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Advanced Mathematical Economics
Paulo B. BritoPhD in Economics: 2020-2021
ISEGUniversidade de Lisboa
Lecture 618.11.2020
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Contents
7 First order quasi-linear partial differential equations 27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2 Scalar equations in the infinite domain and the method of characteristics . . . . . . . 6
7.2.1 Introduction: the method of characteristics . . . . . . . . . . . . . . . . . . . 67.2.2 The two simplest first order PDEs . . . . . . . . . . . . . . . . . . . . . . . . 77.2.3 Linear equation with constant coefficients . . . . . . . . . . . . . . . . . . . . 77.2.4 Semi-linear equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.2.5 Quasi-linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7.3 The linear equation in the semi-infinite domain and Laplace transform methods . . . 177.3.1 Linear equation with zero right-hand side . . . . . . . . . . . . . . . . . . . . 177.3.2 Linear equation with homogeneous right-hand side . . . . . . . . . . . . . . . 20
7.4 Qualitative dynamics of first-order PDEâs . . . . . . . . . . . . . . . . . . . . . . . . 217.5 Evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7.6.1 The transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.6.2 Age-structured population dynamics . . . . . . . . . . . . . . . . . . . . . . . 237.6.3 Cohortâs budget constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.6.4 Interest rate term-structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.6.5 Optimality condition for a consumer choice problem . . . . . . . . . . . . . . 287.6.6 Growth and inequality dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 30
7.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.A Laplace transforms and inverse Laplace transforms . . . . . . . . . . . . . . . . . . . 33
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Chapter 7
First order quasi-linear partialdifferential equations
7.1 Introduction
In this chapter we present introductory results on first-order partial differential equations (PDE)and some applications to demography and economics. Those equations can also be called hyperbolicPDEs, using a classification for second-order equations. They are another example of functionalequations.
A first-order (or hyperbolic) PDEs is a known function of one or more unknown functions ofmore than one independent variable, together with its first-order derivatives. If there is only oneunknown function we call the PDE scalar, and if there are two unknown functions we call it planarPDE.
In physics these equations model advection, travelling or transportation behaviors. They areused in mathematical demography for modelling age-dependent dynamics of population.
In economics, usually one of the independent variables is time and the other independent variableis the support of some distribution. They can be applied in continuous time overlapping generationsmodels, vintage capital models, interest rate term-structure models in continuous time. They alsoprovide an elegant and effective way of modelling the dynamics of distribution in heterogeneousagent economies.
The Hamilton-Jacobi equation for deterministic optimal control problems with a finite horizonand a constraint given by a ordinary differential equation is also usually a non-linear first-orderPDE.
The field is very large in terms of equations studied and methods involved, and is not generallyin the toolbox of economists. We will only present a very brief introduction allowing to study verysimple linear models.
There are two benefits from studying these equations: first, they provide a convenient modelling
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framework for setting up and characterizing the solution for models with heterogeneity, whichis becoming topical in economics, second, they provide a better understanding of the implicitassumptions which are introduced when using ODE (or difference equations ) models for studyingdynamics of heterogeneity.
We assume throughout that there are only two independent variables x = (ð¥1, ð¥2) â X â â2and deal mainly with equations of dimension one, that is mappings ð¢ ⶠX â â.
Definition A first-order partial differential equation in two independent variables x âX â â2 is a known relation ð¹ ⶠð· â â where ð· â â5 involving the unknown function ð¢ ⶠX â âand its gradient
ð¹(x, ð¢(x), âð¢(x)) = 0 (7.1)
where âð¢(x) is the gradient of ð¢(.), i.e.
âð¢(x) = (ðð¢(x)ðð¥1, ðð¢(x)ðð¥2
)â€
(ð¢ð¥1(x), ð¢ð¥2(x))â€
where we use the ð¢ð¥ð(.) =ðð¢(.)ðð¥ð = ðð¥ðð¢(x).
Types of first-order PDE First-order PDE are classified into four categories:
⢠linear PDE: if ð¹(â ) is linear in the derivatives âð¢(x) and ð¢,
ð(x) ð¢ð¥1 + ð(x) ð¢ð¥2 = ð(x) ð¢ + ð(x) (7.2)
where ð(â ), ð(â ), ð(â ) and ð(â ) are differentiable functions of x;
⢠semi-linear PDE: if ð¹(â ) is linear in âð¢(x) and non-linear in ð¢, which only enters into theright-hand side,
ð(x) ð¢ð¥1 + ð(x) ð¢ð¥2 = ð(x, ð¢); (7.3)
⢠quasi-linear PDE: if ð¹(â ) is linear in âð¢ and non-linear in ð¢ as
ð(x, ð¢) ð¢ð¥1 + ð(x, ð¢) ð¢ð¥2 = ð(x, ð¢) (7.4)
⢠non-linear PDE: if ð¹(â ) is non-linear in âð¢(x) and ð¢
ð¹(x, ð¢, âð¢) = 0
where ð¹ is non-linear in ð¢ð¥1 and/or ð¢ð¥2Linear equations can be classified further as non-autonomous or autonomous, if ð, ð, ð and ð
are constants, or non-homogeneous or homogeneous, if function ð(x, ð¢) is homogeneous in ð¢.
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Multi-dimensional first order PDE equations In addition we can consider systems ofhyperbolic equations
F(x, u(x), ð·xu(x) = 0
where x â âð and u ⶠâð â âð, and ð·xu denotes the Jacobian.For, instance a linear planar equation in two independent variables we have
A Dxu(x) = B u(x)
where
u(x) = (ð¢1(x)ð¢2(x)) and Dxu(x) = (
ðð¥1ð¢1(x) ðð¥2ð¢1(x)ðð¥1ð¢2(x) ðð¥2ð¢2(x)
)
and
A = (ð11 ð12ð21 ð22) and B = (ð1ð2
) .
Solutions A solution to a first-order PDE is a differentiable function ð(ð¥, ðŠ) that satisfies thePDE. Existence and uniqueness of solutions for first-order PDE, and for problems involving them,are not guaranteed. Classic solutions are solutions such that ð¢ â ð¶1(X). Otherwise we callgeneralised or weak solutions (i.e, non-differentiable or discontinuous solutions).
There are several methods for obtaining solutions which can be applied to general orspecific problems. The analytical methods for simpler equations are
⢠method of characteristics
⢠transformation methods (in particular application of Laplace transforms)
Those methods simplify the first-order PDE into a system of ODEâs or a parameterised ODE.
Problems involving PDEs There are two main types of problems involving first-order PDE.
1. the Cauchy problem: there is a single constraint on x along a surface Î â X:
â§{âš{â©
ð¹(x, ð¢(x), âð¢(x)) = 0, x â Xð¢ â£Î= ð, x â Î â X
where ð is a constant;
2. problems may involve two constraints, associated with each independent variable, for instance
â§{{âš{{â©
ð¹(x, ð¢(x), âð¢(x)) = 0, x â Xð¢ â£ð¥1=0= ð(ð¥2), (0, ð¥2) â Xð¢ â£ð¥2=0= ð(ð¥1), (ð¥1, 0) â X
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Well-posed problems Existence, uniqueness and properties of solutions vary widely. Again wehave to distinguish existence properties of the PDE and of the problem involving the PDE (i.e., thePDE and the boundary conditions). A problem is ill-posed if, for instance, although the PDEhas a solution the problem involving the PDE may not have a solution, in the same domain as thesolution to the PDE. A problem is well-posed if the general solution to the PDE has a particularsolution satisfying the constraints of the problem.
Linear PDEs, and well-posed problems involving liner PDEs, have explicit solutions.
Qualitative theory Let us consider the case in which time is one of the independent variables,i.e., ð¢ = ð¢(ð¡, ð¥) such that ð¢ ⶠR+ à R â R. Therefore, the solution of a first-order PDE describesa solution of a distribution, or wave, travelling over X across time.
For linear and well-posed PDEâs the distributional dynamics that characterizes their solutioncan be explicitly discovered.
For non-linear PDEs we are unaware of the existence of a qualitative theory as developed asthe qualitative theory for ODEâs. In particular, a Grobmann-Hartmann theorem for PDE doesnot seem to be available. There are phenomena that do not exist in ODEâs: traveling waves, frontwaves, for instance.
An important case of first-order PDEâs are equations of type
ð¢ð¡ + ð(ð¢)ð¥ = 0
that satisfy a conservation law
â«X
ð¢(ð¡, ð¥) ðð¥ = ï¿œÌï¿œ for every ð¡ â â+.
where ï¿œÌï¿œ is a constant. Given an initial function ð¢(0, ð¥) = ð(ð¥) such that
â«X
ð(ð¥) ðð¥ = ï¿œÌï¿œ for ð¡ = 0,
the solution ð¢(ð¡, ð¥) conserves the same mass throughout time. Assuming that the solution exists,we may be interested in characterizing the asymptotic behavior of the distribution limð¡ââ ð¢(ð¡, ð¥).Thisis the analog of studying the long-run behavior for ODEâs. However, while the solution of an ODEcan converge asymptotically to a steady state (a point) the solution of a PDE can converge asymp-totically to a function (or a steady state distribution).
In the rest of the chapter, in section 7.2 we solve scalar linear equations with an infinite domainthrough the method of characteristics. In section 7.3 we solve some linear equations in the semi-infinite domain by using Laplace transform methods. In section 7.4 we discsuss the qualitatitiveanalysis of first-order PDEâs when time is one independent variable. In Section 7.6 has severalapplications to economics and demography are presented.
