. Ps) The Laplace’s Equation is given by A problem in which we are required to find ‘Ψ’ such that this equation is satisfied in a region of space V and also such that ‘Ψ’ satisfies certain conditions on the boundary ‘S ‘ of V is called a BVP for the Laplace Equation. There are several types of BVPs.

INTERIOR DIRICHLET PROBLEM: If ‘f’ is a continuous function described on the boundary ‘S’ of some finite region ‘V’, determine a function ‘Ψ’ that satisfies within V and Ψ = f on S. Eg; Finding the temperature distribution within a body in the steady state when each point of the surface is kept at a prescribed steady temperature. The solution of an interior Dirichlet problem, if it exists, is unique.

EXTERIOR DIRICHLET PROBLEM: If ‘f’ is a continuous function described on the boundary ‘S’ of some finite simply connected region ‘V’, determine a function ‘Ψ’ that satisfies outside V and Ψ = f on S. Eg; determining the distribution of potential outside a body whose surface potential is prescribed. The solution of an exterior Dirichlet problem is not unique unless some restriction is placed on the behaviour of Ψ as r → ∞ For instance, in the 3 dimensional case, the solution is unique provided |Ψ(x, y, z)| < C/r where C is a constant. Another example: In the 2 dimensional case, if Ψ is bounded at ∞, the solution is unique.

Note 1: When the region V is bounded, the solution of the exterior Dirchlet problem can be deduced from a corresponding interior Dirichlet Problem as follows: Choose a spherical surface ‘C’ within V with centre ‘O’ and radius ‘a’. A point ‘P’ outside V is mapped to a point ‘π’ inside C such that OP.Oπ = a2.Thus we can map the region exterior to the body surface S of V to a region V* lying inside C. Let Ψ*(π) be the solution of the Interior Dirichlet Problem 2 Ψ = 0 within V*, Ψ* = f*(π) for π ϵ S* and f*(π) = [a f(P)] /Oπ . Then, Ψ(P) = [a. Ψ*(π)]/OP is the solution of the exterior Dirichlet Problem outside V, Ψ = f(P) for P on S.

Illustration:

Note 2: Consider the potential Ψ(x, y, z) = Ψ0(x, y, z) – 2z log (x2 + y2) due to a charge kz on the segment . , Ψ0 taking value 1 at the origin. On the surface x2 + y2 = e –c/2z, the second term takes the value ‘c’. Any equipotential surface on which Ψ = 1+c passes through the origin, so the potential at the origin is undefined. Thus, the Dirichlet Problem may not possess a solution assuming prescribed values at all points of the boundary. This was shown by Lebesgue.

INTERIOR NEUMANN PROBLEM: If ‘f’ is a continuous function defined uniquely at each point of the boundary ‘S’ of a finite region V, the Interior Neumann Problem is to determine Ψ(x, y, z) such that within V and coincides with f at every point of S.

NECESSARY CONDITION FOR THE EXISTENCE OF SOLUTIONS TO THE INTERIOR NEUMANN PROBLEM: By Gauss Theorem, Putting so that and putting the normal, we get On the boundary, Hence, . Thus, So the necessary condition for the existence of a solution is that the integral of f over the boundary S should vanish.

EXTERIOR NEUMANN PROBLEM: If ‘f’ is a continuous function prescribed at each point of the smooth boundary S of a bounded simply connected region V, the problem is to find a function Ψ(x, y, z) satisfying the following conditions: (i) outside V and (ii) on S.

Note: In the 2D case, the interior Neumann Problem can be reduced to the Interior Dirichlet Problem. Illustration: Let S be a plane region and C its boundary. Let Ψ be a solution of the Interior Neumann Problem. Then, (i) within S and (ii) on C. Construct a function φ which satisfies the Cauchy Riemann Equations within S and also on C. Then ‘Ψ+iφ’ is an analytic function and φ is defined uniquely except for the constant term. It can also be shown that φ is harmonic. Thus if we know φ on C, we can find φ within S. Using the Cauchy Riemann Equations; we can determine Ψ within S. This is precisely the solution to the Interior Dirichlet Problem.

INTERIOR CHURCHILL PROBLEM: If ‘f’ is a continuous function prescribed on the boundary ‘S’ of a finite region ‘V’, determine a function Ψ(x, y, z) satisfying the following conditions: (i) within V and (ii) at every point of S. An exterior Churchill Problem can be defined similarly.