Introductory Excel for Scientists and Engineers

Explore solving differential equations with Excel using Euler's method, the foundational approach introduced in this section, with a review of calculus concepts for solid grounding.

Learn how to use computers to solve differential equations by understanding derivatives, gradients, and slopes on curved graphs, and recognize why straight-line approximations fail for real-world functions.

Compute the gradient of a function across a region by partitioning the domain into small segments, using dy/dx as the derivative, and tabulating the results in Excel.

Explore how Euler's method approximates differential equations by stepping y along with small h in Excel, highlighting the derivative, numerical analysis, and trade-offs between accuracy and computation.

Apply Euler's method in Excel to solve differential equations from an initial condition, using x and y columns, a fixed step size h, and cell references.

Explore how to implement Euler's method in Excel, extend calculations with drag-down, plot y versus x, and analyze how step size and accumulated errors affect convergence toward an analytical solution.

Explore Euler's method in Excel by solving dy/dx equals x^2 with starting value, using the recurrence y next equals y plus h times x squared, compare with the analytical solution.

Derive the derivative of power functions using a limit; the lecture establishes d/dx x^n = n x^{n-1} and shows d/dx (x^3/3) = x^2 with y = x^3/3.

Explore Euler's method in Excel by computing x cubed over three, compare predicted x values with analytical solutions, and examine how smaller step sizes improve accuracy through error plots.

Explore evaluating Euler's method in Excel by plotting simulated versus analytical solutions, measuring errors, and refining with smaller step sizes to improve accuracy.

Examine how relative error, not absolute error, governs the accuracy of Euler's method in Excel, and learn to compute, color-code, and chart absolute and relative errors over time.

Analyze how step size affects accuracy in Euler's method in Excel and how residuals indicate issues. Examine how Taylor series expansions offer faster convergence at the cost of added complexity.

Explore finite difference equations and extend the method to any differential equation using an Excel-based naively implemented algorithm, including a mass–on–a–spring example and relaxation methods.

Explore finite difference methods for solving differential equations on a computer using a discrete x-grid and spacing h to approximate derivatives, especially in one dimension within Excel.

Derive a second-order finite difference model of a mass on a spring with damping or driving forces and build an algorithm updating x with m, c, k, and h.

Explore finite difference methods in Excel, deriving first and second derivatives from data points and understanding how interval choices affect solution behavior in implicit and multi-step schemes.

Solve a second-order simple harmonic oscillator with initial conditions using time-step finite difference, converting to a system of first-order equations in Excel with m, c, k.

Factor constants into A and B and tabulate them to simplify complex formulas. Apply finite difference methods to propagate X with clear initial conditions for easier debugging in Excel.

Explore solving differential equations in Excel using a simple finite-difference approach, visualize damping, amplitude, and frequency, and assess accuracy against analytical predictions.

Solve the logistic equation, a first-order population model with carrying capacity K and growth rate R, using the finite difference method in Excel to update P over time.

Convert the logistic differential equation to a finite difference scheme and solve it numerically, then verify results through computational physics, experiments, and analytical solutions.

Set up the logistic equation in Excel with cell references and absolute references, propagate formulas, and explore how R and K affect the model while validating against analytical predictions.

Explore solving Laplace and Helmholtz-type equations in multiple dimensions with the finite difference method on a grid, showing that a point equals the average of its neighbors and noting spacing.

Explore the relaxation method in finite difference grids, showing that the central value f_i_j equals the average of its four neighboring cells, with generalization to multiple dimensions.

Use the relaxation method in a square heat-conduction domain in Excel to compute interior temperatures from fixed boundary conditions via a four-neighbor average.

Apply the relaxation method to solve a 2D finite-difference temperature distribution problem in rectangular Cartesian domains using Excel, exploring boundary conditions, grid refinement, and intuitive visualization.

Reframe higher order differential equations as a system of linked first-order equations to simplify solving with computers, enabling robust handling of multi-dimensional and higher-derivative problems.

Transform higher order differential equations into systems of first-order equations to enable stable numerical solutions, using methods like Runge-Kutta for ordinary and partial differential equations.