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7.2 Scalar equations in the infinite domain and the method ofcharacteristics
7.2.1 Introduction: the method of characteristics
In this section we solve hyperbolic PDE in the infinite domain. We denote the independent variablesby x and assume that the domain of x is the whole set X = â2. We consider scalar functionð¢ ⶠâ2 â â as our dependent variable and consider quasi-linear equations of type 1
ð(x, ð¢) ð¢ð¥1(x) + ð(x, ð¢) ð¢ð¥2(x) = ð(x, ð¢(x)), x â X
and ð(.), ð(.), and ð(.) are known functions.One useful method to solve the hyperbolic PDE in the infinite domain is the method of
characteristics.The following definition is useful
Definition 1. Directional derivative Consider a function ð(x), the derivative of ð in thedirection given by vector v = (ð£ð¥1 , ð£ð¥2)†is
âvð(x) = limââ0ð(ð¥1 + â ð£ð¥1 , ð¥2 + â ð£ð¥2) â ð(x)
â
if the limit exists.
If function ð(ð¥, ðŠ) is differentiable, the directional derivative of ð in the direction given byvector v = (ð£ð¥, ð£ðŠ)†is equal to the dot product2
âvð(x) = âð(x) â v = (ðð¥1 , ðð¥2) â (ð£ð¥1 , ð£ð¥2) = ðð¥1(x)ð£ð¥1 + ðð¥2(x)ð£ð¥2 .
We start with simple linear PDE to illustrate their solution using the method of character-
istics. A characteristic is a curve in the domain x, which we can write as x = X(ð) such thatsuch that ð¢(X(ð)) behaves like an ODE. This means that for every ð â â the solution of the PDEbehaves like and ODE.
It is very important to remember that we assume, in all this section, that there are no restrictionson the domain of the independent variables, ð¥1 and ð¥2 in this section, that is, we assume x â â2.In the next sections we introduce constraints on the domain X.
1The following notation sometimes is more convenient
ð(x, ð¢)ðð¥ð¢(x) + ð(x, ð¢)ððŠð¢(x) = ð(x, ð¢(x)), x â X.
2Observe there is a relationship with the total differential. Let ð§ = ð(ð¥, ðŠ), where ð(.) is differentiable. The total
differential is ðð§ = ðð¥(ð¥, ðŠ)ðð¥ + ððŠ(ð¥, ðŠ)ððŠ. If we write ðð¥ = ð£ð¥â and ððŠ = ð£ðŠâ then âð(ð¥, ðŠ) â v = limââ0 ðð§â .
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7.2.2 The two simplest first order PDEs
We start with the two simplest first-order PDE: ð¢ð¥1(x) = 0 and ð¢ð¥2(x) = 0.
Proposition 1. The equation ð¢ð¥1(x) = 0, x â X = â2
has the general solutionð¢(x) = ð(ð¥2)
where ð â ð¶1(â) is an arbitrary differentiable function.
Proof. First observe that the solution to equation ð¢ð¥ = 0 is any function that remains constantalong direction ð£ = (1, 0)â€. This can be proved by observing that the directional derivative alongthat direction is zero,
âð¢(ð¥, ðŠ) â (1, 0) = ð¢ð¥ à 1 + ð¢ðŠ à 0 = ð¢ð¥ = 0.
This is equivalent to any function function, ð(.), that remains unchanged along any variationparallel to the ð¥-axis, that is ð(ðŠ).
In order to have a better intuition on this result, consider an ODE ð¢ð¥(ð¥) = 0, where ð¢(ð¥) is anunknown function of single independent variable ð¢ ⶠâ â X â â. This equation has the solutionð¢(ð¥) = ð where ð is an arbitrary point in the domain of ð¢(.), X â â. In the case of the PDEð¢ð¥1(x) = 0 the solution is ð¢(x) = ð(ð¥2) where ð(ð¥2) is an arbitrary differentiable function over â.
Proposition 2. The equationð¢ð¥2(x) = 0, x â X = â2
has the general solutionð¢(x) = ð(ð¥1)
where ð â ð¶1(â) is an arbitrary differentiable function.
Proof. Not the PDE solution is constant along the direction ð£ = (0, 1)â€, because it is equivalent tothe directional derivative along that direction being equal to zero,
âð¢(ð¥1, ð¥2) â (0, 1) = ð¢ð¥2 = 0.
In this case, the solution is any function function, ð(.), that remains unchanged along any variationparallel to the ðŠ-axis, that is ð(ð¥1).
From those two previous results we can understand more general linear first order scalar PDEâsas being constant along particular directions, which are called characteristics.
7.2.3 Linear equation with constant coefficients
Next we consider linear equations without side constrains and Cauchy problems for linear hyperbolicequations defined in the infinite domain.
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A. Free boundary problems
Consider the first order linear autonomous PDE
ð¢ð¥1(x) + ð ð¢ð¥2(x) = 0, x â X = â2 (7.5)
where ð â 0 is an arbitrary constant.
Proposition 3. The general solution of PDE (7.5) is
ð¢(x) = ð(ð¥2 â ð ð¥1),
where ð â ð¶1(â) is an arbitrary differentiable function.
Proof. First, observe that the PDE (7.5) determines a function ð¢(ð¥, ðŠ) which is constant along thedirection ð£ = (1, ð)â€, because
âð¢ â (1, ð) = ð¢ð¥ + ðð¢ðŠ = 0.
To interpret this geometrically consider the two-dimensional surface
ð â¡ { (ð¥, ðŠ, ð¢(ð¥, ðŠ))} â â3.
A particular solution of the PDE (ð¥0, ðŠ0, ð¢(ð¥0, ðŠ0)) belongs to the surface ð. But, next we showthat a solution to the PDE traces out a curve ð¶ over the surface ð in which ð¢ remains constant.We call this curve a characteristic curve.
In order to determine curve ð¶ we parametrize the two independent variables as ð¥ = ð(ð ),ðŠ = ð (ð ), where ð â â. Then, we get a parameterized value for ð¢, as ð¢ = ð(ð ) = ð¢(ð(ð ), ð (ð )). Acharacteristic curve is defined as
ð¶ = { (ð1(ð ), ð2(ð ), ð(ð )) ⶠð(ð ) = constant}.
Taking derivatives to ð¢ = ð(ð ) = ð¢(ð1(ð ), ð2(ð )) we find
ðððð =
ðð¢(ð1(ð ), ð2(ð ))ðð = ð¢ð¥1
ðð1ðð + ð¢ð¥2
ðð2ðð
The PDE will hold if and only if the following conditions hold:
⢠the characteristic system
ðð1ðð = 1
ðð2ðð = ð
⢠the compatibility conditionðððð = 0
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Solving the characteristic system and the compatibility equation we get
ð¥1 = ð1(ð ) = ð + ð1ð¥2 = ð2(ð ) = ðð + ð2ð¢ = ð(ð ) = ð(0)
where ð1 and ð2 are arbitrary constants, and ð(0) is an arbitrary function, say ð(ð). We can setð1 = 0 and ð2 = ð. Eliminating ð in the first two equations we find
ð = ð¥1 =ð¥2 â ð
ð .
This implies that there, a characteristic curve is a straight line in (ð¥1, ð¥2) with a constant valueð¥2 â ð ð¥1 = ð is constant. Then we find the general solution for (7.5) to be constant along thedirection (1, ð)â€,
ð¢(ð¥, ðŠ) = ð(ð ) = ð(ð) = ð(ð¥2 â ð ð¥1), where ð is an arbitrary ð¶1 function.
Verification: In order to check that this is a solution, assume that ð¢(ð¥1, ð¥2) = ð(ð¥2 â ð ð¥1)for ð(â ) â ð¶1(R) Then
ð¢ð¥1(ð¥1, ð¥2) + ð ð¢ð¥2 (ð¥1, ð¥2) = âððâ²(ð¥2 â ð ð¥1) + ðð
â²(ð¥2 â ð ð¥1) = 0
which is equation (7.5).We call projected characteristic to the line ð¥2 = ð + ð ð¥1, where ð â â is arbitrary, and we
call ð(ð¥2 â ð ð¥1) the first integral of the PDE.Figure 7.1 depicts projected characteristic lines for cases ð > 0 and ð < 0. These curves
correspond to the projection in the space (ð¥1, ð¥2) of the solution curves of the PDE (7.5) overwhich ð¢(ð¥1, ð¥2) is constant.
Figure 7.1: Characteristic lines for equation (7.5) for ð > 0 (left figure) and ð < 0 (right figure)
Linear right hand side Next we introduce the linear first-order PDE with an homogeneousright-hand side
ð¢ð¥1 + ð ð¢ð¥2 = ð ð¢, x â â2 (7.6)where ð â 0 and ð â 0 are constants.
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Proposition 4. The general solution of PDE (7.6) is
ð¢(x) = ð(ð¥2 â ð ð¥1) ðð ð¥1
where ð(.) is an arbitrary ð¶1 function.
