Solution of the Laplace and Poisson Equations

Example 6.1 Solution of the Laplace and Poisson Equations. Write a general MATLAB function to determine the numerical solution of a two-dimensional elliptic partial differential equation of the general form  

The flat sides of the plate are insulated so that no heat is transferred through these sides. Calculate the temperature profiles within the plate. 

Calculate the temperature profiles within the plate and compare these with the results of 

Method of Solution We solve this problem by matrix inversion because matrix operations are much faster than element-by-element operations in M ATLAB, especially when a large number of equations are to be solved. In order to solve the set of equations in matrix format, u j values have to be rearranged as a column vector. Therefore, we put in order all the dependent variables in a vector and renumber them from to (p+ ) q + 1), as illustrated in Fig. E6.1 2. Using this single numbering system, Eq. (6.50) can be written as 

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The numerical solution of the Laplace and Poisson elliptic partial differential equations is demonstrated in Example 6.1. 

The main specificity of the lEF method is that, instead of starting from the boundary conditions as in the DPCM, it defines the Laplace and Poisson equations describing the specific system under scrutiny, here including also anisotropic dielectrics, ionic solutions, liquids with a flat surface boundary, quadrupolar liquids, and it introduces the relevant specifications by proper mathematical operators. The fundamental result is that the lEF formalism manages to treat structurally different systems within a common integral equation-like approach. In other words, the same considerations exploited in the isotropic DPCM model leading to the definition of a surface cheurge density a(s) which completely describes the solvent reaction response, are still valid here, also for the above mentioned extensions to non-isotropic systems. 

Methods bcused on apparent surface charges These methods have become very popular mainly due to the seminal work of the Pisa group [21,40-51]. In these techniques the reaction field generated in the solvent by the presence of the solute is treated by a set of apparent charges spread over the solute cavity. At the classical level the electrostatic contribution to solvation is determined by equation 14, which is rigorously derived from Laplace and Poisson equations. 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

Chapter 5 investigates the shape of liquid drops, bubbles, and the liquid surface in the vicinity of a solid surface, using the Laplace-Young equation. The last chapter. Chapter 6, contains a number of interesting properties and applications, such as the vibrational oscillations of soap film membranes and the application of soap films to the analogue solutions of the differential equations of Poisson and Laplace. 

Analogue Solutions to the Differential Equations of Laplace and Poisson... 

Solution to Example 6.1. This program calculates and plots % the temperature profiles of a rectangular plate by solving % Laplace or Poisson equation by finite difference method,... 

If U and U2 are solutions of Poisson s equation and obey the boundary condition (1.94), their difference = U2—U1 satisfies Laplace s equation and the condition on the surface S ... 

For this formulation, the solution to the inverse problem is unique [7] however, there still exists the problem of continuity of the solution on the data. The linear algebraic counterpart to the elliptic boundary value problem is often useful in discussing this problem of noncontinuity. The numerical solution to all elliptic boundary value problems (such as the Poisson and Laplace problems) can be formulated in terms of a set of linear equations, = b. For the solution of Laplace s equation, the system can be reformulated as ... 

Special linear systems arise from the Poisson equation, d uldx + d uldy = f x, y) on a rectangle, 0 Laplace equation of Section II.A is a special case where fix, y) = 0.] If finite differences with N points per variable replace the partial derivatives, the resulting linear system has equations. Such systems can be solved in 0(N log N) flops with small overhead by special methods using fast Fourier transform (FFT) versus an order of AC flops, which would be required by Gaussian elimination for that special system. Storage space also decreases from 2N to units. Similar saving of time and space from O(N ) flops, 2N space units to 0(N log N) flops and space units is due to the application of FFT to the solution of Poisson equations on a three-dimensional box. 

At all channel wall surfaces, the no-slip boundary condition is applied to the velocity field (the Navier-Stokes equation), the fixed zeta-potential boundary condition is imposed on the EDL potential field (the Poisson-Boltzmann equation), and the insulation boundary condition is assigned to the applied electric field (the Laplace equation), and the no-mass penetration condition is specified for the solute mass concentration field (the mass transport equation). In addition, the third-kind boundary condition (i. e., the natural convection heat transfer with the surrounding air) is applied to the temperature field at all the outside surfaces of the fabricated channels to simultaneously solve the energy equation for the buffer solution together with the conjugated heat conduction equation for the channel wall. 

By insertion of Eq. [52b] for j(r, ) into this equation, it is easy to see that a Poisson-type equation is obtained (actually a Laplace equation, with the rhs equal to zero). If the diffusion coefficient is a constant over the domain, then that variable drops out. Notice that, in Eq. [52b], an effective dielectric constant [exp(—pct)(r))] appears, and this expression can vary over orders of magnitude due to large variations in the potential. Finally, if we seek a solution to the time-dependent problem, we must solve the full Smoluchowski equation iteratively. The particle density then depends explicitly on time. 

The property of the surface configuration to reach its equilibrium value, with minimum area, in seconds is used to obtain the solution of mathematical minimization problems in the following two chapters, and in Chapter 6 it is applied to the solution of Laplace s and Poisson s differential equation. 

So, indeed, Laplace s equation results from the minimization of the electrostatic energy of the system. This result can easily be extended to all problems that require the solution of Laplace s equation, and can be generalized to include Poisson s equation. However this latter analysis is beyond the scope of this book. 

Scalar and vector fields that depend on more than one independent variable, which we write in the notation T fx, y), TCx, t), TCr), TCr, t), etc., are very often obtained as solutions to PDEs. Some classic equations of mathematical physics that we will consider are the wave equation, the heat equation, Laplace s equation, Poisson s equation, and the SchrOdinger equation for some exactly solvable quantum-mechanical problems. 

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