Learn how Taylor series provide accurate equation solutions with quantifiable error, and master the calculus requirements, with the instructor guiding you through analytical differentiation and practical examples.

Explore how Taylor series provide robust, more accurate approximations in numerical analysis, expressed as constant plus progressively higher power terms, offering reliable results over short distances.

Explore how Taylor and Maclaurin series express functions via derivatives at zero, with factorial coefficients, and apply these ideas to differential equations to approximate y(x).

Taylor's series shows how a function changes over small steps using prior y values and derivatives for higher accuracy. Error ties to the next term and shrinks with step sizes.

Explore solving a differential equation with a Taylor series by differentiating the function multiple times, using initial conditions, and validating results by hand and with tools.

Apply the product rule to differentiate complex functions like x e^x, and master higher derivatives, including second and third derivatives, by breaking the function into manageable parts.

Apply Taylor series to approximate derivatives, simplify complex expressions by grouping terms, and compare with Runge-Kutta methods, using Excel-ready formulas and tools like Wolfram Alpha.

Recognize and correct mistakes in derivation steps, as the lecture demonstrates spotting errors and fixing them, turning missteps into teachable moments in Introductory Excel for scientists and engineers.

Build Taylor series based models in Excel by inputting derivatives, using e^x, and locking references with dollar signs to drag formulas, then compare to the analytical solution.

Learn to model in Excel with a Taylor series approximation, using derivatives and spacing h to update y, and examine higher order terms and dimensional scaling.

Explore Taylor series modelling in excel by inputting and dragging formulas, defining a safe x-domain, and checking numerical results against experiment and the analytical solution.

Demonstrate Taylor series modelling in Excel to obtain an analytical solution for y(x) with a given initial condition, using the exponential e and chain rule for differentiation.

Examine Taylor series behavior in modeling by varying step size, showing how higher derivatives and factorial terms affect convergence toward the analytical solution and error trends.

Learn convergence analysis to detect mistakes, like a Taylor series error, and identify the exact computational level where problems arise, guiding you to locate and fix them in Excel.

Evaluate how well the simulator matches analytic results by analyzing convergence: test local and accumulated errors as spacing halves, verify fifth- and fourth-power scaling, and identify higher-order tail issues.

Explore convergence analysis by comparing absolute and relative errors against analytical solutions across different spacings, and identify how order of accuracy and derivative terms affect the results.

Perform convergence analysis of Taylor series across orders, fix a missing factor of two and related errors, and verify second- and first-order terms converge as theory predicts.

Demonstrates how to perform a convergence study to detect errors in a tailor series method for a differential equation, using error comparisons and convergence rate.

Explore the Runge-Kutta family of algorithms for solving differential equations on a computer, understand their practicality, convergence, and accuracy for large step sizes.

Understand why Runge-Kutta methods offer greater accuracy than naive finite spacing or Taylor series approaches in solving differential equations, especially for higher dimensions, with controllable error and practical computer costs.

Explore Runge cutting methods to solve differential equations by inputting F(x,y) into tailor series formulas, enabling a second order ranch cutter formula without any differentiation.

Explore the Runge-Kutta method, including second-order truncations and the popular fourth-order scheme, using Taylor expansions, derivatives, and k1–k4 to solve differential equations efficiently.

Learn to implement the Runge-Kutta 4 method in Excel, using k1-k4 and the RK4 update y_x+h = y_x + (k1+2k2+2k3+k4)/6, with f(x,y) = (1+y^2) sin x.

Apply Runge-Kutta to x and y values, drag across cells, and observe blow-up at large step sizes. Use a limited range and graphs to study convergence without an analytical solution.

Analyze the convergence of Runge-Kutta results by comparing y at x=1 across decreasing step sizes h, and assess absolute and relative errors to verify near fourth-order convergence.

Investigate how round-off errors, floating-point limits, and accumulated errors influence Runge-Kutta convergence, highlighting step counts, error ratios, and numerical checks with potential extrapolation.

Explore stiff equations and numerical stiffness, why prior methods fail without tiny step sizes, and learn implicit and predictive-corrective techniques to solve such systems, with applications in fluid dynamics.