Proof. To solve it by using the method of characteristics we parameterize again both the indepen-dent variables, ð¥1 = ð1(ð ) and ð¥2 = ð2(ð ), and the unknown function ð¢ = ð¢(ð1(ð ), ð2(ð )) = ð(ð )and solve the system
ðð1ðð = 1,
ðð2ðð = ð,
ðððð = ðð
which have solutions
ð¥1 = ð1(ð ) = ð , ð¥2 = ð2(ð ) = ð ð + ð, ð¢ = ð(ð ) = ð(ð) ððð .
where ð is an arbitrary constant and ð(ð) is an arbitrary function. Then ð = ð¥1 and the projectedcharacteristic if again ð¥2 â ð ð¥1 = ð and ð¢ = ð(ð)ððð1ðð ð¥1 = ð(ð)ðð ð¥1 .
This equation has the same projected characteristics as shown in figure 7.1 but now, the valueof ð¢(.) will not remain constant along the characteristics, as in the case of equation (7.5): it willgrow or decay along the characteristic at the rate ð, respectively, if ð > 0 or if ð < 0.
Cauchy problems
Consider again equation (7.5) and assume that we know the distribution for ð¥2 for a particularvalue of ð¥1, say ð¥1 = 0. If ð¥1 is interpreted as time, and ð¥2 as another independent variable, wecall the problem an initial-value problem (which is a particular case of the Cauchy problem)
â§{âš{â©
ð¢ð¥1(x) + ð ð¢ð¥2 (x) = 0, x â â+ à âð¢ = ð(ð¥2), x â {ð¥1 = 0} à â
(7.7)
where ð is a known ð¶1 function. We can write the initial condition as ð¢(0, ð¥2) = ð(ð¥2) whereð(.) is known.
Proposition 5. The general solution to the Cauchy problem (7.7) is
ð¢(ð¥1, ð¥2) = ð(ð¥2 â ð ð¥1), (ð¥1, ð¥2) â X = â2
Proof. In the three-dimensional surface ð, previously presented, the constraint defines a curve (0, ð¥2, ð(ð¥2)) that has a projection in the (ð¥1, ð¥2) space characterized by a curve passing throughpoint { (0, ð¥2)}. Using the same method that we used to determine the characteristic curve ð¶, weparameterize the constraint Î by a new variable ð, such that it defines a direction Î = { (0, ð)}.
Introducing the two parameterizations (associated to the characteristic curve and the initialcondition) we define
ð¥1 = ð1(ð , ð), ð¥2 = ð2(ð , ð)
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implyingð¢ = ð¢(ð1(ð , ð), ð2(ð , ð)) = ð(ð , ð).
The characteristic system and the compatibility condition become the system of parameterized(by ð) intitial value problems where the ODEâs have the independent variable ð
ðð1(ð , ð)ðð = 1, s.t ð1(0, ð) = 0,
ðð2(ð , ð)ðð = ð, s.t ð2(0, ð) = ð
ðð(ð , ð)ðð = 0, s.t ð(0, ð) = ð(ð).
The solution to the three ODE initial value problems allows us to obtain a relationship betweenthe initial independent variables and the parameters related to the characteristic and the initialcondition
ð¥1 = ð1(ð , ð) = ð (7.8)ð¥2 = ð2(ð , ð) = ðð + ð (7.9)
andð¢ = ð(ð , ð) = ð(0, ð) = ð(ð), for any ð â ð
To get the solution in the original independent variables, we have to obtain the reversed re-
lationships, say ð = ð(ð¥, ðŠ) and ð = ð (ð¥, ðŠ). In order to get it, observe that the solution forthe characteristic system can be written as (ð¥1, ð¥2) = ðº(ð , ð). If this system is invertible then(ð , ð) = ðºâ1(ð¥1, ð¥2). The system (7.8)-(7.9) can provide this solution:
(ð¥1ð¥2) = (1 0ð 1) (
ð ð) â (
ð ð) = (
1 0âð 1) (
ð¥1ð¥2
) = ( ð¥1ð¥2 â ð ð¥1)
Therefore, ð = ð¥1 and ð = ð¥2 â ð ð¥1. Then ð¢(ð¥1, ð¥2) = ð(ð , ð) = ð(ð) = ð(ð¥2 â ð ð¥1)
Example If the initial distribution is ð¢(0, ð¥2) = ð(ð¥2) = ðâð¥22 , then the solution to the Cauchy
problem (7.7) isð¢(ð¥1, ð¥2) = ðâ(ð¥1âð¥2)
2 .
Figure 7.2 illustrates this case. The projected characteristics are again as those depicted in figure7.1.
Next, consider an equation (7.6) and the associated Cauchy problem
â§{âš{â©
ð¢ð¥1(x) + ð¢ð¥2(x) = ð ð¢(x), x â â2
ð¢ = ð(ð¥2), x â { ð¥1 = 0} à â(7.10)
where ð â 0 is a constant.
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Figure 7.2: Solution for the problem ð¢ð¥ + ð¢ðŠ = 0 and ð¢(0, ð¥2) = ðâð¥22 , 3d plot and 2d plot for
ð¥1 â {â1, â0.5, 0, 0.5, 1}
Proposition 6. The solution to problem (7.10) is
ð¢(x) = ð(ð¥2 â ð¥1)ðð ð¥1
Exercise: prove this.In Figure 7.3 we present an illustration. Observe that for ð > 0 the solution has both a advection
(i.e., transport) and a growing behavior.
Figure 7.3: Solution for the problem ð¢ð¥ + ð¢ðŠ = 0.7ð¢ and ð¢(0, ðŠ) = ðâðŠ2 , 3d plot and 2d plot for
ð¥ â {â1, â0.5, 0, 0.5, 1}
Next, we will see what we can learn from the application of the method of characteristics tosolving the semi-linear and the quasi-linear equations.
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7.2.4 Semi-linear equation
We consider first one simple semi-linear equation that can be solved by transformation to a linearequation. Next we present conditions for the existence of solutions to more general semi-linearequations, with or without a zero right-hand side, i.e, with ð(x, ð¢) = 0 or ð(x, ð¢) â 0.
Semi-linear equation with zero right-hand-side
We consider the problem for a more general case, in which the coefficient functions are not specified
â§{âš{â©
ð(x)ð¢ð¥1 + ð(x)ð¢ð¥2 = 0, x â â2
ð¢ â£Î= ð, x â Î â â2(7.11)
where ð(.), ð(.) and ð(.) are ð¶1 functions in â2, and there is a constraint given by curve Î. Thesolution to this problem depends on the form of the constraint surface Î.
Let us consider the points in the constrained set parameterised by ð, write Î = { (ð¥1, ð¥2) =(ðŸ1(ð), ðŸ2(ð))} and define
ðŽ(ð) â¡ ð(ðŸ1(ð), ðŸ2(ð))ðµ(ð) â¡ ð(ðŸ1(ð), ðŸ2(ð))
We say that the constraint Î is characteristic if it is tangent to the projected characteristic
and Î is non-characteristic if it is not tangent to the projected characteristic.We will see next that, Î is characteristic if
ðŽ(ð)ðµ(ð) =
ðŸâ²1(ð)ðŸâ²2(ð)
and Î is non-characteristic ifðŽ(ð)ðµ(ð) â
ðŸâ²1(ð)ðŸâ²2(ð)
. (7.12)
Proposition 7. Consider the Cauchy problem (7.11). A unique solution exists if Î is non-characteristic in all its domain. The local solution to the problem exist and is unique, and can bewritten as
ð¢(ð¥1, ð¥2) = ð(ðº|â1Î (ð¥1, ð¥2)).
where det ðº|Î â 0.
Proof. In order to see this we proceed in two phases.
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Paulo Brito Advanced Mathematical Economics 2020/2021 14
⢠First: applying the same method as before, we introduce the change in coordinates ð¥1 =ð1(ð , ð), ð¥2 = ð2(ð , ð), implying ð¢ = ð¢(ð¥1, ð¥2) = ð¢(ð1(ð , ð), ð2(ð , ð)) = ð(ð , ð). The charac-teristic system and the compatibility condition become
ðð1(ð , ð)ðð = ð(ð1(ð , ð), ð2(ð , ð))
ðð2(ð , ð)ðð = ð(ð1(ð , ð), ð2(ð , ð))
ðð(ð , ð)ðð = 0
and the constraints on their values introduced by Î that we associate with ð = 0 are
ð1(0, ð) = ðŸ1(ð)ð2(0, ð) = ðŸ2(ð)ð(0, ð) = ð(ð).
If we solve the ODE characteristic system together with the initial conditions we obtain thetransformation (ð¥1, ð¥2) = ðº(ð , ð), where
ð¥1 = ð1(ð , ð) (7.13)ð¥2 = ð2(ð , ð). (7.14)
In order to obtain the solution satisfying ð¢ = ð(0, ð) = ð(ð) we need to solve system (7.13)-(7.14), that is, we need to find (ð , ð) = ðºâ1(ð¥1, ð¥2).
