Partial Differential Equations Study Notes

Definition and Notation of Partial Differential Equations

• Definition: A Partial Differential Equation (PDE) is a differential equation involving partial derivatives of one or more dependent variables with respect to one or more independent variables.

• Standard Notations:

  • $p = \frac{\partial z}{\partial x}$

  • $q = \frac{\partial z}{\partial y}$

  • $r = \frac{\partial^2 z}{\partial x^2}$

  • $s = \frac{\partial^2 z}{\partial x \, \partial y}$

  • $t = \frac{\partial^2 z}{\partial y^2}$

Formation of Partial Differential Equations

• By Elimination of Arbitrary Constants: Differentiating an equation $f(x, y, z, a, b) = 0$ partially with respect to $x$ and $y$ and eliminating the constants $a$ and $b$ yields a first-order PDE of the form $f(x, y, z, p, q) = 0$

• By Elimination of Arbitrary Functions: Eliminating arbitrary functions $f$ from an expression $f(u, v) = 0$ (where $u$ and $v$ are functions of $x, y, z$) yields a first-order linear PDE $P p + Q q = R$

Types of Integrals / Solutions of PDEs

• Complete Integral: A solution containing as many arbitrary constants as independent variables ($\phi(x, y, z, a, b) = 0$)

• Particular Integral: A solution obtained by assigning specific numerical values to arbitrary constants in the complete integral

• Singular Integral: Represents the envelope of the family of surfaces given by the complete integral, obtained by eliminating $a$ and $b$ from $\phi = 0$, $\frac{\partial \phi}{\partial a} = 0$, and $\frac{\partial \phi}{\partial b} = 0$

• General Integral: A solution of the form $f(u, v) = 0$ containing arbitrary functions

Lagrange's Solution of First-Order Linear PDEs

• Form: $P p + Q q = R$, where $P, Q, R$ are functions of $x, y, z$

• Lagrange's Auxiliary Equations:   $\frac{dx}{P} = \frac{dy}{Q} = \frac{dz}{R}$

• Method: Solve the auxiliary equations to find two independent integrals $u(x, y, z) = c_1$ and $v(x, y, z) = c_2$. The general solution is expressed as $f(u, v) = 0$

Standard Forms for Non-Linear First-Order PDEs

• Standard Form I: $f(p, q) = 0$

  • Complete Integral: $z = ax + by + c$, where $b = \phi(a)$ from $f(a, b) = 0$

• Standard Form II: $f(p, q, z) = 0$

  • Method: Substitute $X = x + ay$, which converts the PDE into an ordinary differential equation (ODE) $f\left(\frac{dz}{dX}, a\frac{dz}{dX}, z\right) = 0$

• Standard Form III: $f(p, x) = g(q, y)$

  • Method: Equate both sides to an arbitrary constant $a$ ($f(p, x) = a$ and $g(q, y) = a$), solve for $p$ and $q$, and integrate $dz = p\,dx + q\,dy$

• Standard Form IV (Clairaut's Form): $z = px + qy + f(p, q)$

  • Complete Integral: $z = ax + by + f(a, b)$

Charpit's Method for General First-Order Non-Linear PDEs

• Charpit's Auxiliary Equations:   $\frac{dp}{\frac{\partial f}{\partial x} + p \frac{\partial f}{\partial z}} = \frac{dq}{\frac{\partial f}{\partial y} + q \frac{\partial f}{\partial z}} = \frac{dz}{-p \frac{\partial f}{\partial p} - q \frac{\partial f}{\partial q}} = \frac{dx}{-\frac{\partial f}{\partial p}} = \frac{dy}{-\frac{\partial f}{\partial q}}$

• Method: Solve for $p$ and $q$ using the auxiliary equations, then integrate $dz = p\,dx + q\,dy$

Homogeneous Linear PDEs with Constant Coefficients

• General Form: $F(D, D')z = f(x, y)$, where $D = \frac{\partial}{\partial x}$ and $D' = \frac{\partial}{\partial y}$

