Courses taught at this level traditionally focus on some of the more basic topics, such as Fourier series and simple boundary value problems. In order to make it possible to achieve some of the more advanced methods in one semester, treatment is necessarily streamlined in certain ways.
Partial Differential Equations
Example: d ’ Alembert ’ s Wave Equation
Types of Equations
Linear equations are further classified according to the properties of terms with the highest orders of derivatives, since this determines many qualitative properties of solutions. Elliptic equations of the second order are associated with an operatorL of the form (1.6), so that the eigenvalues of the symmetric matrix[ai j] are strictly positive at every point in the domain. By default, we will use these terms in the homogeneous sense, meaning that the boundary values of the function or the derivative are set equal to zero.
Well Posed Problems
Note that hyperbolic and parabolic equations return to elliptic equations in the spatial variables if the solution is independent of time. For evolution equations, we also impose initial conditions that specify the values of u and possibly its time derivative at some point in time.
Approaches
For the numerical simulations, exact solutions for a cylindrical pipe were used to provide boundary data at the point where the pipe meets the aorta. Theoretical analysis also plays an important role here, in that the regularity theory for the fluid equations is used to predict the accuracy of the simulation.
Real Numbers
The latter must be elements of a set and therefore may not exist.
Complex Numbers
Leonhard Euler, perhaps the most influential mathematician of the 18th century, published this identity in 1748. The calculus rules for differentiating and integrating exponentials derive from power series expansion, and thus extend to the complex case.
Domains in R n
This is because the boundary of the sequence contained in the string must be a point in the string or a boundary point. Closed and open sets are related in the sense that the complement of an open set is closed, and vice versa.
Differentiability
This means that the support does not exclude points where the function merely "passes" through zero. The piecewise boundary assumption C1 ensures that the points of each component of the boundary have well-defined tangent spaces and normal directions.
Ordinary Differential Equations
It can be shown that the limit of this sequence exists and solves (2.6) under some general assumptions on F, leading to a proof of the following result. Given the solution to the growth/decay equation in Example 2.3, it is reasonable to start with an exponential solution as a guess.
Vector Calculus
With these definitions, we turn to the divergence theorem, which relates the flow of a vector field through a closed surface to the divergence of the field inside. On the surface side, the integrand becomes the directional derivative with respect to the outer normal of the unit that is labeled.
Exercises
In this case, calculate both sides of the formula from Theorem 2.6 and show that the result reduces to an application of the main theorem of calculus in dexj. In this exercise we show that the volume of the unit sphere inR is given by. A).
Model Problem: Oxygen in the Bloodstream
For the bloodstream model, we can reasonably assume that the velocity is independent of oxygen concentration (because oxygen makes up a relatively small part of the total density). As we noted above, it is reasonable for the bloodstream model to assume a linear relationship.
Lagrangian Derivative and Characteristics
By Theorem 3.2, the PDE (3.7) can be effectively reduced to a pair of ODEs, namely the characteristic equation (3.8) and the Lagrange derived equation (3.10). Assuming x≥0, it is natural to index the features with the starting time t0 so that x(t0)=0.
Higher-Dimensional Equations
As in the one-dimensional case, we now see that since (3.19) holds for an arbitrary region R, the integrand must vanish. Conservation of mass is reflected in the fact that the area of the spot is indeterminate.
Quasilinear Equations
Contrary to the characteristic equation (3.8) in the linear case, the equation for x(t) here depends on the initial condition u(0,x0). Example 3.9 In order to explicitly solve the transport equation, we simplify the initial condition to a piecewise linear function.
Exercises
Show that all features originating from x0∈ [0,1] meet at the same point (thus creating a hit). As we noted in Section 1.2, d'Alembert's derivation of the wave equation in the 18th century was an early milestone in the development of PDE theory.
Model Problem: Vibrating String
To develop an equation for the string, we apply Newton's laws of motion to the segments of the discretization as if they were individual particles. With respect to the angles labeled in Fig.4.2, the net vertical force on a single segment is ΔF(t,xj)=Tsinαj+Tsinβj.
Characteristics
By Theorem 3.2, the unique solution with an initial condition is w(0,x)=w0(x). 4.10) We will refer back to the original conditions and in a moment. To highlight the role that the characteristic lines play in the solution of Theorem 4.1, consider the functions.
Boundary Problems
The solution of the wave equation onR given in Theorem 4.1 can be adapted to the boundary conditions (4.15). The idea is to expand,htoRin in such a way that formula (4.8) gives a solution that satisfies the boundary conditions for everything.
