Partial Differential Equations

Introduction to Partial Differential Equations (PDEs)

A Partial Differential Equation (PDE) is an equation that involves an unknown multivariable function and its partial derivatives with respect to two or more independent variables. The order of a PDE is the order of the highest partial derivative it contains.

PDEs are fundamental in modeling a vast array of phenomena in science and engineering, including:

• Heat flow and diffusion: The Heat Equation, $\frac{\partial u}{\partial t} = \alpha \nabla^2 u$, describes how temperature $u$ changes over time $t$ in a given region.
• Wave propagation: The Wave Equation, $\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$, models phenomena like vibrating strings, sound waves, and electromagnetic waves.
• Electrostatics and steady-state phenomena: Laplace's Equation, $\nabla^2 u = 0$, and Poisson's Equation, $\nabla^2 u = f$, describe potentials in fields where conditions are not changing over time.
• Fluid dynamics: The Navier-Stokes equations are a set of PDEs describing the motion of viscous fluid substances.
• Quantum mechanics: The Schrödinger equation, $i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi$, is a linear PDE that describes how the quantum state of a physical system changes over time.

Unlike Ordinary Differential Equations (ODEs), which typically have general solutions involving arbitrary constants, the solutions to PDEs often involve arbitrary functions. To obtain a unique solution for a specific physical problem, PDEs must be supplemented with boundary conditions (specifying the value of the solution or its derivatives on the boundary of the domain) and/or initial conditions (specifying the state of the system at an initial time).

Method 1: Direct Partial Integration

Some of the simplest PDEs can be solved by performing direct partial integration with respect to one variable at a time. This method is analogous to solving simple ODEs like $y' = f(x)$ by direct integration.

Key Idea: When integrating with respect to one variable (say, $x$), all other variables (say, $y, z, \dots$) are treated as constants. The "constant of integration" that arises from this process will not be a simple constant $C$, but rather an arbitrary function of the other variable(s) that were held constant during the integration.

Example 1: First-order PDE

Consider the PDE: $\frac{\partial u}{\partial x} = f(x,y)$

To find $u(x,y)$, we integrate both sides with respect to $x$, holding $y$ constant:

Explanation:

• The term $\int f(x,y) \, \partial x$ represents the partial antiderivative of $f(x,y)$ with respect to $x$.
• $G(y)$ is an arbitrary function of $y$. This is because if we differentiate $G(y)$ with respect to $x$ (treating $y$ as constant), we get $\frac{\partial}{\partial x} G(y) = 0$. So, $G(y)$ behaves like a constant during the partial differentiation with respect to $x$.

Example 2: Second-order PDE (Mixed Partial)

Consider the PDE: $\frac{\partial^2 u}{\partial x \partial y} = f(x,y)$, which can be written as $\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial y}\right) = f(x,y)$.

Step 1: Integrate with respect to $x$ (treating $y$ as constant).

Here, $G(y)$ is an arbitrary function of $y$, acting as the "constant" of integration for the $x$-integration.

Step 2: Integrate the result from Step 1 with respect to $y$ (treating $x$ as constant).

Explanation:

• The term $\int \left( \int f(x,y) \, \partial x \right) \partial y$ is the result of integrating $f(x,y)$ first with respect to $x$, then with respect to $y$.
• $\int G(y) \, \partial y$ is the integral of the arbitrary function $G(y)$ from the first integration. Let's call this $G_1(y)$. Since $G(y)$ was arbitrary, $G_1(y)$ is also an arbitrary function of $y$.
• $H(x)$ is a new arbitrary function of $x$, which acts as the "constant" of integration for the $y$-integration.

So, the general solution can be written as:

The specific forms of $G_1(y)$ and $H(x)$ are determined by boundary or initial conditions.

Method 2: Separation of Variables

The method of separation of variables is a powerful and widely used technique for solving certain types of linear homogeneous PDEs with appropriate boundary conditions. The core idea is to assume that the solution can be expressed as a product of functions, where each function depends on only one of the independent variables.

For a function $u(x,y)$, we might assume $u(x,y) = X(x)Y(y)$. For $u(x,t)$, we assume $u(x,t) = X(x)T(t)$.

General Procedure:

1. Assume a Separable Solution:

   Postulate that the solution $u$ can be written as a product of functions of single variables. For example, if $u = u(x,t)$, assume $u(x,t) = X(x)T(t)$, where $X(x)$ is a function of $x$ only, and $T(t)$ is a function of $t$ only.