⢠Second: The system is locally invertible to ð = ð(ð¥, ðŠ) and ð = ð (ð¥, ðŠ) if we can apply theinverse function theorem (ð , ð) = ðºâ1(ð¥1, ð¥2). This is possible if the Jacobian of ðº has anon-zero determinant evaluated at points (0, ð).The Jacobian of system (7.13)-(7.14) evaluated at point (ð , ð) = (0, ð) is
ð·(ðº)|Î =âââââââ
ðð1ðð (0, ð)
ðð1ðð (0, ð)
ðð2ðð (0, ð)
ðð2ðð (0, ð)
âââââââ
= (ð(ðŸ1(ð), ðŸ2(ð)) ðŸâ²1(ð)
ð(ðŸ1(ð), ðŸ2(ð)) ðŸâ²2(ð)
) = (ðŽ(ð) ðŸâ²1(ð)
ðµ(ð) ðŸâ²2(ð))
Then det (ð·(ðº|Î)) â 0 if condition (7.12) holds, and, using the inverse function theorem, we can(at least locally) determine (ð , ð)|ð =0 = ðºâ1(ð¥1, ð¥2), and the solution will have the generic formð¢(ð¥, ðŠ) = ð(ðºâ1(ð¥1, ð¥2))
This means that, geometrically, the solution will propagate not along parallel characteristiclines but along lines which can change slope depending on the values of ð¥1 and ð¥2.
Exercise Consider next the special case of (7.11)
â§{âš{â©
ð¢ð¥1 + ð ð¥2 ð¢ð¥2 = 0, x â â2
ð¢(0, ð¥2) = ð(ð¥2) x â { ð¥1 = 0} à â
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Paulo Brito Advanced Mathematical Economics 2020/2021 15
where ð(â ) is an arbitrary ð¶1(â) function.Show that the solution is
ð¢(x) = ð(ð¥2 ðâð ð¥1) for any x â â2.
In this case we can observe that the characteristics are
ð¥2 = ð ðð ð¥1
they are still parallel as in the linear case, but behave differently depending on the sign of ð:1. if ð > 0 then ð¥2 diverges exponentially for increasing values of ð¥12. if ð = 0 then ð¥2 is constant for any values of ð¥13. if ð < 0 then ð¥2 converges exponentially to zero for increasing values of ð¥1.
General semi-linear equation
The Cauchy problem for a semi-linear equation and an associated boundary in a surface Î isâ§{âš{â©
ð(x)ð¢ð¥1(x) + ð(x)ð¢ð¥2(x) = ð(x, ð¢(x)), x â â2
ð¢ â£Î= ð, x â Î â â2(7.15)
where ð(.) and ð(.) are ð¶1 functions in â2 and ð(.) is a ð¶1 function in â3. Observe that the functionð¢ enters, possibly in a non-linear from, in the right-hand side.
Again we introduce a parameterisation associated with the characteristic surface and the bound-ary surface by a pair (ð , ð) and set ð¥1 = ð1(ð , ð) and ð¥2 = ð2(ð , ð) and ð¢ = ð(ð , ð) = ð¢(ð1(ð , ð), ð2(ð , ð))
In this case the characterstic equation system and the compatibility condition becomeðð1(ð , ð)
ðð = ð (ð1(ð , ð), ð2(ð , ð))ðð2(ð , ð)
ðð = ð (ð1(ð , ð), ð2(ð , ð))ðð(ð , ð)
ðð = ð (ð1(ð , ð), ð2(ð , ð), ð(ð , ð))
and the constrains on their values introduced by Î that we associate with ð = 0
ð1(0, ð) = ðŸ1(ð)ð2(0, ð) = ðŸ2(ð)ð(0, ð) = ð(ð)
We observe again that from the solution of the two first ODEâs we get a relationship (ð¥1, ð¥2) =ðº(ð , ð) and if Î is non-characteristic we get, at least locally (ð , ð) = ðºâ1(ð¥1, ð¥2), which allows foruniqueness and existence of solutions for the PDE problem. The only difference is related to thefact that now the right hand side of the compatibility condition for ð depends on ð .
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Paulo Brito Advanced Mathematical Economics 2020/2021 16
7.2.5 Quasi-linear equations
Let us consider the semi-linear equation and an associated boundary in a surface Î
â§{âš{â©
ð(x, ð¢)ð¢ð¥1 + ð(x, ð¢)ð¢ð¥2 = ð(x, ð¢), x â â2
ð¢ â£Î= ð, x â Î â â2
where ð(.), ð(.) and ð(.) is a ð¶1 functions in â3.Again we introduce a parameterisation associated with the characteristic surface and the bound-
ary surface by a pair (ð , ð) and set ð¥1 = ð1(ð , ð) and ð¥2 = ð2(ð , ð) and ð¢ = ð(ð , ð) = ð¢(ð1(ð , ð), ð2(ð , ð))In this case the characterstic equation system and the compatibility condition become
ðð1(ð , ð)ðð = ð (ð1(ð , ð), ð2(ð , ð), ð(ð , ð))
ðð2(ð , ð)ðð = ð (ð1(ð , ð), ð2(ð , ð), ð(ð , ð))
ðð(ð , ð)ðð = ð (ð1(ð , ð), ð2(ð , ð), ð(ð , ð)) .
This system, differently from the previous cases, lost their recursive structure, in the sense thatwe cannot separate the determination of the solutions for ð1(.) and ð2(.) from ð(.): the twoindependent variables, ð1 and ð2, and the dependent variable, ð , are jointly determined. In orderto solve the system, Î provides the boundary conditions for ð = 0:
ð1(0, ð) = ðŸ1(ð)ð2(0, ð) = ðŸ2(ð)ð(0, ð) = ð(ð)
Now the non-characteristic conditions for (Î, ð) are more involved because all three differentialequations depend on (ð1, ð2, ð) and the conditions for the application of the non-characteristiccondition may not hold.
The geometric meaning is the following: while for linear and semi-linear PDE the characteristiclines are parallel and do not cross, for the quasi-linear case this may not be the case. At singularitypoints the uniqueness and even the existence of solutions may break down.
A well known quasi-linear first-order PDE is the inviscid Burgerâs equation (see https://en.wikipedia.org/wiki/Burgers%27_equation)
â§{âš{â©
ð¢ð¡ + ð¢ð¢ð¥ = 0, (ð¡, ð¥) â â2
ð¢(0, ð¥) = ð(ð¥) (ð¡, ðŠ) â { ð¡ = 0} à â
It can be proved that the characteristic equations can intersect which implies that the solutionscannot be unique at those singular points. Introducing some solvability conditions, gives birth toshock waves, which is a type of behavior not presented in linear hyperbolic PDEâs.
https://en.wikipedia.org/wiki/Burgers%27_equationhttps://en.wikipedia.org/wiki/Burgers%27_equation
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Paulo Brito Advanced Mathematical Economics 2020/2021 17
7.3 The linear equation in the semi-infinite domain and Laplacetransform methods
In the previous cases we assumed that the independent variables were defined in the space X = â2.The solution of the first-order PDE and/or of the associated problems varies both in terms of theexistence and of the methods of determination if the domain is different, that is X â â2. In thiscase we may have as solutions not functions (single-valued continuous mappings) but generalizedfunctions (also called weak solutions).
7.3.1 Linear equation with zero right-hand side
To illustrate this, assume that X = â2++, that is ð¥ > 0 and ðŠ > 0 and consider the problem
â§{{âš{{â©
ð¢ð¥ + ðð¢ðŠ = 0, (ð¥, ðŠ) â â2++ð¢(ð¥, 0) = ð(ð¥), (ð¥, ðŠ) â â++ à {ðŠ = 0}ð¢(0, ðŠ) = ð(ðŠ), (ð¥, ðŠ) â {ð¥ = 0} à â++Ã
(7.16)
A convenient way to solve this equation is to use Laplace transforms instead of the methodof characteristics (see the Appendix ). In order to do this we pick one of the independent variablesas a parameter (for instance ð¥) and keep one variable as an independent variable (for instanceðŠ)3. Laplace transforms are convenient because the domain of transformation is the semi-infiniteinterval [0, â).
The method of solution follows the steps:
1. First, we apply Laplace transforms to go from the PDE into a parameterized ODE
2. Second, we solve the ODE and apply the transforms of the boundary conditions
3. Finally, we apply inverse Laplace transforms to obtain the solution
Proposition 8. The solution to Cauchy problem (7.16) is
ð¢(ð¥, ðŠ) = ð(ðŠ â ðð¥)ð»(ðŠ â ðð¥) + ð ââ1 [ â«ð¥
0ð(ð )ðâðð(ð¥âð )ðð ] (ðŠ) (7.17)
where
ð»(ð§) =â§{âš{â©
0, if 𧠆01, if ð§ > 0
is the Heaviside generalized function and ââ1[ð(ð¥)] (ðŠ) is the inverse Laplace transform. There-fore
ð(ðŠ â ðð¥)ð»(ðŠ â ðð¥) =â§{âš{â©
0 if ðŠ â ð𥠆0ð(ðŠ â ðð¥) if ðŠ â ðð¥ > 0.
3The choice can be done in a way to simplify the solution of the problem, given the constraints.
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Paulo Brito Advanced Mathematical Economics 2020/2021 18
Proof. Let ð(ð¥, ð) be the Laplace transform of ð¢(ð¥, ðŠ) taking variable ð¥ as a parameter, that is
ð(ð¥, ð) â¡ â[ð¢(ð¥, ðŠ)](ð) = â«â
0ðâððŠð¢(ð¥, ðŠ)ððŠ
where ð > 0. The Laplace transforms of ð¢ð¥(â ) and ð¢ðŠ(â ) are
â[ð¢ð¥(ð¥, ðŠ)](ð) = â«â
0ðâððŠð¢ð¥(ð¥, ðŠ)ððŠ = ðð¥(ð¥, ð)
and â[ð¢ðŠ(ð¥, ðŠ)](ð) = â«
â
0ðâððŠð¢ðŠ(ð¥, ðŠ)ððŠ = ðð(ð¥, ð) â ð¢(ð¥, 0) = ðððŠ(ð¥, ð) â ð(ð¥).