• Auxiliary Equation (A.E.): Formed by substituting $D = m$ and $D' = 1$ into $F(D, D') = 0$

• Complementary Function (C.F.):

  • Distinct real roots $\alpha_1, \alpha_2, \dots, \alpha_n$: $C.F. = f_1(y + \alpha_1 x) + f_2(y + \alpha_2 x) + \dots + f_n(y + \alpha_n x)$

  • Repeated root $\alpha$ (twice): $C.F. = f_1(y + \alpha x) + x f_2(y + \alpha x)$

Classification of Second-Order Linear PDEs

• General Equation: $R r + S s + T t + f(x, y, z, p, q) = 0$

• Classification Criterion (based on discriminant $S^2 - 4RT$):

  • Hyperbolic: $S^2 - 4RT > 0$

  • Parabolic: $S^2 - 4RT = 0$

  • Elliptic: $S^2 - 4RT < 0$

Classical Second-Order Partial Differential Equations

• Wave Equation (1D): $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$

  • D'Alembert's Solution: $u(x, t) = \frac{1}{2}[\phi(x - ct) + \phi(x + ct)] + \frac{1}{2c} \int_{x - ct}^{x + ct} \psi(s)\,ds$

• Heat Equation (1D): $\frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2}$

  • Solved using the method of separation of variables yielding Fourier sine/cosine series solutions.

• Laplace Equation (2D): $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$

  • Functions satisfying Laplace's equation are called harmonic functions.

Sturm-Liouville Problems and Green's Function

• Sturm-Liouville Equation: $\frac{d}{dx}\left(p(x) \frac{dy}{dx}\right) + q(x) y = \lambda y$

• Properties: Eigenvalues $\lambda_n$ are real, ordered, and countable; corresponding eigenfunctions $y_n(x)$ are orthogonal.

• Green's Function: Used to construct integral solutions for non-homogeneous boundary value problems $L y = f(x)$ in the form $y(x) = \int_a^b G(x, t) f(t)\,dt$

Definition and Notation of Partial Differential Equations

• Definition: A Partial Differential Equation (PDE) is a differential equation involving partial derivatives of one or more dependent variables with respect to two or more independent variables.

• Standard Notations (Monge's Notation):

  • $p = \frac{\partial z}{\partial x}$

  • $q = \frac{\partial z}{\partial y}$

  • $r = \frac{\partial^2 z}{\partial x^2}$

  • $s = \frac{\partial^2 z}{\partial x \,\partial y}$

  • $t = \frac{\partial^2 z}{\partial y^2}$

Formation of Partial Differential Equations

• By Elimination of Arbitrary Constants:

  • Differentiating an equation $f(x, y, z, a, b) = 0$ partially with respect to $x$ and $y$ gives equations involving $p$ and $q$.

  • Eliminating constants $a$ and $b$ yields a first-order PDE $f(x, y, z, p, q) = 0$.

  • If the number of arbitrary constants equals the number of independent variables, a first-order PDE is obtained.

• By Elimination of Arbitrary Functions:

  • Eliminating arbitrary functions $f$ from an expression $f(u, v) = 0$ (where $u = u(x, y, z)$ and $v = v(x, y, z)$) yields a first-order linear PDE of Lagrange's form $P p + Q q = R$.

Types of Integrals / Solutions of PDEs

• Complete Integral: A solution $\phi(x, y, z, a, b) = 0$ containing as many arbitrary constants ($a, b$) as independent variables ($x, y$).

• Particular Integral: A solution obtained by assigning specific numerical values to the arbitrary constants in the complete integral.

• Singular Integral: Represents the envelope of the family of surfaces given by the complete integral. It is obtained by eliminating $a$ and $b$ from:

  • $\phi(x, y, z, a, b) = 0$

  • $\frac{\partial \phi}{\partial a} = 0$

  • $\frac{\partial \phi}{\partial b} = 0$

• General Integral: Obtained by setting $b = \phi(a)$ in the complete integral to get $\phi(x, y, z, a, \phi(a)) = 0$, differentiating with respect to $a$, and eliminating $a$. It corresponds to the form $f(u, v) = 0$.

