Fundamentals of Partial Differential Equations

1 Introduction

Partial Differential Equations (PDEs) are equations that involve an unknown function of several variables and their partial derivatives. PDEs are used to formulate many applied problems in physics, engineering and optimization including heat transfer, wave propagation, fluid flow and elasticity. In this series of notes we will cover the fundamentals of PDEs including their classification, solution methods and applications.

The study of PDEs is quite complicated and draws on many areas of mathematics. It is recommended to have studied the following topics before tackling PDEs:

• Multivariate Calculus
• Ordinary Differential Equations
• Multidimensional Integration

Some additional topics that are helpful but not strictly necessary include:

• Measure Theory
• Functional Analysis

In making these study notes I have used the following resources:

1. Bright Side of Mathematics: Partial Differential Equations Series.
2. Lecture Notes by Prof. Viktor Grigoryan, Fall 2010, UC Santa Barbara.
3. Lecture Notes by Prof. Denis Labutin, Fall 2026, UC Santa Barbara.

2 Partial Derivatives

First recall the following definition of a partial derivative from our study of multivariate calculus.

A partial derivative of a function \(u:\mathbb{R}^n \to \mathbb{R}\) in variables \(x_1, x_2, \ldots, x_n\) with repect to \(j\)-th variable is the derivative of \(u\) with respect to \(x_j\) while keeping all other variables fixed

\[ \partial_j u = \frac{\partial u}{\partial x_j}, \quad j = 1, 2, \ldots, n, \]

also written as \(D_j u\), \(\partial_{x_{j}}\) or \(u_{x_j}\).

We also write \(\partial_j^k\) (or \(D_j^k\)) for the \(k\)-th partial derivative with respect to the \(j\)-th variable.

A multi-index \(\alpha = (\alpha_1, \ldots, \alpha_n)\) is an \(n\)-tuple of non-negative integers. The order of a multi-index is given by \(\| \alpha \| = \alpha_1 + \ldots + \alpha_n\). We define

\[ D^\alpha = \partial_1^{\alpha_1} \ldots \partial_n^{\alpha_n} = \frac{\partial^{\| \alpha \|}}{\partial x_1^{\alpha_1} \ldots \partial x_n^{\alpha_n}} \]

Consider the function \(u(x,y) = 2x^2 + 4y^3 + 2xy\). The partial derivative of \(f\) with respect to \(x\) and \(y\) respectively are given by

\[ u_x = 4x + 2y, \quad u_y = 12y^2 + 2x. \]

3 Partial Differential Equations

A partial differential equation (PDE) is an equation of an unknown function \(u:\Omega \to \mathbb{R}\) (where typically \(\Omega \subseteq \mathbb{R}^n\) open, \(n \ge 2\)) involving some partial derivative of \(u\)

\[ F(x, u(x), (D^\alpha u)(x)) = 0, \]

where \(F\) is a given function, \(\alpha\) is a multi-index, and \(D^\alpha u\) denotes the partial derivative of \(u\) corresponding to the multi-index \(\alpha\).

A function \(u:\Omega\to\mathbb{R}\) is a (classical) solution of a PDE if all partial derivatives appearing in the PDE are well-defined and the function \(u\) satisfies the PDE at all points in \(\Omega\).

A simple example of a PDE for function \(u(x,y)\) written in different notations

\[ \begin{align*} \cos (u) + 3u_{x} + 2u\cdot u_{xy} & = 0 \\ \cos (u) + 3 \partial_ x u + 2 u\cdot \partial _{xy} u & = 0\\ \cos (u) + 3D_x u + 2u\cdot D_x D_y u & = 0. \end{align*} \]

The order of a PDE is the order of the highest partial derivative in the equation.

The order of our last example is 2 since the highest order partial derivative is \(u_{xy}\).

4 Differential Operators

To write PDEs compactly we think of differentiation as an operation applied to a function.

A differential operator of order \(m\) is a map \(L\) that takes a sufficiently differentiable function \(u\) to the function \[ Lu(x)=\sum_{\|\alpha\|\le m}a_\alpha(x)\,(D^\alpha u)(x), \] where the \(a_\alpha\) are given functions called the coefficients.

On functions \(u(x,y)\) the operators

\[ Lu=u_x+xu_y \quad \& \quad Mu=u_{xx}+3u_{xy}+2u_x+5u \]

are differential operators of orders \(1\) and \(2\) with coefficients \(a_{(1,0)}=1,a_{(0,1)}=x\) and \(a_{(2,0)}=1,a_{(1,1)}=3,a_{(1,0)}=2,a_{(0,0)}=5\) respectively. The Laplacian

\[ \Delta=\sum_i\partial_{x_i}^2 \]

is the second-order operator with all coefficients \(1\) (see Laplace Operator).

A map \(L:V\to W\) between vector spaces is a linear operator if for all \(u,v\in V\) and scalars \(c\) \[ L(cu+v)=cL(u)+L(v). \]

We show the operator \(Mu=u_{xx}+3u_{xy}+2u_x+5u\) above is linear. Differentiation is linear, \(\partial(cu+v)=c\,\partial u+\partial v\), and multiplying by a function \(a(x)\) is linear, so each term \(a_\alpha D^\alpha u\) is linear in \(u\), and a sum of linear maps is linear. Directly, \[ M(cu+v)=c\,u_{xx}+v_{xx}+3(cu_{xy}+v_{xy})+2(cu_x+v_x)+5(cu+v)=cMu+Mv. \]

By contrast, \(Nu=u_x+u^2\) is not linear since \(N(2u)=2u_x+4u^2\neq2Nu\), and neither is \(Nu=u_x+1\) since \(N(0)\neq0\).

