# Preface

This section presents solutions to the Laplace and Helmholtz equations with mixed boundary conditions in rectangular coordinates.

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Introduction to Linear Algebra with *Mathematica*

## Glossary

# Mixed Problems for Laplace and Helmholtz Equations

In this section, we consider boundary value problems for Laplace's and Helmholtz equations subject to the boundary condition of the **third**. This is a type of boundary condition that involve both the functions and its normal derivative in the boundary conditions:

*a*and

*b*are nonzero functions or constants, not simultaneously zero. The third type boundary conditions are variously designated, but frequently are called Robin's boundary conditions, which is mistakenly associated with the French mathematical analyst Victor Gustave Robin (1855--1897) from the Sorbonne in Paris. Actually, Robin never used this boundary condition as it follows from the historical research article:

Karl Gustafson and Takehisa Abe. The third boundary condition—was it robin’s?

*The Mathematical Intelligencer*, March 1998, Volume 20, Issue 1, pp. 63--71.

Suppose we want to find the steady-state temperature *u*(*x,y*) in a rectangular plate with two insulated boundaries. When no heat escapes from the lateral faces of the plate, we solve Laplace's equation

*x*(\( u_x (0,y) =0 \mbox{ and } u_x (a,) =0 \) ) and represented as \( u(x,y) = X(x)\,Y(y) . \) Substituting

*u*=

*X Y*into Laplace's equation, we get

*X*:

*Y*, we have a similar differential equation

*n*= 0. Adding products of all partial solutions, we represent the required solution in the form of infinite series:

*y*, we have

*a*'s:

**Laplace's equation** is a second-order
partial differential equation named after Pierre-Simon
Laplace who, beginning in 1782, studied its properties while investigating the gravitational attraction of arbitrary bodies in space. However, the equation first appeared in 1752 in a paper by
Euler on hydrodynamics. Laplace's equation is often written as:

*f*is a given smooth function, is called the Poisson's equation. The equation is named after the French mathematician, geometer, and physicist Siméon Denis Poisson (1781--1840). In two dimensions, the Laplace equation in rectangular coordinates becomes

*f*is a given smooth function. The Laplacian occurs in differential equations that describe many physical phenomena, such as electric and gravitational potentials, the diffusion equation for heat and fluid flow, wave propagation, and quantum mechanics. The Laplacian represents the flux density of the gradient flow of a function. For instance, the net rate at which a chemical dissolved in a fluid moves toward or away from some point is proportional to the Laplacian of the chemical concentration at that point.

Since there is no time dependence in the Laplace's equation or Poisson's equation, there is no initial conditions to be satisfied by their solutions. However, there should be certain boundary conditions on the boundary curve or surface \( \partial\Omega \) of the region Ω in which the differential equation is to be solved. Typically, there are known three types of boundary conditions.

The problem of finding a solution of Laplace's equation that takes on given boundary values is known as a **Dirichlet problem**. On the other hand, if the values of the normal derivative are prescribed on the boundary, the problem is said to be a **Neumann problem**. Physically, it is plausible to expect that three types of boundary conditions will be sufficient to determine the solution completely. Indeed, it is possible to establish the existence and uniqueness of the solution of Laplace's (and Poisson's) equation under the first and third type boundary conditions, provided that the boundary \( \partial\Omega \) of the domain Ω is smooth (have no corner or edge). However, we avoid explicit statements and their proofs because the material is beyond the scope of the tutorial. Instead, we concentrate our attention on some typical problems that could be solved by means of separation of variables.

## Neumann Problem for a Rectangle

The general interior Neumann problem for Laplace's equation for rectangular domain \( [0,a] \times [0,b] , \) in Cartesian coordinates can be formulated as follows. Find solution of Laplace's equation*f*

_{0},

*f*

_{b},

*g*

_{0}, and

*g*

_{a}are give functions. Here we use the shortcut notation

*u*

_{x}and

*u*

_{y}for partial derivatives with respect to

*x*and

*y*, respectively.

Using superposition principle, we can break the given Neumann problems into four similar problems when flux source comes only from one side of the rectangle, and other three sides are isolated. Hence, we can represent the solution as the sum of four functions

aA = Graphics[Text[Style["A", FontSize -> 16], {-0.4, -0.4}]];

a0 = Graphics[ Text[Style[ "\!\(\*SubscriptBox[\(u\), \(x\)]\)(a,y) = 0"], {1.5, -0.2}]];

a1 = Graphics[ Text[Style["\!\(\*SubscriptBox[\(u\), \(y\)]\)(x,0) = 0"], {3.5, 1.0}]];

a2 = Graphics[ Text[Style["\!\(\*SubscriptBox[\(u\), \(y\)]\)(x,b) = 0"], {1.5, 2.2}]];

ax = {Arrowheads[Large], Arrow[{{-0.3, 0}, {3.6, 0}}]};

ax1 = Graphics[{Thick, Black, ax}];

ay = {Arrowheads[Large], Arrow[{{0, -0.3}, {0, 2.6}}]};

ay1 = Graphics[{Thick, Black, ay}];

txt1 = Graphics[Text[Style["x", FontSize -> 14], {3.4, 0.25}]];

txt2 = Graphics[Text[Style["y", FontSize -> 14], {0.25, 2.4}]];

left = Graphics[ Text[Style[ "\!\(\*SubscriptBox[\(u\), \(x\)]\)(0,y) = \ \!\(\*SubscriptBox[\(g\), \(0\)]\)(y)"], {-0.7, 1}]];

delta = Graphics[ Text[Style["\[CapitalDelta] u = 0", FontSize -> 16], {1.5, 1.0}]];