From the continuity and differentiability properties of ð¢(â ), ð¢ð¥(ð¥, ðŠ) + ðð¢ðŠ(ð¥, ðŠ) = 0 holds if andonly if
â«â
0ðâððŠ (ð¢ð¥(ð¥, ðŠ) + ðð¢ðŠ(ð¥, ðŠ)) ððŠ = 0.
Then the PDE, in Laplace transforms, is equivalent to the linear ODE in the variable ð¥, parame-terized by the transformed variable ð 4
ðð¥(ð¥, ð) + ð(ðð(ð¥, ð) â ð(ð¥)) = 0.
In order to solve the Cauchy problem, we also need to introduce the Laplace transform of ð(ðŠ),that is
â[ð¢(0, ðŠ)](ð) = â«â
0ðâððŠð(ðŠ)ððŠ = Ί(ð).
Then we get an initial-value problem for the parameterized (by ð) ODE
â§{âš{â©
ðð¥(ð¥, ð) = âððð(ð¥, ð) + ðð(ð¥), ð¥ > 0ð(0, ð) = Ί(ð), ð¥ = 0.
The solution isð(ð¥, ð) = Ί(ð)ðâððð¥ + ð â«
ð¥
0ð(ð )ðâðð(ð¥âð )ðð .
The inverse Laplace transform is
ð¢(ð¥, ðŠ) = ââ1 [ ð(ð¥, ð)] (ðŠ) = 12ðð limð ââ â«ðŸ+ðð
ðŸâðððððŠ ð¹ (ð§)ðð§.
4We can obtain the same functional equation if we evaluate directly the integral, because
â«â
0ðâððŠ (ð¢ð¥(ð¥, ðŠ) + ðð¢ðŠ(ð¥, ðŠ)) ððŠ = â«
â
0ðâððŠ ðð¢ðð¥ (ð¥, ðŠ)ððŠ + ð â«
â
0ðâððŠ ðð¢ððŠ (ð¥, ðŠ)ððŠ =
= ðð¥(ð¥, ð) + ð (â«â
0ðâððŠð¢(ð¥, ðŠ) â â«
â
0ð¢(ð¥, ðŠ) ðððŠ (ð
âððŠ) ððŠ) =
= ðð¥(ð¥, ð) â ðð¢(ð¥, 0) + ððð(ð¥, ð) = 0
applying integration by parts.
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Paulo Brito Advanced Mathematical Economics 2020/2021 19
Therefore
ð¢(ð¥, ðŠ) = ââ1[ð(ð¥, ð)] =
= ââ1[Ί(ð)ðâððð¥] + ð ââ1[ â«ð¥
0ð(ð )ðâðð(ð¥âð )ðð ].
The solution is (7.16). This can be proved by the fact that
ðâðð¥ðΊ(ð) = ðâðð¥ð â«â
0ðâðð ð(ð )ðð = â«
â
0ðâð(ð +ðð¥)ð(ð )ðð
= â«â
ðð¥ðâð ðŠð(ðŠ â ð ð¥)ððŠ, (ð + ðð¥ = ðŠ)
= â«â
0ðâð ðŠð(ðŠ â ð ð¥) ð»(ðŠ â ðð¥)ððŠ =
= â[ ð(ðŠ â ð ð¥) ð»(ðŠ â ðð¥)] (ð)
and ââ1 [ â[ ð(ðŠ â ð ð¥) ð»(ðŠ â ðð¥)] (ð)] = ð(ðŠ â ð ð¥) ð»(ðŠ â ðð¥).
Example 1. In order to have an intuition on the solution consider the case: ð > 0, ð(ðŠ) = 0and ð(ð¥) = ð, a constant. In this case
ð(ð¥, ð) = ðð â«ð¥
0ðâðð(ð¥âð )ðð = ð (1 â ð
âððð¥
ð ) .
Thenð¢(ð¥, ðŠ) = ðââ1 [ 1 â ð
âððð¥
ð ] (ðŠ) = ðð»(ðð¥ â ðŠ)
That is the solution is
ð¢(ð¥, ðŠ) =â§{âš{â©
ð for 0 < ðŠ < ð ð¥0 for ðŠ ⥠ð ð¥
In this case the solution takes a constant value for {(ð¥, ðŠ) ⶠ0 < ðŠ < ðð¥} where ð > 0 and ð¥ > 0 ,and it is equal to zero elsewhere.
Example 2. If, instead, we had the case ð(ðŠ) = ðððŠ and ð(ð¥) = 0 we would have
ð¢(ð¥, ðŠ) = ð(ðŠ â ðð¥)ð»(ðŠ â ðð¥) = ðð(ðŠâðð¥) ð»(ðŠ â ðð¥)
ð¢(ð¥, ðŠ) =â§{âš{â©
0 for 0 < ðŠ †ðð¥ðð(ðŠâðð¥) for ðŠ > ðð¥.
In this case the projected characteristics are as in figure :
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Paulo Brito Advanced Mathematical Economics 2020/2021 20
Figure 7.4: Characteristic lines for (7.16) for ð > 0 ð(ð¥) = 0 and ð(ðŠ) = ðððŠ
7.3.2 Linear equation with homogeneous right-hand side
Now consider the problemâ§{{âš{{â©
ð¢ð¥ + ð¢ðŠ = ðð¢, ð¥ > 0, ðŠ > 0ð¢(ð¥, 0) = ððð¥, ð¥ > 0ð¢(0, ðŠ) = 0, ðŠ > 0
Using the same method we find the solution
ð¢(ð¥, ðŠ) = ð ðð (ðð¥âðŠ)(1 â ð»(ðŠ â ð ð¥)).
or, equivalently,
ð¢(ð¥, ðŠ) =â§{âš{â©
ð ðð (ðð¥âðŠ), if 0 < ðŠ < ð ð¥0, if ðŠ ⥠ð ð¥
To prove this we can go back to the Laplace transform of equation (7.17),
ð(ð¥, ð) = ð â«ð¥
0ðð ð ðâðð(ð¥âð )ðð = ð ð
ð ð¥ â ðâð ð ð¥ð + ðð ,
applying the inverse Laplace transform, we find
ð¢(ð¥, ðŠ) = ââ1[ ð(ð¥, ð)](ðŠ) = ð ðð (ðð¥âðŠ)(1 â ð»(ðŠ â ð ð¥)).
A graphical depiction of the solution for ð = 1 and ð = 2 presented in Figure 7.5. The projected
characteristics are as in Figure 7.4 but, differently from that case where ð¢ is constant, now thesolution growth at the rate ð along the characteristic lines.
7.4 Evolution equations
In this section we assume that the independent variables are time and one set of characteristics ð¥,and that the PDE takes the form
ð¢ð¡ + ðð¥ð(ð¢) = 0 (ð¡, ð¥) â â+ à â.
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Paulo Brito Advanced Mathematical Economics 2020/2021 21
Figure 7.5: Solution for the problem ð¢ð¥ + ð¢ðŠ = ð¢, ð¢(0, ðŠ) = 0, ð¢(ð¥, 0) = ð2ð¥ and defined for ð¥ > 0and ðŠ > 0
The solution for generic non-linear first order PDEâs can display several types of behavior (see
(Dafermos, 2000, p.13) :
(1) blow-up if the the solution becomes infinite when a certain level for ð¥ is reached in finitetime,
(2) globally bounded solutions while ð¥ goes to infinite in infinite time;
(3) progressive concentration along time tending asymptotically to a degenerate distribution con-centrated at a finite value for ð¥, ð¥â, in infinite time;
(4) shock-waves such that, after a surface Î(ð¡, ð¥) is reached, the solution becomes non-smoothand multivalued; or
(5) rarefaction waves such that the distribution becomes increasingly dispersed.
From the above results we can conclude the following as regards the linear case
ð¢ð¡ + ð(ð¥) ð¢ð¥ = ðð¢ (ð¡, ð¥) â â+ à â.
(see (Olver, 2014, p.29)). The characteristic curves have the following generic properties:
1 for each point (ð¡, ð¥) â â2, there is a unique characteristic passing through that point
2 characteristic curves cannot cross each other
3 if ð¥ = â(ð¡) is a characteristic curve, then ð¥ = â(ð¡) + ð is also a characteristic curve for ð â â,if ð = 0 or ð¥ = ð ðâðð¡
4 the path traced out by a characteristic curve ð¥ = â(ð¡) for increasing values of ð¡ always movesin the same direction and cannot change the direction of propagation
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Paulo Brito Advanced Mathematical Economics 2020/2021 22
5 as ð¡ â â the characteristic curve either converges to a fixed point, ð¥(ð¡) â ð¥â where ð(ð¥â) = 0or go to ±â either in finite or infinite time.
To illustrate the last point consider the problem
â§{âš{â©
ð¢ð¡ + ðœ(ð¥ â ð¥â) = 0, for (ð¡, ð¥) â â+ à âð¢(0, ð¥) = ð(ð¥), for (ð¡, ð¥) â { ð¡ = 0} à â.
The solution to this Cauchy problem is
ð¢(ð¡, ð¥) = ð(ð¥â + (ð¥ â ð¥â) ðâðœ ð¡, (ð¡, ð¥) â â+ à â.
