Numerical Methods for PDEs

Numerical Methods for Solving PDEs

• Analytical solutions are not always available for PDEs, necessitating numerical methods.

Finite Difference Method (FDM)

• Employs difference quotients to approximate derivatives.
• Utilizes a discrete grid to represent the domain.
• First-order derivative approximation: $\frac{\partial u}{\partial x} \approx \frac{u_{i+1,j} - u_{i,j}}{\Delta x}$
• Second-order derivative approximation: $\frac{\partial^2 u}{\partial x^2} \approx \frac{u_{i+1,j} - 2u_{i,j} + u_{i-1,j}}{(\Delta x)^2}$

Heat Equation Example

• The heat equation is represented by: $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
• Discretization yields: $\frac{u_{i,n+1} - u_{i,n}}{\Delta t} = \alpha \frac{u_{i+1,n} - 2u_{i,n} + u_{i-1,n}}{(\Delta x)^2}$
• Solving for $u_{i,n+1}$: $u_{i,n+1} = u_{i,n} + \alpha \frac{\Delta t}{(\Delta x)^2} (u_{i+1,n} - 2u_{i,n} + u_{i-1,n})$
• Stability is affected by the choice of $\Delta t$ and $\Delta x$.
• Stability condition for the heat equation: $\alpha \frac{\Delta t}{(\Delta x)^2} \leq \frac{1}{2}$

Finite Element Method (FEM)

• The domain is divided into smaller, simpler elements.
• Approximates solutions within each element.
• The method follows a series of steps:
• Weak Formulation: Convert the PDE into an integral form.
• Discretization: Divide the domain into elements.
• Element Equations: Derive equations for each element.
• Assembly: Combine element equations to form a global system.
• Solve: Solve the global system of equations.

Poisson Equation Example

• Poisson's equation is: $-\nabla \cdot (c \nabla u) = f \quad \text{in } \Omega$
• Boundary condition: $u = 0 \quad \text{on } \partial \Omega$
• Weak form requires $u \in H_0^1 (\Omega)$ such that $\int_\Omega c \nabla u \cdot \nabla v , dx = \int_\Omega fv , dx \quad \forall v \in H_0^1 (\Omega)$

Real-world Applications

• Used in heat transfer analysis.
• Used in fluid dynamics simulations.
• Applicable to electromagnetics problems.
• Used for structural mechanics calculations.

Software Solutions

• COMSOL Multiphysics is a software.
• ANSYS is a software.
• FEATool Multiphysics (MATLAB) is a software.
• FEniCS (Python) is a software.

FDM vs FEM Comparison

• FDM uses regular grids; FEM uses irregular grids.
• FDM is suitable for simple geometries; FEM is suitable for complex ones.
• FDM is easier to implement; FEM is more complex.
• FDM has lower accuracy for complex geometries; FEM has higher accuracy.
• FDM has more difficult boundary handling for complex boundaries; FEM's boundary handling is more natural.

Hypothesis Testing Basics

• A hypothesis is a statement about a population parameter.

Types of Hypotheses

• Null Hypothesis ($H_0$): Claims "no effect" or "no difference". It's what's being disproved.
• Alternative Hypothesis ($H_1$ or $H_a$): Contradicts the null hypothesis. It is what you're trying to prove.

Hypothesis Examples

• Example 1:
• $H_0: \mu = 100$ (population mean is 100)
• $H_1: \mu \neq 100$ (population mean is not 100)
• Example 2:
• $H_0: p = 0.5$ (population proportion is 0.5)
• $H_1: p > 0.5$ (population proportion is greater than 0.5)

Test Statistic

• It is computed from sample data.
• It helps determine if the null hypothesis can be rejected.
• Examples are the z-statistic and t-statistic.

P-value

• It represents the probability of observing a test statistic as extreme as, or more extreme than, the one computed from the sample data.
• It assumes the null hypothesis is true.

Significance Level ($\alpha$)

• Probability of rejecting the null hypothesis when it's true (Type I error).
• Common values: 0.05 and 0.01.

Decision Rule

• Reject $H_0$ if p-value $\le \alpha$.
• Fail to reject $H_0$ if p-value $> \alpha$.

Types of Errors

• Type I Error (False Positive): Rejecting a true $H_0$; probability is $\alpha$.
• Type II Error (False Negative): Failing to reject a false $H_0$; probability is $\beta$.

Hypothesis Tests for the Mean ($\mu$)

z-test

• Assumptions:
• Population standard deviation ($\sigma$) is known.
• Normally distributed population or large sample size (n $\ge$ 30).
• Test Statistic:
• $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$
• $\bar{x}$ is the sample mean.
• $\mu_0$ is the hypothesized population mean.
• $\sigma$ is the population standard deviation.
• $n$ is the sample size.

t-test

• Assumptions:
• Population standard deviation ($\sigma$) is unknown.
• Normally distributed population or large sample size (n $\ge$ 30).
• Test Statistic:
• $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$
• $\bar{x}$ is the sample mean.
• $\mu_0$ is the hypothesized population mean.
• $s$ is the sample standard deviation.
• $n$ is the sample size.
• Degrees of Freedom: $df = n - 1$

Hypothesis Tests for the Proportion ($p$)

z-test for Proportion

• Assumptions:
• $np_0 \ge 10$ and $n(1 - p_0) \ge 10$
• Test Statistic:
• $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$
• $\hat{p}$ is the sample proportion.
• $p_0$ is the hypothesized population proportion.
• $n$ is the sample size.

Steps in Hypothesis Testing

• Step 1: State the hypotheses (Formulate $H_0$ and $H_1$).
• Step 2: Determine the test statistic (z or t) and calculate its value.
• Step 3: Determine the P-value associated with the test statistic.
• Step 4: Make a decision by comparing the p-value to the significance level ($\alpha$).
• Step 5: Interpret the results in the context of the problem.

Example: z-test for the Mean

• A researcher tests if the average IQ score of students is greater than 100.
• Sample of 50 students, mean IQ score of 105.
• Population standard deviation is 15, $\alpha = 0.05$.
• Hypotheses:
• $H_0: \mu = 100$
• $H_1: \mu > 100$
• Test Statistic:
• $z = \frac{105 - 100}{15 / \sqrt{50}} \approx 2.357$
• P-value:
• $P(z > 2.357) \approx 0.0092$
• Decision:
• Reject $H_0$ because $0.0092 < 0.05$.
• Interpretation:
• Evidence supports the conclusion that the average IQ score of students is greater than 100.