   Why? This assumption simplifies the PDE by allowing us to treat derivatives with respect to one variable as operations on only one of the product functions.

2. Substitute into the PDE:

   Replace $u$ and its partial derivatives in the original PDE with the assumed product form. For instance, if the PDE involves $u_t$ and $u_{xx}$:

   • $\frac{\partial u}{\partial t} = X(x)T'(t)$ (where $T'(t) = \frac{dT}{dt}$)
   • $\frac{\partial^2 u}{\partial x^2} = X''(x)T(t)$ (where $X''(x) = \frac{d^2X}{dx^2}$)

   Why? This transforms the PDE involving multiple variables into an equation involving functions of single variables and their ordinary derivatives.

3. Separate the Variables:

   Algebraically manipulate the substituted equation to group all terms involving one independent variable (and its corresponding function) on one side of the equation, and all terms involving the other independent variable on the other side. This often involves dividing through by $X(x)T(t)$ (assuming $X(x)T(t) \neq 0$, which is generally true for non-trivial solutions).

   For example, you might arrive at an equation like: $\frac{X''(x)}{X(x)} = \frac{1}{\alpha} \frac{T'(t)}{T(t)}$ (from the Heat Equation).

   Why? This step is crucial. If an expression depending only on $x$ is equal to an expression depending only on $t$ for all values of $x$ and $t$, then both expressions must be equal to the same constant.

4. Introduce the Separation Constant:

   Since the left side of the separated equation depends only on $x$ and the right side depends only on $t$ (or other respective variables), and they are equal, they must both be equal to a constant value. This constant is called the separation constant (often denoted by $\lambda$, $k^2$, $-k^2$, $m^2$, etc., chosen for convenience based on the expected form of solutions).

   So, from the example above: $\frac{X''(x)}{X(x)} = \lambda$ and $\frac{1}{\alpha} \frac{T'(t)}{T(t)} = \lambda$.

   Why? This step decouples the original PDE into a set of Ordinary Differential Equations (ODEs), each involving only one independent variable.

5. Solve the ODEs:

   Solve each of the resulting ODEs for $X(x)$, $T(t)$, etc. The solutions to these ODEs will typically involve arbitrary constants and will depend on the value and sign of the separation constant $\lambda$. Different ranges of $\lambda$ (e.g., $\lambda > 0$, $\lambda = 0$, $\lambda < 0$) might lead to different forms of solutions (exponential, trigonometric, linear).

   Why? ODEs are generally easier to solve than PDEs. Their solutions provide the building blocks for the PDE's solution.

6. Apply Boundary/Initial Conditions (Superposition):

   The individual solutions $X(x)T(t)$ obtained for different values of the separation constant (eigenvalues) are often combined using the principle of superposition (for linear homogeneous PDEs) to form a general solution. This usually takes the form of an infinite series (e.g., a Fourier series).

   The arbitrary constants in the ODE solutions and the specific values of the separation constants are then determined by applying the given boundary conditions and initial conditions of the problem.

   Why? Boundary and initial conditions are necessary to find a unique solution that matches the specific physical problem being modeled. Superposition allows us to construct more complex solutions from simpler ones.

The solver below demonstrates steps 1-4, showing how a PDE is separated into a system of ODEs.

Some Famous PDEs

Several PDEs appear frequently in engineering and physics, each modeling distinct physical phenomena:

• Wave Equation (1D): Describes the propagation of waves, such as the vibrations of a string, sound waves, or light waves.

  Form: $ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \quad \text{or} \quad u_{tt} = c^2 u_{xx} $

  Here, $u(x,t)$ is the displacement of the wave at position $x$ and time $t$, and $c$ is the wave speed.

• Heat Conduction Equation (1D): Describes how temperature distributes or how a substance diffuses over time.

  Form: $ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} \quad \text{or} \quad u_{t} = \alpha u_{xx} $

  $u(x,t)$ is the temperature (or concentration) at position $x$ and time $t$, and $\alpha$ is the thermal diffusivity (or diffusion coefficient).

• Laplace's Equation (2D): Describes steady-state phenomena where there is no change over time, such as steady-state temperature distributions, electrostatic potentials in charge-free regions, or certain ideal fluid flows.

  Form: $ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \quad \text{or} \quad u_{xx} + u_{yy} = 0 \quad (\text{often written as } \nabla^2 u = 0) $

  $u(x,y)$ represents the quantity (e.g., temperature, potential) at position $(x,y)$.