The asymptotic dynamics depend on the sign of ðœ:
1. if ðœ > 0 we have a rarefaction wage, i, e. limð¡ââ ð¢(ð¡, ð¥) = ð(ð¥â) which is a constant for allð¥ â ð
2. if ðœ < 0 we have a rarefaction wage, i, e. limð¡ââ ð¢(ð¡, ð¥) = ð¿ð¢(ð¥â) is a degenerate distributionwhose mass is concentrated at ð¥ = ð¥â.
7.5 Applications
7.5.1 The transport equation
We consider a simple example called the transport equation. To simplify assume that theindependent variables are (ð¡, ð¥) and that their domain is unbounded, i.e., (ð¡, ð¥) â â2 5 :
â§{âš{â©
ðð¡ð¢(ð¡, ð¥) + ðð¥ (ð ð¥ ð¢(ð¡, ð¥)) = 0, (ð¡, ð¥) â â+ à âð¢(0, ð¥) = ð(ð¥), (ð¡, ð¥) â {ð¡ = 0} à â
(7.18)
The transport equation describes a conservation phenomenon. If, for instance,
â«â
ââð(ð¥)ðð¥ = 1
Proposition 9. The solution to the transport equation Cauchy problem (7.18) is
ð¢(ð¡, ð¥) = ðâðð¡ð (ð¥ðâðð¡) .
5We use the notation ðð¥ð ð(x) =ðð(x)ðð¥ð
, where x = (ð¥, ⊠, ð¥ð, ⊠ð¥ð) â âð.
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Paulo Brito Advanced Mathematical Economics 2020/2021 23
Proof. The PDE can be equivalently written as
ð¢ð¡(ð¡, ð¥) + ð ð¥ ð¢ð¥(ð¡, ð¥) + ð ð¢(ð¡, ð¥) = 0.
Let us consider a change in variables: ð¥ = ð(ðŠ) = ðððŠ and
ð£(ð¡, ðŠ) = ððð¡ ð¢(ð¡, ð(ðŠ)).
Taking derivatives for ð¡ and ðŠ and applying the relationship in equation (7.18) we find that
ð£ð¡(ð¡, ðŠ) + ð£ðŠ(ð¡, ðŠ) = 0
if and only if ð¢ð¡(ð¡, ð¥) + ðð¥ð¢ð¥(ð¡, ð¥) + ðð¢(ð¡, ð¥) = 0. This equation has the form of (7.5), with ð = 1.As ð£(0, ðŠ) = ð¢(0, ð(ðŠ)) = ð(ð(ðŠ)) we can use the solution to Cauchy problem (7.7) to obtain
ð£(ð¡, ðŠ) = ð(ð(ðŠ â ð¡)) = ð (ðð(ðŠâð¡)) = ð (ð(ðŠ)ðâðð¡) .
To obtain the solution we just need to transform back to the original function ð¢(.) and substituteð(ðŠ) = ð¥.
The projected characteristics are as in Figure 7.6 for ð > 0. Differently from the previous caseswe see that the characteristics are non-linear, and, in this case exponentially growing.
Figure 7.6: Characteristic lines for (7.18) for ð > 0
7.5.2 Age-structured population dynamics
The exponential model for population dynamics, ï¿œÌï¿œ = ðð, where ð is the difference between thefertility rate and the mortality rate, has the solution ð(ð¡) = ð(0)ððð¡,
Although this model may be a good approximation asymptotically, in the shorter run there isa large deviation. One of the reasons for the deviation is related to the fact that both fertility andmortality rates are age-dependent. If we introduce age-dependent mortality and fertility rates thedynamics of the population is governed by a first-order PDE.
Let ð(ð, ð¡) be the number of females of age ð at time ð¡ in a population. The total populationat time ð¡ is
ð(ð¡) = â«ðððð¥
0ð(ð, ð¡) ðð
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Paulo Brito Advanced Mathematical Economics 2020/2021 24
Defining the proportion of population with age ð by ð(ð, ð¡) = ð(ð, ð¡)/ð(ð¡) we have
â«ðððð¥
0ð(ð, ð¡) ðð = 1.
The proportion of population, measured by the proportion of females, between ages ð1 andð2 > ð1 at time ð¡ is
ð([ð1, ð2], ð¡) = â«ð2
ð1ð(ð, ð¡)ðð.
If there is no mortality, the instantaneous change in ð([ð1, ð2], ð¡) is
ððð¡ â«
ð2
ð1ð(ð, ð¡)ðð = ð(ð1, ð¡) â ð(ð2, ð¡)
where ð(ð1, ð¡) and ð(ð2, ð¡) is the flow of females which is just entering and leaving the inter-val [ð1, ð2]. As
ððð¡ â«
ð2ð1
ð(ð, ð¡)ðð = â«ð2ð1 ð¢ð¡(ð, ð¡)ðð and, using the fundamental theorem of calculusð(ð1, ð¡) â ð(ð2, ð¡) = â â«
ð2ð1
ð¢ð(ð, ð¡)ðð then
ððð¡ â«
ð2
ð1ð(ð, ð¡)ðð â (ð(ð1, ð¡) â ð(ð2, ð¡)) = â«
ð2
ð1(ðð¡(ð, ð¡) + ðð(ð, ð¡)) ðð = 0
and there is a conservation law. However, introducing mortality, the population in the interval[ð1, ð2] will not remain constant
â«ð2
ð1(ðð¡(ð, ð¡) + ðð(ð, ð¡)) ðð â â«
ð2
ð1ð(ð, ð¡)ð(ð, ð¡)ðð = 0
Therefore we have the equation for an age-dependent population
ðð¡(ð, ð¡) + ðð(ð, ð¡) = ð(ð, ð¡)ð., for (ð, ð¡) â [0, ðððð¥] à â+.
The McKendry model further assumes an initial population distribution and an age-dependentfertility
â§{{âš{{â©
ðð¡ + ðð = âð(ð, ð¡)ð, (ð, ð¡) â (0, X) Ã (0, â)ð(ð, 0) = ð0(ð), (ð, ð¡) â (0, X) Ã {ð¡ = 0}ð(0, ð¡) = ð(ð¡), (ð, ð¡) â {ð = 0} Ã (0, â)
(7.19)
X = max{ð}, the maximum age of the population, ð0(ð) is the initial age-distribution of thepopulation, and the number of newborns is
ð(ð¡) = â«ðððð¥
0ðœ(ð, ð¡)ð(ð, ð¡)ðð
where ðœ(ð, ð¡) is the age-distribution of fertility at time ð¡. If we compared to the PDE alreadypresented, the McKendrick model has two new features:
1. first, it has two boundary conditions: an initial distribution for the population (at ð¡ = 0) andfor the population at age ð = 0;
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Paulo Brito Advanced Mathematical Economics 2020/2021 25
2. second, the boundary condition referring to the newborns is non-local, that is, it depends onthe distribution of the total population. This last feature implies that it is hard to solve,requiring the solution of an integral equation.
Assuming away that global nature of fertility, the Mc-Kendrick equation features a differenttype of dynamics depending in the difference between ð and ð¡: for ð < ð¡ the dynamics dependson the newborns, i.e., population with age ð = 0, while for ð > ð¡ the dynamics is governed bythe initial age-distribution of the population. Of course, asymptotically the first type of behaviorprevails.
Consider the caseâ§{{âš{{â©
ðð¡ + ðð = âðð, (ð, ð¡) â (0, X) Ã (0, â)ð(ð, 0) = ð0(ð), (ð, ð¡) â (0, X) Ã {ð¡ = 0}ð(0, ð¡) = ð(ð¡), (ð, ð¡) â {ð = 0} Ã (0, â)
where ð0(ð) is the initial distribution of population, and ð(ð¡) is the number of offspring hereassumed as exogenous, i.e., independent of the distribution of population.
Prove that the solution is
ð(ð, ð¡) =â§{âš{â©
ð(ð¡ â ð)ð(ð), if ð †ð¡ð0(ð â ð¡) ð(ð)ð(ðâð¡) , if ð ⥠ð¡
whereð(ð) = ðâðð
is the probability of survival until age ð. Therefore, ð(ð)ð(ðâð¡) = ðâðð¡.A simpler version of this model, assumes that there is only one fertile age 0 < ðŒ < X, leading
toð(ð¡) = â«
ðððð¥
0ðœ ð¿(ð â ðŒ) ð(ð, ð¡)ðð = ðœð(ðŒ, ð¡).
The solution displays an âecho effectâ with period equal to ðŒ.Reference: McKendrick (1926) and for a recent textbook presentation Kot (2001)
7.5.3 Cohortâs budget constraint
Let ð€(ð, ð¡) be the financial wealth of an agent with age ð at time ð¡. The budget constraint is
ð€ð¡ + ð€ð = ð (ð, ð¡) + ðð€(ð, ð¡) (7.20)
where ð (ð, ð¡) is the savings at age ð at time ð¡ and ð is the interest rate. If we assume that theinitial stock of wealth is unbounded then ð€ ⶠ(0, ðŽ) à (0, ð ) â â and the initial wealth distributionis ð€(0, ð¡) = 0.