Forcing Terms
In the solution formula (4.8), u(t,x) depends only on the values of the range and base of the triangle, Dt,x∩ {t =0}. The existence of the dependency domain is a constraint imposed by the propagation speed c.
Model Problem: Acoustic Waves
Since σ/ρ0 is assumed to be very small, we can linearize by taking a first-order Taylor approximation on the right-hand side. Note that the "acceleration" term on the right is the Lagrangian derivative of the velocity field.
Integral Solution Formulas
Theorem 4.10 shows that the influence interval of the point (t0,x0) is the forward light cone. The claimed two-dimensional solution follows by substituting this formula for and the corresponding result for in the Kirchhoff formula from theorem 4.10.
Energy and Uniqueness
If the constant is reset as in (4.30), then the range of influence is limited to the points in spacetime that can be reached with a velocity less than or equal toc. Proof Ifu1andu2are solutions of the equation with the same initial conditions, then w:=u1−u2 satisfies (4.30) with the initial conditions.
Exercises
This places all derivatives and coefficients dependent on t on the left and all involving terms on the right. In some cases, one or both of the reduced equations is an ODE that can be solved explicitly.
Model Problem: Overtones
Helmholtz Equation
From (5.8) we can derive the fundamental tone of the string, as predicted by d'Alembert's wave equation model. Example 5.3 The one-dimensional wave equation can be used to model the fluctuations in air pressure inside a clarinet.
Circular Symmetry
In contrast to strings, the model predicts that the clarinet spectrum will contain only odd multiples of the fundamental frequency ω1. Lemma 5.1 reduces the problem of determining drum frequencies to the Helmholtz equation.
Spherical Symmetry
Related to this set of Legendre functions are functions of angle variables called spherical harmonics. Since the term 1/r is radial, separation of the radial and angular variables is possible in (5.34).
Exercises
In particular, show that Imω>0 ifγ>0, which implies that the solutions decay exponentially with time. Use this to write the solution hl(s) in terms of Jk. d) Express the eigenvaluesλ in terms of Bessel zeros with fractional values ofk.
Model Problem: Heat Flow in a Metal Rod
The higher dimensional form of the heat equation can be derived by an argument similar to that given above. The importance of the heat equation as a model extends far beyond its original thermodynamic context.
Scale-Invariant Solution
Surprisingly, the function representing the random distribution of displacements plays no role in the final equation, except for the value of the constant D. In the next section, we will generalize this convolution formula to a higher dimension and check that they give a solution of the heat equation that satisfies the desired initial condition.
Integral Solution Formula
In combination, Theorems 6.2 and 6.3 show that a bounded solution of the heat equation opRn with continuous initial data satisfies (6.21). Another interesting feature of the heat kernel is the fact that Ht(x) is strictly positive for all >0 andx ∈Rn.
Inhomogeneous Problem
We can see the origin of the infinite propagation speed in Einstein's diffusion model of Sect.6.1. Rn Hε(y)f(t−ε,x−y)dny The corresponding result for (6.29) has no boundary terms due to the compact support of f, t.
Exercises
We have seen in previous chapters that separation of variables can produce families of product solutions for certain PDEs. Trigonometric series PDE solutions were extensively studied in the 18th century by d'Alembert, Euler, Bernoulli and others.
Inner Products and Norms
For an inner product space, the definition of Euclidean length in terms of the dot product implies that the function. We will explain how to determine that the norm does not come from the inner product in the exercises.
Lebesgue Integration
With this basic picture in mind, we will ask the reader to accept some important consequences of Lebesgue's definition without further reasoning. In the examples and exercises, we will restrict our attention to functions for which ordinary Riemannian integrals exist.
L p Spaces
The sensitivity of p to the spread of the function decreases as p increases, as shown by the fact that . The estimation on the right limits this amplitude in terms of the mass and decays as a function of time.
Convergence and Completeness
This property does not necessarily hold in a general normed vector space, as the following shows. In functional analysis, a complete normed vector space is called a Banach space, and a complete inner product space is called a Hilbert space.
Orthonormal Bases
Thus, the partial sums of the series cj[v]2 are bounded and the terms are all positive. To complete the proof, note that Sn[v] →vinH means that the limit asn→ ∞ of the left-hand side of (7.21) is zero.
Self-adjointness
The proper definition of self-adjoint in functional analysis involves a more precise specification of the domain on which and (7.24) holds. The cancellations occur as a result of the oscillations, just as for sine functions, as Fig.7.4 illustrates.