Lagrange's Solution of First-Order Linear PDEs

• Form: $P p + Q q = R$, where $P, Q, R$ are functions of $x, y, z$.

• Lagrange's Auxiliary Equations:   $\frac{dx}{P} = \frac{dy}{Q} = \frac{dz}{R}$

• Method of Solution:

  1. Grouping Method: Pair any two ratios to integrate directly and find $u(x, y, z) = c_1$.

  2. Method of Multipliers: Find multipliers $(l, m, n)$ such that $l P + m Q + n R = 0$, implying $l\,dx + m\,dy + n\,dz = 0$, which integrates to $v(x, y, z) = c_2$.

• General Solution: Expressed as $f(u, v) = 0$ or $u = \phi(v)$.

Standard Forms for Non-Linear First-Order PDEs

• Standard Form I: $f(p, q) = 0$

  • Complete Integral: $z = a x + b y + c$, where $b$ is determined from $f(a, b) = 0$.

• Standard Form II: $f(p, q, z) = 0$

  • Method: Substitute $X = x + a y$, transforming the PDE into an ordinary differential equation (ODE):     $f\left(\frac{dz}{dX}, a \frac{dz}{dX}, z\right) = 0$

• Standard Form III: $f(p, x) = g(q, y)$

  • Method: Equate both sides to an arbitrary constant $a$ ($f(p, x) = a \Rightarrow p = h_1(x, a)$ and $g(q, y) = a \Rightarrow q = h_2(y, a)$), then integrate $dz = p\,dx + q\,dy$ to get:     $z = \int h_1(x, a)\,dx + \int h_2(y, a)\,dy + b$

• Standard Form IV (Clairaut's Form): $z = p x + q y + f(p, q)$

  • Complete Integral: $z = a x + b y + f(a, b)$.

Charpit's Method for General First-Order Non-Linear PDEs

• Applicable to general first-order equations $f(x, y, z, p, q) = 0$.

• Charpit's Auxiliary Equations:   $\frac{dp}{\frac{\partial f}{\partial x} + p \frac{\partial f}{\partial z}} = \frac{dq}{\frac{\partial f}{\partial y} + q \frac{\partial f}{\partial z}} = \frac{dz}{-p \frac{\partial f}{\partial p} - q \frac{\partial f}{\partial q}} = \frac{dx}{-\frac{\partial f}{\partial p}} = \frac{dy}{-\frac{\partial f}{\partial q}}$

• Method: Solve for $p$ and $q$ using auxiliary equations and the original PDE, then integrate $dz = p\,dx + q\,dy$

Homogeneous and Non-Homogeneous Linear PDEs with Constant Coefficients

• General Form: $F(D, D') z = f(x, y)$, where $D = \frac{\partial}{\partial x}$ and $D' = \frac{\partial}{\partial y}$.

• Auxiliary Equation (A.E.): Formed by substituting $D = m$ and $D' = 1$ into $F(D, D') = 0$

• Complementary Function (C.F.):

  • Distinct real roots $\alpha_1, \alpha_2, \dots, \alpha_n$:     $C.F. = f_1(y + \alpha_1 x) + f_2(y + \alpha_2 x) + \dots + f_n(y + \alpha_n x)$

  • Repeated root $\alpha$ ($m$ times):     $C.F. = f_1(y + \alpha x) + x f_2(y + \alpha x) + \dots + x^{m-1} f_m(y + \alpha x)$

• Particular Integral (P.I.):

  • For $f(x, y) = \exp(a x + b y)$: $P.I. = \frac{1}{F(a, b)} \exp(a x + b y)$ provided $F(a, b) \neq 0$.

  • For $f(x, y) = \sin(a x + b y)$ or $\cos(a x + b y)$: Substitute $D^2 \rightarrow -a^2$, $D D' \rightarrow -a b$, and ${D'}^2 \rightarrow -b^2$.