The general solution of equation (7.20) is
ð€(ð, ð¡) = (â«ð
0ð (ð§, ð§ â ð + ð¡)ðâðð§ðð§ + ð(ð¡ â ð)) ððð
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Paulo Brito Advanced Mathematical Economics 2020/2021 26
for an arbitrary ð(.).If we assume that there are no bequests, that is no wealth at birth, ð€(ð, ð¡) = 0 and ð (ð, ð¡) =
ððð(ðŸâð)+ðð¡ â ð the solution becomes
ð€(ð, ð¡) =âð2â
ð(Ί (ðŸð + ð â ð
2â
ð ) â Ί ((ðŸ â 2ð)ð + ð â ð
2â
ð )) ð ðŸ
2ð2((2ðŸ+4(ð¡âð))ðâ2(ðŸâ2ð)ð)ð+(ðâð)24ð â
â ðð (1 â ððð) (7.21)
where Ί(ð¥) = erf(ð¥) = (2/âð) â«ð¥0 ðâð§2ðð§. Figure ?? illustrates equation (7.21). It displays a
life-cycle behavior of savings: the agent tends to be a net borrower at young age and lender at olderages, although it dissaves later in life.
Figure 7.7: Illustration of equation (7.21) for ðŸ = 88, ð = 0.0029, ð = 0.02, ð = 0.02 and ð = 50.
7.5.4 Interest rate term-structure
We consider an economy in which there is perfect foresight (i.e, there is a deterministic setting) andin which there are two types of assets: a bank account and a continuum of bonds with maturitydates ranging from zero to infinity, ð¥ â [0, â). We show that, if there is absence of arbitrageopportunities, the forward rate at time ð¡ for a bound maturing at time ð¡ + ð¥ perfectly anticipatesthe spot interest rate for time ð¡ + ð¥.
First, the bank accountâs balance, for an initial deposit of ðµ(0), follows the process
ðµ(ð¡) = ðµ(0)ðâ«ð¡
0 ð(ð)ðð .
where ð(ð¡) is the spot interest rate. Therefore, the rate of return for a bank account isððµ(ð¡)
ðð¡1
ðµ(ð¡) = ð(ð¡), for any ð¡ ⥠0.
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Paulo Brito Advanced Mathematical Economics 2020/2021 27
Second, the price, at time ð¡, for a bond6 that matures at time ð¡ + ð¥,
ð(ð¡, ð¥) = ðâ â«ð¥
0 ð(ð¡,ðŠ)ððŠ (7.22)
where ð(ð¡, ð¥) is the forward interest rate for maturity ð¥ ⥠0. The spot interest rate and theforward rate with maturity zero are the same ð(ð¡, 0) = ð(ð¡).
The change in the price of the bond is
ðð(ð¡, ð¥) = ðð¡ð(ð¡, ð¥) ðð¡ + ðð¥ð(ð¡, ð¥) ðð¥ == (ðð¡ð(ð¡, ð¥) â ðð¥ð(ð¡, ð¥)) ðð¡
because in we move ðð¡ on time the duration to maturity reduces such as ðð¥ + ðð¡ = 0. Takingderivatives in equation (7.22) we have
ðð¡ð(ð¡, ð¥) = âð(ð¡, ð¥) â«ð¥
0 ðð¡ð(ð¡, ðŠ)ððŠ
ðð¥ð(ð¡, ð¥) = âð(ð¡, ð¥) ð(ð¡, ð¥). Therefore, the rate of return for a bond with maturity ð¥
ðð(ð¡, ð¥)ðð¡
1ð(ð¡, ð¥) = ð(ð¡, ð¥) â â«
ð¥
0ðð¡ð(ð¡, ðŠ) ððŠ.
Using the mean value theorem ð(ð¡, ð¥) â ð(ð¡, 0) = â«ð¥0 ðð¥ ð(ð¡, ðŠ) ððŠ, and noting that ð(ð¡, 0) = ð(ð¡)we find
ðð(ð¡, ð¥)ðð¡
1ð(ð¡, ð¥) = â â«
ð¥
0(ðð¡ð(ð¡, ðŠ) â ðð¥ð(ð¡, ðŠ)) ððŠ + ð(ð¡)
Third, If there are no arbitrage opportunities, the instantaneous rate of return for any twoinvestments should be equal for every point in time ð¡. In particular, the rate of return for the bankaccount should be equal to the rate of return for a bond of any maturity ð¥ ⥠0,
ððµ(ð¡)/ðð¡ðµ(ð¡) =
ðð(ð¡, ð¥)/ðð¡ð (ð¡, ð¥) .
This condition is equivalent to
â«ð¥
0(ðð¡ð(ð¡, ðŠ) â ðð¥ð(ð¡, ðŠ)) ððŠ = 0, for every ð¥ ⥠0.
Therefore the forward rate, in a deterministic setting, solves the following Cauchy problemâ§{âš{â©
ðð¡ð(ð¡, ð¥) â ðð¥ð(ð¡, ð¥) = 0, (ð¡, ð¥) â â2+ð(ð¡, 0) = ð(ð¡), (ð¡, ð¥) â â+ à {ð¡ = 0} .
Because the PDE has the general solution ð(ð¡, ð¥) = â(ð¡ + ð¥), the solution to the Cauchy problemis
ð(ð¡, ð¥) = ð(ð¡ + ð¥) the forward rate, at time ð¡, for a bond maturing at time ð¡ + ð¥ is a perfect predictor to the spotrate at time ð¡ + ð¥.
6It is implicitly assumed that the bond has face value equal to one and does not pay coupons.
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Paulo Brito Advanced Mathematical Economics 2020/2021 28
7.5.5 Optimality condition for a consumer choice problem
Consider the following general consumer problem
maxð1,ð2
ð¢(ð1, ð2)
subject to the following constraints
â§{{âš{{â©
ðž(ð1, ð2) = ð1ð1 + ð2ð2 †ð0 †ð1 †Ìð10 †ð2 †Ìð2
Assume that ð¢(.) is continuous, differentiable, increasing and concave in both arguments. FormingThe Lagrangean
â = ð¢(ð1, ð2) + ð(ð â ðž(ð1, ð2)) â ð1ð1 â ð2ð2 + ð1( Ìð1 â ð1) + ð2( Ìð2 â ð2).
The solution (which always exists) (ðâ1, ðâ2) verifies the Karush-Kuhn-Tucker conditions
ð¢ðð(ð1, ð2) â ððð â ðð â ðð = 0, ð = 1, 2ðððð = 0, ðð ⥠0, ðð ⥠0, ð = 1, 2ðð( Ìðð â ðð) = 0, ðð ⥠0, ðð †Ìðð, ð = 1, 2ð(ð â ðž(ð1, ð2)) = 0, ð ⥠0, ðž(ð1, ð2) †ð
We have three cases:First, an interior solution, ðâ1 â (0, Ìð1) and ðâ2 â (0, Ìð2), with the optimality conditions
ð2ð¢ð1(ðâ1, ðâ2) = ð1ð¢ð2(ðâ1, ðâ2) (7.23)ðž(ðâ1, ðâ2) = ð (7.24)
Equation (7.23) is a first-order partial differential equation with solution
ð¢(ðâ1, ðâ2) = ð£ (ð1ðâ1 + ð2ðâ2
ð1)
and, if we use equation (7.24) and define ð€ â¡ ð/ð1 in the optimum, we have
ð¢(ðâ1, ðâ2) = ð£(ð€)
If the utility function is strictly concave then with very weak conditions (differentiability) wehave an unique interior optimum. It is clear that the budget set, in real terms, is a projectedcharacteristic.
Second, the first corner solution for ð1 is ðâ1 = 0 and an interior solution for ð2 ðâ2 â (0, Ìð2) andthe budget constraint be saturated, and satisfies the conditions
ð2ð¢ð1(ðâ1, ðâ2) = ð1ð¢ð2(ðâ1, ðâ2) â ð2ð1 (7.25)ðž(ðâ1, ðâ2) = ð (7.26)
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Paulo Brito Advanced Mathematical Economics 2020/2021 29
Figure 7.8: Interior optimum
Figure 7.9: Corner solution: zero consumption
Equation (7.25) is a first-order partial differential equation with solution
ð¢(ðâ1, ðâ2) =ð1ðâ2ð1
+ ð£ (ð1ðâ1 + ð2ðâ2
ð1)
If we use equation (7.28) in the optimum we have
ð¢(ðâ1, ðâ2) = âð1ð2ðâ2
ð1 + ð£(ð€) < ð£(ð€)
then the indirect utility level is smaller than for the unconstrained caseThird, the second corner solution for ð1: lLet ðâ1 = Ìð1 and ðâ2 â (0, Ìð2) and let the budget
constraint be saturated; It verifies the conditions
ð2ð¢ð1(ðâ1, ðâ2) = ð1ð¢ð2(ðâ1, ðâ2) + ð2ð1 (7.27)ðž(ðâ1, ðâ2) = ð (7.28)
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Paulo Brito Advanced Mathematical Economics 2020/2021 30
Equation (7.27) is a first-order partial differential equation with solution
ð¢(ðâ1, ðâ2) = âð1ðâ2ð1
+ ð£ (ð1ðâ1 + ð2ðâ2
ð1)
and if we use equation (7.28) in the optimum we have
ð¢(ðâ1, ðâ2) =ð1ð2ðâ2
ð1 + ð£(ð€) < ð£(ð€)
then the indirect utility level is smaller than for the unconstrained caseWe verify that the interior optimum has characteristics given by ð1 + ð1ð2 ð2, and in general they
coincide with the budget constraint.We can repeat the same exercise for the corner solutions for ð2.