Explain why explicit methods struggle with stiff equations and numerical stiffness when x and y vary at very different rates. Explore implicit methods and predictor-corrector approaches to solve stiff problems.

Compare implicit and explicit Euler methods for the equation y' = -20 x y in Excel, highlighting stability, algebraic steps, and the analytical solution e^{-20x}.

Explore implicit, explicit, and Crank-Nicholson methods for solving differential problems, comparing accuracy, stability, and reliability, and learn when averaging explicit and implicit terms improves results.

Explore predictor correct methods that blend explicit guesses with implicit refinements to improve accuracy. Prime implicit steps with explicit results using trapezoid or crank Nicholson approaches for iterative refinement.

Explore predictor-corrector methods, including the crank Nicholson formula and the trapezoid formulas, to solve y' = -0.5 y with y(0)=1, illustrating predictor and corrector cycles and accuracy improvements.

Learn how to apply statistical methods in Excel to analyze experimental data, using spreadsheet tools to perform powerful analyses while avoiding misuses like cherry picking and improper testing.

Explore distribution functions as analytical tools that link real-world data to paper and computer analyses. Focus on the normal distribution, its mean and standard deviation.

Explore the normal distribution and how uncorrelated measurements follow a bell-shaped curve. See its graph with mean, and examples from dice rolls, gas energies, and piston lengths.

Explore the normal distribution by examining dice experiments, summarize probabilities, and identify that the mean centers the curve while the standard deviation governs its width.

Study how the normal distribution arises from its analytic probability function, with mean and standard deviation shaping the curve and about two thirds of the area within one standard deviation.

Define the standard deviation sigma as the square root of the average squared deviations from the mean, showing how data spread around the mean with the dice example.

Explore how to simulate a normal distribution with 10 six-sided dice in Excel, generate data using rand and randbetween, create a histogram, and fit the results to a normal model.

Show how rolling ten six-sided dice creates a frequency histogram that resembles a normal distribution, and estimate its mean and standard deviation with Excel using a maximum likelihood approach.

Plot a normal distribution using the mean and standard deviation, compare it to cumulative frequencies with the probability distribution function and cumulative distribution function, and normalize for overlay.

Explore the viable distribution as a flexible tool for failure analysis of manufactured components. Learn to interpret its two constants, lambda (scale) and beta (shape), and assess time-to-failure behavior.

This lecture contrasts the normal distribution with the Weibull distribution, detailing its probability density function, shape parameter k, and lambda, and its use in reliability and time-to-failure analysis.

Explore how the Weibull distribution models time-to-failure with different shape parameters, fit to data in Excel, and interpret lambda as scale and k as the shape to estimate reliability.

Explore the Weibull distribution for reliability analysis, adjusting shape and scale parameters, and visualize its probability density function and cumulative distribution function with Excel.

Demonstrate how the Weibull shape and scale parameters shape time-to-failure, producing Gaussian or normal-like and exponential forms, and relate half-life to average lifespan under a constant failure rate.

Explore the Weibull distribution and how shape parameters shape the failure rate, early-life failures, and reliability, with cumulative distribution function behavior and implications for out of the box faults.

Apply the students t test to compare two data sets and test statistical hypotheses in experimental science, while understanding hypothesis testing pitfalls and data quality impacts on interpretation.

Apply the student t test to compare sample means from treated and untreated data, testing null versus alternative hypotheses to decide if a treatment changes hardness beyond random chance.

Explore the student’s t test to compare sample and population means using Excel. Understand one-tailed and two-tailed tests, paired samples, p-values, and the null hypothesis; replicate experiments to avoid bias.

Source gaussian distributed random numbers from the internet, using random.org, and import the data into Excel by copying and pasting with a normal distribution setup (mean 5, std dev 5.6).

Learn to perform a two-tailed student's t test in Excel on two data sets, compare Gaussian distributions, assess means and variances, and avoid bias through replication.

Apply the Student's t-test in Excel to compare two data sets, interpret results against sigma rules, and plan pre-specified decision ranges to avoid misinterpreting the null hypothesis.

Congratulate yourself on completing the introductory excel for scientists and engineers course, and tackle the challenging material for your career or research.