Partial Differential Equations with
Fourier Series and Boundary Value
Problems (3rd Edition, 2016) | Solutions
Manual PDF
Multi-index - answer-A multi-index, α, is an n-tuple of non-negative integers, and the modulus
of α is defined to be the sum of these integers.
Linear PDE - answer-A PDE is linear if it only involves partial derivatives of u(x) with functional
coefficients.
Homogeneous PDEs - answer-A PDE is homogeneous if the term without u(x) is 0.
Linear Operator - answer-An operator, L, is linear if for any given functions u and v, and
constants, s, and t, we have that L[su+tv] = sL[u] + t L[v]
Laplace Operator - answer-The Laplace operator, ∆u, is given by the sum of the second partial
derivatives of u with respect to each component of x.
Linear PDE using operators - answer-A PDE can be written in the form L[u] = f(x), where L is an
operator. If L is a linear operator, then the PDE is linear.
Superposition - answer-If u and v are solutions of a linear homogeneous PDE, then for any
constants s and t, su+tv is also a solution.
1
, Linear Transport equation - answer-The linear transport equation is given by u_t + cu_x = 0, and
it is a first order linear homogeneous PDE.
Laplace's equation - answer-The Laplace equation is given by ∆u=0, and is a second order linear
homogeneous PDE.
Poisson's equation - answer-Poisson's equation is given by ∆u = f(u), and is a second order PDE.
It is linear only if f is linear, and if so, then it is inhomogeneous.
Heat equation - answer-The heat equation is given by u_t - γ∆u=0, and it is a second order linear
homogeneous PDE.
Wave equation - answer-The wave equation is given by u_{tt} - c^2 ∆u = 0, and is a second
order linear homogeneous PDE.
Differentiability Classes - answer-A function, u(x) with domain U is called C^k(U) if all partial
derivatives of u up to order k exist and are continuous. If it is C^k for all integers k, then is called
C infinity, or smooth.
Classification of second order linear PDEs - answer-The class of a second order linear PDE is
determined by the coefficients of the second order partial derivative terms. Let B be the
coefficient of the mixed second order partial derivative term, and A and C be the other two
coefficients. Then the PDE is
1. Elliptic if B^2-4AC< 0 (e.g. the Laplace equation)
2. Parabolic if B62 - 4AC = 0 (e.g. heat equation)
3. Hyperbolic if B^2-4AC >0 (e.g. wave equation)
2
Fourier Series and Boundary Value
Problems (3rd Edition, 2016) | Solutions
Manual PDF
Multi-index - answer-A multi-index, α, is an n-tuple of non-negative integers, and the modulus
of α is defined to be the sum of these integers.
Linear PDE - answer-A PDE is linear if it only involves partial derivatives of u(x) with functional
coefficients.
Homogeneous PDEs - answer-A PDE is homogeneous if the term without u(x) is 0.
Linear Operator - answer-An operator, L, is linear if for any given functions u and v, and
constants, s, and t, we have that L[su+tv] = sL[u] + t L[v]
Laplace Operator - answer-The Laplace operator, ∆u, is given by the sum of the second partial
derivatives of u with respect to each component of x.
Linear PDE using operators - answer-A PDE can be written in the form L[u] = f(x), where L is an
operator. If L is a linear operator, then the PDE is linear.
Superposition - answer-If u and v are solutions of a linear homogeneous PDE, then for any
constants s and t, su+tv is also a solution.
1
, Linear Transport equation - answer-The linear transport equation is given by u_t + cu_x = 0, and
it is a first order linear homogeneous PDE.
Laplace's equation - answer-The Laplace equation is given by ∆u=0, and is a second order linear
homogeneous PDE.
Poisson's equation - answer-Poisson's equation is given by ∆u = f(u), and is a second order PDE.
It is linear only if f is linear, and if so, then it is inhomogeneous.
Heat equation - answer-The heat equation is given by u_t - γ∆u=0, and it is a second order linear
homogeneous PDE.
Wave equation - answer-The wave equation is given by u_{tt} - c^2 ∆u = 0, and is a second
order linear homogeneous PDE.
Differentiability Classes - answer-A function, u(x) with domain U is called C^k(U) if all partial
derivatives of u up to order k exist and are continuous. If it is C^k for all integers k, then is called
C infinity, or smooth.
Classification of second order linear PDEs - answer-The class of a second order linear PDE is
determined by the coefficients of the second order partial derivative terms. Let B be the
coefficient of the mixed second order partial derivative term, and A and C be the other two
coefficients. Then the PDE is
1. Elliptic if B^2-4AC< 0 (e.g. the Laplace equation)
2. Parabolic if B62 - 4AC = 0 (e.g. heat equation)
3. Hyperbolic if B^2-4AC >0 (e.g. wave equation)
2