Every differential operator \(Lu=\sum a_\alpha D^\alpha u\) whose coefficients depend only on \(x\) is linear. This is the case of interest in the next section.

5 Linear and Nonlinear PDEs

A PDE is linear of order \(m\) if it can be written as \[ Lu=\sum_{\|\alpha\|\le m}a_\alpha(x)\,(D^\alpha u)(x)=f(x),\qquad x\in\Omega, \] where \(L\) is a linear differential operator and \(a_\alpha\), \(f\) are given functions. A PDE which is not linear is called nonlinear.

A linear PDE is homogeneous if \(f\equiv0\), that is, it can be written as \[ Lu=0. \] Equivalently, every term contains the unknown function \(u\) or one of its derivatives. Otherwise the PDE is inhomogeneous.

Linearity gives the important superposition principle: if \(Lu_1=0\) and \(Lu_2=0\) then \(L(c_1u_1+c_2u_2)=0\), so the solutions of a homogeneous linear PDE form a vector space. If \(u_p\) is any solution of \(Lu=f\) then all solutions are \(u=u_p+u_h\) with \(Lu_h=0\).

Nonlinear PDEs are divided according to how nonlinear the highest-order derivatives are:

• Semilinear: the highest-order derivatives appear linearly with coefficients depending only on \(x\), e.g. \(\Delta u=u^3\).
• Quasilinear: the highest-order derivatives appear linearly but the coefficients may depend on \(u\) and lower-order derivatives, e.g. Burgers’ equation \(u_t+uu_x=0\).
• Fully nonlinear: nonlinear in the highest-order derivatives, e.g. the eikonal equation \(u_x^2+u_y^2=1\).

Order and type of \(u_t-u_{xx}+xu=0\): second order, linear homogeneous. Of \(u_t-u_{xx}+1=0\): second order, linear inhomogeneous (\(f=-1\)). Of \(u_t+u_{xxx}+\sqrt{1+u}=0\): third order, semilinear (hence nonlinear).

6 Classification of PDEs

There are two lenses through which to view PDEs: the mathematical structure and the physical phenomena they model.

6.1 Classification by Mathematical Structure

The classification below is for second-order linear PDEs (it extends to quasilinear ones by freezing the coefficients at a given solution). Write \[ Lu=\underbrace{\sum_{i,j=1}^{n}a_{ij}(x)\,u_{x_ix_j}}_{\text{principal part}}+\underbrace{\sum_ib_i(x)u_{x_i}+c(x)u}_{\text{lower-order terms}}, \qquad A=(a_{ij})_{i,j=1}^{n}\ \text{symmetric}. \] The principal part consists of the highest-order terms, which are the ones that determine the qualitative behaviour. The type at a point \(x\) depends on the Eigenvectors and Eigenvalues of \(A(x)\):

1. Elliptic - all eigenvalues of \(A\) are non-zero and have the same sign.
2. Parabolic - one eigenvalue of \(A\) is zero and the rest are non-zero with the same sign.
3. Hyperbolic - one eigenvalue of \(A\) has a different sign than the rest (and none is zero).

In two dimensions, the classification can be determined by the discriminant of the principal part. For a second-order PDE in two variables \(x\) and \(y\) of the form \[ a u_{xx} + b u_{xy} + c u_{yy} + (\ \text{lower order terms}) = 0, \] the PDE is classified as follows: \[ \begin{cases} b^2 - 4ac < 0 & \text{elliptic} \\ b^2 - 4ac = 0 & \text{parabolic} \\ b^2 - 4ac > 0 & \text{hyperbolic}. \end{cases} \]

(This is the eigenvalue test for \(A=\begin{pmatrix}a&b/2\\b/2&c\end{pmatrix}\), whose determinant is \(ac-b^2/4\).) The type is tied to the existence of characteristic curves, see PDE Characteristics.

The prototypical examples of each type of PDE are given in Table 1 below.

Type	Prototypical Example	Typical Behavior
Elliptic	Laplace’s Equation: \(\Delta u = 0\), Poisson’s Equation: \(-\Delta u = f\).	Equilibrium or steady state, boundary value problems, smooth solutions, every point influences each other.
Parabolic	Heat Equation: \(u_t = \Delta u\)	Smoothing, irreversible in time, infinite propagation speed, initial-boundary value problems.
Hyperbolic	Wave Equation: \(u_{tt}=c^2\Delta u\)	Finite propagation speed, real characteristics, singularities travel, time-reversible.

See Laplace Equation, Heat Equation and Wave Equation.

6.2 Classification by Physical Phenomena

Alternatively, PDEs can be classified by asking what a solution does as time evolves. A good way to see the distinction is to test each equation on a plane wave \(u=\exp (i(kx - \omega t))\), which gives a dispersion relation \(\omega = \omega (k)\).

Phenomenon	Model Equation	Dispersion Relation	Observations
Transport (Advection)	\(u_t + c u_x = 0\)	\(\omega = ck\)	Profile moves rigidly at speed \(c\): \(u = f(x-ct)\).
Diffusion	\(u_t = D u_{xx}\)	\(\omega = -iDk^2\)	Modes decay like \(\exp(-Dk^2t)\), so high frequencies die fastest and the profile smooths and spreads.
Wave Propagation	\(u_{tt} = c^2 u_{xx}\)	\(\omega = \pm ck\)	Two transport waves, profile moves rigidly at speed \(c\): \(u = f(x-ct) + g(x+ct)\).
Dispersion (Schroedinger)	\(i u_t = -\frac{1}{2} u_{xx}\)	\(\omega = \frac{1}{2} k^2\)	Modes rotate in the complex plane at different speeds, so the profile spreads and oscillates.

See Transport Equation.

Backlinks

• Calculus & Differential Equations