Cauchy's Problem and Existence / Uniqueness

• Initial Value Problem: Consists of finding a solution surface passing through a given initial space curve $x = x_0(t)$, $y = y_0(t)$, $z = z_0(t)$.

• Uniqueness Condition: For $P p + Q q = R$, a unique solution exists if:   $P(x_0, y_0, z_0) \frac{dy_0}{dt} - Q(x_0, y_0, z_0) \frac{dx_0}{dt} \neq 0$

• No Solution / Infinite Solutions: If $P \frac{dy_0}{dt} - Q \frac{dx_0}{dt} = 0$, then:

  • Infinite Solutions if $R \frac{dx_0}{dt} = P \frac{dz_0}{dt}$.

  • No Solution if $R \frac{dx_0}{dt} \neq P \frac{dz_0}{dt}$.

Classification and Canonical Forms of Second-Order Linear PDEs

• General Form: $R r + S s + T t + f(x, y, z, p, q) = 0$

• Classification Criterion (discriminant $\Delta = S^2 - 4 R T$):

  • Hyperbolic: $S^2 - 4 R T > 0$ (Two real characteristic curves). Canonical form: $\frac{\partial^2 z}{\partial u \,\partial v} = \phi(u, v, z, z_u, z_v)$.

  • Parabolic: $S^2 - 4 R T = 0$ (One real characteristic curve). Canonical form: $\frac{\partial^2 z}{\partial v^2} = \phi(u, v, z, z_u, z_v)$.

  • Elliptic: $S^2 - 4 R T < 0$ (Complex conjugate characteristic curves). Canonical form: $\frac{\partial^2 z}{\partial \xi^2} + \frac{\partial^2 z}{\partial \eta^2} = \phi(\xi, \eta, z, z_\xi, z_\eta)$.

Classical Second-Order Partial Differential Equations

• Wave Equation (1D): $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$

  • D'Alembert's Solution for infinite domain ($-\infty < x < \infty$):     $u(x, t) = \frac{1}{2}\left[\phi(x - c t) + \phi(x + c t)\right] + \frac{1}{2 c} \int_{x - c t}^{x + c t} \psi(s)\,ds$

  • Fourier Series Solution for finite domain ($0 \le x \le l$) with $u(0, t) = u(l, t) = 0$:     $u(x, t) = \sum_{n=1}^{\infty} \sin\left(\frac{n \pi x}{l}\right) \left[a_n \cos\left(\frac{n \pi c t}{l}\right) + b_n \sin\left(\frac{n \pi c t}{l}\right)\right]$

• Heat Equation (1D): $\frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2}$

  • Solved via separation of variables: $u(x, t) = \sum_{n=1}^{\infty} c_n \sin\left(\frac{n \pi x}{l}\right) \exp\left(-\frac{c^2 n^2 \pi^2 t}{l^2}\right)$.

• Laplace Equation (2D): $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$

  • Functions satisfying Laplace's equation are called harmonic functions.

  • In polar coordinates: $\frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0$.

Sturm-Liouville Problems and Green's Function

• Sturm-Liouville Equation: $\frac{d}{dx}\left(p(x) \frac{dy}{dx}\right) + q(x) y + \lambda r(x) y = 0$

• Key Properties:

  • Eigenvalues $\lambda_n$ are real, countable, and ordered ($\lambda_1 < \lambda_2 < \dots$).

  • Eigenfunctions $y_n(x)$ corresponding to distinct eigenvalues are orthogonal with weight function $r(x)$.

• Green's Function: Used for non-homogeneous BVP $L[y] = f(x)$:   $y(x) = \int_{a}^{b} G(x, t) f(t)\,dt$   where $G(x, t)$ is continuous at $x = t$, satisfies boundary conditions, and possesses a jump discontinuity in derivative at $x = t$:   $$\left.\frac{\partial G}{\