Show[ax1, ay1, rectangle, txt1, txt2, delta, left, a0, a1, a2, aA, AspectRatio -> 3/4]

right = Graphics[ Text[Style[ "\!\(\*SubscriptBox[\(u\), \(x\)]\)(a,y) = \ \!\(\*SubscriptBox[\(g\), \(a\)]\)(y)"], {3.8, 1}]];

a0 = Graphics[ Text[Style["\!\(\*SubscriptBox[\(u\), \(x\)]\)(0,y) = 0"], {-0.7, 1.0}]];

a1 = Graphics[ Text[Style[ "\!\(\*SubscriptBox[\(u\), \(y\)]\)(x,0) = 0"], {1.5, -0.3}]];

a2 = Graphics[ Text[Style["\!\(\*SubscriptBox[\(u\), \(y\)]\)(x,b) = 0"], {1.5, 2.2}]];

aB = Graphics[Text[Style["B", FontSize -> 16], {-0.4, -0.4}]];

Show[rectangleB, right, txt1, txt2, ax1, ay1, a0, a1, a2, aB]

down = Graphics[ Text[Style[ "\!\(\*SubscriptBox[\(u\), \(y\)]\)(x,0) = \ \!\(\*SubscriptBox[\(f\), \(0\)]\)(x)"], {1.5, -0.3}]];

a0 = Graphics[ Text[Style["\!\(\*SubscriptBox[\(u\), \(x\)]\)(0,y) = 0"], {-0.7, 1.0}]];

a1 = Graphics[ Text[Style["\!\(\*SubscriptBox[\(u\), \(x\)]\)(a,y) = 0"], {3.5, 1.0}]];

a2 = Graphics[ Text[Style["\!\(\*SubscriptBox[\(u\), \(y\)]\)(x,b) = 0"], {1.5, 2.2}]];

aC = Graphics[Text[Style["C", FontSize -> 16], {-0.4, -0.4}]];

Show[rectangleC, down, txt1, txt2, ax1, ay1, a0, a1, a2, aC]

top = Graphics[ Text[Style[ "\!\(\*SubscriptBox[\(u\), \(y\)]\)(x,b) = \ \!\(\*SubscriptBox[\(f\), \(b\)]\)(x)"], {1.5, 2.3}]];

a0 = Graphics[ Text[Style["\!\(\*SubscriptBox[\(u\), \(x\)]\)(0,y) = 0"], {-0.7, 1.0}]];

a1 = Graphics[ Text[Style["\!\(\*SubscriptBox[\(u\), \(x\)]\)(a,y) = 0"], {3.5, 1.0}]];

a2 = Graphics[ Text[Style[ "\!\(\*SubscriptBox[\(u\), \(y\)]\)(x,0) = 0"], {1.5, -0.3}]];

aD = Graphics[Text[Style["D", FontSize -> 16], {-0.4, -0.4}]];

Show[rectangleD, top, txt1, txt2, ax1, ay1, a0, a1, a2, aD, delta]

**Problem A ** consists of the partial differential equation \( \Delta u =0 \) in the rectangle \( (0,a) \times (0,b) \) subject to the boundary conditions:

*y*. This give the familiar Sturm--Liouville problem for

*Y*:

*X*, we get the following differential equation:

*C*is an arbitrary constant. Since each term in the above sum satisfies the homogeneous boundary conditions \( u_y (x,0) = u_y (x,b) =0 , \) we know that the sum has the same property (subject to uniform convergence, which is assumed). To satisfy the boundary conditions in variable

*x*, we have to consider two equations

**Problem B ** consists of the partial differential equation \( \Delta u =0 \) in the rectangle \( (0,a) \times (0,b) \) subject to the boundary conditions:

*y*. To solve problem B, we proceed in exactly the same as in the previous problem: set \( u(x,y) = X(x)\,Y(y) \) and substitute into the Laplace equation and homogeneous boundary conditions in

*y*. This give the familiar Sturm--Liouville problem for

*Y*:

*X*, we get the following differential equation:

*C*is an arbitrary constant. Since each term in the above sum satisfies the homogeneous boundary conditions \( u_y (x,0) = u_y (x,b) =0 , \) we know that the sum has the same property (subject to uniform convergence, which is assumed). To satisfy the boundary conditions in variable

*x*, we have to consider two equations

*A*

_{n}should be chosen so that the identity

*g*, and its coefficients follow from Euler's formulas:

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