Mathematical Modeling and PDEs Introduction

Using math to describe, analyze, and predict real-world phenomena.

A series of steps to create and validate a model: Problem identification, assumptions, formulation, analysis, interpretation, validation, and refinement.

A model where outcomes are precisely determined by known relationships.

A model incorporating random variables and probabilities.

A model describing relationships that don't change over time.

A model describing systems changing over time.

A model where the output is proportional to the input.

A model with complex relationships, not proportional.

Models describing continuous change using derivatives.

Models for systems changing in steps, not continuously.

Models analyzing data and making predictions.

Models analyzing strategic interactions between parties.

Equations involving functions of multiple variables and their partial derivatives.

A PDE describing the flow of heat in a medium.

A PDE describing wave propagation through a medium.

A vector differential operator.

Direction and magnitude of steepest increase in a scalar field.

Measures the source/sink strength of a vector field.

Measures the rotation/circulation in a vector field.

Divergence of the gradient.

Dependent variable and its derivatives appear only to the first power

Higher powers or nonlinear functions of the dependent variable or its derivatives

A PDE with additional conditions specified at boundaries of the domain

Solution value specified at the boundary

Derivative of the solution is specified at the boundary

Specifies the state of a system at a starting time (usually t=0)

Assumptions made about the system to make the mathematical problem easier to solve. These assumptions may be simplifications of real-world complexities, but they are necessary to develop a manageable model.

Testing the model against real-world data to see how well it predicts observed outcomes. This step is crucial to determine the model's accuracy and reliability.

A mathematical model that accounts for random variables and probability distributions, reflecting the inherent uncertainty in real-world events.

A model that describes how a system changes over time, capturing the dynamic nature of real-world processes.

Translating the mathematical results back into the real-world context, making sense of the mathematical solution in terms of the original problem.

An equation that describes the rate of change of a function with respect to a single independent variable (often time). It uses ordinary derivatives like 'd/dt' or 'd/dx'.

An equation that describes the rate of change of a function with respect to multiple independent variables (often time and space). It uses partial derivatives like '∂/∂t' or '∂/∂x'.

A vector differential operator that acts like a multi-dimensional derivative, used to calculate gradients, divergences, and curls.

A mathematical equation that relates the Laplacian of the electric potential to the charge density. It's used to describe the electric field produced by a distribution of charges.

A second-order PDE where its discriminant is greater than zero. Hyperbolic PDEs often describe wave propagation.

A second-order PDE where its discriminant is equal to zero. Parabolic PDEs typically describe diffusion processes.

A second-order PDE where its discriminant is less than zero. Elliptic PDEs usually describe steady-state or equilibrium problems.

A technique for solving linear PDEs by assuming the solution can be expressed as a product of functions, each depending on one variable. It simplifies the problem into smaller parts.

A property of a material that determines how quickly heat can diffuse through it. It depends on the material's thermal conductivity, density, and specific heat capacity.

The solution to a PDE that satisfies the governing equation and boundary conditions but contains arbitrary constants.

A way of expressing a periodic function as an infinite sum of sines and cosines.

Two functions are orthogonal if their inner product is zero.

Different shapes of the solution in space, determined by the sine function in the series.

How the solution changes over time, characterized by exponential decay.

A mathematical tool that converts a function from the time domain to the frequency domain.

A mathematical tool that converts a function from the frequency domain back to the time domain.

A mathematical tool that converts a function from the time domain to the complex frequency domain.

A numerical method to solve PDEs by approximating derivatives with differences between discrete points.

A dimensionless number that affects the stability of a numerical solution.

A constraint on the time step size (dt) to prevent numerical solutions from becoming unstable and diverging. It ensures the solution remains accurate and realistic.

A method for solving PDEs that directly calculates the solution at the next time step using values from the current time step. It's like taking one small step at a time.

A method that solves for the solution at the next time step by solving a system of equations. It uses information from both current and future time steps.

A hybrid method combining features of both explicit and implicit schemes to achieve better accuracy and stability. It uses information from both the current and future time steps, but in a balanced way.

Explicit methods directly calculate the solution at the next time step using known values from the current time step. Implicit methods solve a system of equations to find the solution at the next time step, using values from both the current and future time steps.

The rate at which a wave travels through a medium, determined by the tension and density of the medium.

Constraints imposed on the solution of a partial differential equation at the boundaries of the domain, representing physical restrictions or interactions.

For linear, homogeneous PDEs, you can add together solutions to get another solution.

A mathematical problem where you find special values (eigenvalues) and corresponding functions (eigenfunctions) that satisfy a given equation and boundary conditions.