Exercises
In his study of heat flow in 1807, Fourier made the radical claim that it should be possible to represent all solutions of the one-dimensional heat equation by trigonometric series. Indeed, the difficult problem of convergence of Fourier series provided some of the strongest motivations for the development of these tools.
Series Solution of the Heat Equation
In this chapter we will analyze Fourier series in more detail and show that the Fourier approach provides a general solution for the one-dimensional heat equation. The fact that each term satisfies the heat equation does not guarantee that it does, due to the infinite series summation.
Periodic Fourier Series
In the particular case discussed in Example 8.1, the Fourier series was seen to converge very quickly. The Fourier series calculated in Example 8.2 is a good illustration of some of the different notions of convergence.
Pointwise Convergence
Because the sum (8.19) is finite, it is clear that the Dirichlet kernel is a smooth function onT. Hypothesis (8.16) thus guarantees that h(y) is equivalent to a function bounded on the interval [−ε, ε].
Uniform Convergence
Convergence in L 2
Recall from Section 7.3 that when we say an L2 function is Cm, we mean this only up to equivalence, i.e., the original function may require modification to a set of measure zero to make it Cm. As an application of the weak formulation (10.9), let us return to an issue that arose in the traffic model in Section 3.4.
Regularity and Fourier Coef fi cients
Exercises
From the arguments used in Theorem 8.13, (b) implies that the series (8.48) converges to a solution satisfying the initial conditions. Conclude that the distinct-valued Pk are orthogonal in L2(-1,1). b) Use the result of Exercise 8.5 to show that {Pk} forms an orthogonal basis for L2(−1,1).
Model Problem: The Laplace Equation
According to (9.7), this could be interpreted as a weighted average of f with a weight function that depends on r. Note that (9.9) is not the same as saying that the Fourier series for g converges, which is not necessarily true.
Mean Value Formula
The first of these integrals can be estimated by noting that ∂u/∂r is a directed derivative and thus bounded by the magnitude of e|∇u|. For harmonic functions, Theorem 9.3 gives a generalization of the circle formula (Theorem 9.3) to spherical averages in higher dimensions.
Strong Principle for Subharmonic Functions
Note that the same argument works for any radius < R, so this argument shows that u ≡u(x0) on all B(x0;R). Note that the uniqueness of solutions of the Laplace equation also follows directly from Green's first identity (Theorem 2.10), in the case that Ω has a piecewise C1 boundary enu ∈ C2(Ω;R).
Weak Principle for Elliptic Equations
For the maximum principle, we need a stronger assumption called uniform ellipticity, which states that for some fixed constant κ>0,. Switching to a basis in which A is diagonal, we can write tr(A B) in terms of eigenvalues.
Application to the Heat Equation
For a solution of the heat equation, both ± satisfy the hypothesis of Theorem 9.8, implying that. In Sect.6.3 we stated without proof a uniqueness result for solutions of the heat equation onRn.
Exercises
In Sect.1.2 we observed that d'Alembert's formula for a solution of the wave equation makes sense even when the initial data are not differentiable. Weak solutions turned out to be extremely useful, and eventually a consistent mathematical framework was developed.
Test Functions and Weak Derivatives
We can say that locally integrable functions are satisfyu= f in the given weak sense (10.3) holds for allψ∈Ccpt∞(R). Classical derivatives satisfy criterion (10.7) by integration of parts, so they automatically qualify as weak derivatives.
Weak Solutions of Continuity Equations
Let us check that this defines a weak solution forg∈L1loc(R). 10.10) To evaluate the integral, introduce the variables. For certain initial conditions, the definition (10.9) of a weak solution is not sufficient to determine the solution uniquely.
Sobolev Spaces
If ∂Ω is C1 piecewise, then for functions onH1(Ω) it is possible to define boundary constraints onL2(∂Ω) that generalize the boundary constraint of a continuous function. The theory of boundary constraints is too technical for us to cover here, but we can at least show how this works in the one-dimensional case.
Sobolev Regularity
Weak Formulation of Elliptic Equations
Weak Formulation of Evolution Equations
Exercises
Model Problem: The Poisson Equation
Dirichlet ’ s Principle
Coercivity and Existence of a Minimum
Elliptic Regularity
Eigenvalues by Minimization
Sequential Compactness
Estimation of Eigenvalues
Euler-Lagrange Equations
Exercises
Model Problem: Coulomb ’ s Law
The Space of Distributions
Distributional Derivatives
Fundamental Solutions
Green ’ s Functions
Time-Dependent Fundamental Solutions
Exercises
Fourier Transform
Tempered Distributions
The Wave Kernel
The Heat Kernel
Exercises