7.5.6 Growth and inequality dynamics
Consider an economy in which the households are heterogenous as regards their capital endowments,let the endowments belong to the continuum ð â [ð(ð¡), ð(ð¡)] â â+, and let ð(ð¡, ð) be the numberof people having an asset endowment ð, at time ð¡.
If we denote the total population by ð(ð¡) then
ð(ð¡) = â«ð(ð¡)
ð(ð¡) ð(ð¡, ð)ðð.
We can denote the population density by ð(ð¡, ð) = ð(ð¡, ð)/ð(ð¡). In this case
â«ð(ð¡)
ð(ð¡)ð(ð¡, ð)ðð = 1.
Assume that the capital accumulates in linearly as
ðððð¡ = ðŸð(ð¡).
Then, using Leibniz rule
ððð¡ â«
ð(ð¡)
ð(ð¡)ð(ð¡, ð)ðð = â«
ð(ð¡)
ð(ð¡)ðð¡(ð¡, ð)ðð + ð(ð¡, ð(ð¡))
ðð(ð¡)ðð¡ â ð(ð¡, ð(ð¡))
ðð(ð¡)ðð¡
= â«ð(ð¡)
ð(ð¡)ðð¡(ð¡, ð)ðð + ð(ð¡, ð(ð¡))ðŸð(ð¡) â ð(ð¡, ð(ð¡))ðŸð(ð¡) =
= â«ð(ð¡)
ð(ð¡)ðð¡(ð¡, ð) +
ð (ðŸðð(ð¡, ð))ðð ðð
by the mean-value theorem. Then the density satisfies the PDE
ðð¡(ð¡, ð) + ðŸððð(ð¡, ð) + ðŸð(ð¡, ð) = 0
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Paulo Brito Advanced Mathematical Economics 2020/2021 31
0 10 20 30 40 50
0.0
00
.04
0.0
8
k
n(k
,t)
Figure 7.10: Density for several increasing dates
which has the form of the transport equation (7.18).Given an initial density ð0(ð) the solution is then
ð(ð¡, ð) = ðâðŸð¡ð0 (ððâðŸð¡) , (ð¡, ð) â â+.
Assuming a log-normal distribution
ð(ð) = (2ðð2ð2)â12 exp (â(log (ð) â ð)
2
2ð2 )
thenð(ð¡, ð) = (2ðð2ð2)â
12 exp (â(log (ð) â ðŸð¡ â ð)
2
2ð2 )
Figure ?? presents the dynamics of capital distribution, that is, the dynamics of the density ofpopulation for several levels of capital
Growth can be seen as travelling of the density of population for higher levels of capital wealth.We can compute several statistics to characterize the growth and distributional facts from thissimple model (see Figure 7.11:
1. There is unbounded growth, with a constant growth rate ðŸ(ð¡) = ðŸ
2. The distribution is non-ergodic (the average and variance tends to infinity)
ï¿œÌï¿œ(ð¡) = ððŸð¡+ð+ ð2
2 , ðð(ð¡) = (ï¿œÌï¿œ(ð¡))2 (ðð2 â 1)
3. But the inequality measures are constant: Gini and Theil indices
ðº(ð¡) = erf (ð2 ) , Th(ð¡) =ð22
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Paulo Brito Advanced Mathematical Economics 2020/2021 32
0 20 40 60 80 100
15
20
25
30
t
Kav
(a) Increasing average
0 20 40 60 80 100
15
25
35
t
stK
(b) Increasing st. dev.
0 20 40 60 80 100
0.0
0.4
0.8
t
Gin
i K
(c) Constant Gini coefficient
0 20 40 60 80 100
0.0
0.4
0.8
t
Th
eil K
(d) Constant Theil index
Figure 7.11: Linear accumulation function for ðŸ > 0 and an initial log-normal distribution
4. Ratio of the quantiles is also constant
ð90ð10
= âðâ
2 [erfâ1 (1 â 2 910) â erfâ1 (1 â 2 110)]
7.6 References
⢠Mathematics of PDE: introductory Olver (2014) more advanced (Evans, 1998, ch 3) orChechkin and Goritsky (2009)
⢠Applications to economics: Hritonenko and Yatsenko (2013)
⢠Application to mathematical demography: (Kot, 2001, ch. 23)
⢠A useful site: http://eqworld.ipmnet.ru/en/solutions/fpde/fpdetoc1.htm
http://eqworld.ipmnet.ru/en/solutions/fpde/fpdetoc1.htm
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Paulo Brito Advanced Mathematical Economics 2020/2021 33
7.A Laplace transforms and inverse Laplace transforms
Consider function ð(ð¥) where ð¥ > 0. The Laplace transform of ð(ð¥) is
â[ð(ð¥)](ð ) = â«â
0ðâð ð¥ð(ð¥)ðð¥ = ð¹(ð ).
The application of Laplace transforms to the solution of differential equations is convenientbecause it allows for the transformation of a ODE into an non-differential equation and the trans-formation of a PDE into an ODE.
The Laplace transform of ð â²(ð¥) = ðð(ð¥)/ðð¥ is
â[ð â²(ð¥)](ð ) = ððð¥ ( â«â
0ðâð ð¥ð â²(ð¥)ðð¥) = ð â«
â
0ðâð ð¥ð(ð¥)ðð¥ + ðâð ð¥ð(ð¥) â£âð¥=0= ð ð¹(ð ) â ð(0)
if the function ð(.) is bounded.Example: consider the differential equation
ð â²(ð¥) = ðð(ð¥).
Applying the Laplace transform to both sides, yields
â[ð â²(ð¥)](ð ) = ðâ[ð(ð¥)](ð ),
which is equivalent to the the algebraic equation in ð¹(ð )
ð ð¹(ð ) â ð(0) = ðð¹(ð ).
Thereforeð¹(ð ) = ð(0)ð â ð.
To go back to the solution as a function of the independent variable ð¥, we apply the inverseLaplace transform
ââ1[ð¹ (ð )](ð¥) = ð(0)ââ1[ 1ð â ð](ð¥). But
ð(ð¥) = ââ1[ð¹ (ð )](ð¥) = â«â
0ðâð¥ð ð¹(ð )ðð
andââ1 ( 1ð â ð) (ð¥) = ð
ðð¥
Therefore,ð(ð¥) = ð(0) ððð¥.
Some Laplace transforms used in the main text are presented in Table 7.1The Laplace transform and the inverse Laplace transforms are tabulated in many textbooks on
calculus or in the web, see http://tutorial.math.lamar.edu/pdf/Laplace_Table.pdf. We cancompute them using Mathematica http://reference.wolfram.com/language/ref/LaplaceTransform.html and http://reference.wolfram.com/language/ref/InverseLaplaceTransform.html.
http://tutorial.math.lamar.edu/pdf/Laplace_Table.pdfhttp://reference.wolfram.com/language/ref/LaplaceTransform.htmlhttp://reference.wolfram.com/language/ref/LaplaceTransform.htmlhttp://reference.wolfram.com/language/ref/InverseLaplaceTransform.html
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Paulo Brito Advanced Mathematical Economics 2020/2021 34
Table 7.1: Laplace transforms and inverse Laplace transforms
ð(ð¥) ð¹(ð )
ð 1ð
ð¥ 1ð 2
ððð¥ 1ð âð
ð»(ð â ð¥) ðâðð ð ð > 0
ð(ð¥ â ð) ð»(ð¥ â ð) ð¹(ð ) ðâðð ð > 0
-
Bibliography
Chechkin, G. A. and Goritsky, A. Y. (2009). S.n. kruzhkovâs lectures on first-order quasilinear pdes.In Emmrich, E. and Wittbold, P., editors, De Gruyter Proceedings in Mathematics, Analyticaland Numerical Aspects of Partial Differential Equations, pages pp. 1â68. De Gruyter. traduit durusse par B. Andreianov.
Dafermos, C. M. (2000). Hyperbolic Conservation Laws in Continuum Physics, volume 325 ofGrundlehren der mathematischen Wissenschaften. Springer.
Evans, L. C. (1998). Partial Differential Equations, volume 19 of Graduate Series in Mathematics.American Mathematical Society, Providence, Rhode Island.
Hritonenko, N. and Yatsenko, Y. (2013). Mathematical Modeling in Economics, Ecology and theEnvironment. Springer Optimization and Its Applications 88. Springer US.
Kot, M. (2001). Elements of Mathematical Biology. Cambridge.
McKendrick, A. G. (1926). Applications of mathematics to medical problems. Proceedings of theEdinburgh Mathematical Society, 44:98â130.
Olver, P. J. (2014). Introduction to Partial Differential Equations. Undergraduate Texts in Math-ematics. Springer International Publishing, 1 edition.
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First order quasi-linear partial differential equationsIntroductionScalar equations in the infinite domain and the method of characteristicsIntroduction: the method of characteristicsThe two simplest first order PDEsLinear equation with constant coefficientsSemi-linear equationQuasi-linear equations
The linear equation in the semi-infinite domain and Laplace transform methodsLinear equation with zero right-hand sideLinear equation with homogeneous right-hand side
Qualitative dynamics of first-order PDE'sEvolution equationsApplicationsThe transport equationAge-structured population dynamicsCohort's budget constraintInterest rate term-structureOptimality condition for a consumer choice problemGrowth and inequality dynamics
ReferencesLaplace transforms and inverse Laplace transforms