FOLLOW THE PHYSICAL PROBLEM - HOLISTIC NUMERICAL METHODS INSTITUTE - MATHCAD - MAPLE

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FOLLOW THE PHYSICAL PROBLEM

To illustrate numerical methods, modeling examples are taken from seven different majors. You can follow any example of your liking to learn numerical methods.

PHYSICAL PROBLEMS OF NONLINEAR EQUATIONS

Chemical Engineering: Having been asked to develop a scale to measure the volume of oil in a spherical tank requires the solution of a nonlinear equation.

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Civil Engineering: How do you find the maximum sagging in a bookshelf? Using strength of materials and statics fundamentals, see how the problem results in a nonlinear equation.

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Computer Engineering: Supercomputers skip on a divide unit to be more efficient. To find the inverse of a real number, they solve a nonlinear equation without using division.

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Electrical Engineering: Thermistor are temperature measuring devices based on that resistance of materials changes with temperature. To find whether the resistor is calibrated properly, one needs to solve a problem of a nonlinear equation.

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General Engineering: The float in a commode submerges to a certain depth in water. Using Newton's third law of motion and Archimedes principle, this problem of finding the depth to which the float submerges results in a nonlinear equation.

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Industrial Engineering: A Mom and Pop shop wants to find the number of computers they would have to sell if they want to break even. Modeling this problem requires solution of a nonlinear equation.

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Mechanical Engineering: To shrink fit a cylinder into a hollow hub, the cylinder needs to be cooled down. To find out how much cooling needs to be done, you need to solve a nonlinear equation.

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PHYSICAL PROBLEMS OF SIMULTANEOUS LINEAR EQUATIONS

Chemical Engineering: Liquid- Liquid Extraction requires the solution of simultaneous linear equation.

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Civil Engineering: How much pressure can a compounded cylinder take results in requiring solution of simultaneous linear equations.

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Computer Engineering: To infer the surface shape from images requires solution of simultaneous linear equations.

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Electrical Engineering: Finding the unbalance in a 3-phase system requires solution of simultaneous linear equations.

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General Engineering: Finding the velocity of a rocket requires solution of simultaneous linear equations.

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Industrial Engineering: Maximizing the production schedule requires solution of simultaneous linear equations.

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Mechanical Engineering: Finding constants of a regression model for coefficient of thermal expansion vs temperature requires solution of simultaneous linear equations.

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PHYSICAL PROBLEMS OF INTERPOLATION

Chemical Engineering: A physical problem of finding how much time it would take water to boil in a kettle. It requires interpolating specific heat of saturated water vs. temperature data.

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Civil Engineering: Bass fishing never got so technical. Find how interpolation can help you to have a great catch and let you tell others a true fish story.

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Computer Engineering: A robot arm path needs to be developed over several points on a flat plate. The interpolated path needs to be smooth to avoid sudden jerky motion and at the same time needs to be short.

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Electrical Engineering: Thermistors measure temperature based on the principle that resistance of thermistor material changes with temperature. Hence, a manufacturer supplies a resistance vs. temperature calibration curve and it is developed using interpolation.

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General Engineering: The upward velocity of a rocket is given as a function of time. To find the velocity of the rocket at some other time, the problem requires interpolation. Choose the preferred computational system -

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Industrial Engineering: To program a milling machine to make a cam profile, one needs to use interpolation to develop the path of the profile.

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Mechanical Engineering: Find the thermal expansion coefficient of steel at a specific temperature to find out whether a steel shaft will cool down enough to shrink fit into a hollow hub. The thermal expansion coefficient is to be found by using interpolation from a given table of thermal expansion coefficient of steel as a function of temperature.

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PHYSICAL PROBLEMS OF REGRESSION

Chemical Engineering: Interpreting infraraed spectroscopy requires regression.

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Civil Engineering: Finding the longitudinal Young's modulus of a composite requires regression.

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Computer Engineering: Extraction of road networks in images requires regression.

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Electrical Engineering: Finding the appropriate value of resistance requires regression.

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General Engineering: Finding the best mousetrap for a mousetrap car requires regression.

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Industrial Engineering: When should machine be stopped requires regression.

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Mechanical Engineering: Calculating the contraction of a trunnion requires regression.

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PHYSICAL PROBLEMS OF INTEGRATION

Chemical Engineering: What is time required for oxygen to be consumed in the fuel cell? One of the problems affecting the performance and hence commercialization of the direct methanol fuel cell (DMFC) is the depolarization of the oxygen reduction electrode caused by methanol cross-over. In an attempt to understand the mechanism of the depolarization process, an electro-kinetic model for mixed oxygen-methanol current on platinum requires integration of a function.

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Civil Engineering: In a class X528 wells, you need to use integration to find if the concentration of benzene is below the toxicity level at a certain distance from the point of contamination.

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Computer Engineering: To infer the surfaces from vector fields requires integration.

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Electrical Engineering: All electrical components, especially off-the-shelf components do not match their nominal value. Variations in materials and manufacturing as well as operating conditions can affect their value. Suppose a circuit is designed such that it requires a specific component value, how confident can we be that the variation in the component value will result in acceptable circuit behavior? To solve this problem a probability density function will need to be integrated to determine the confidence interval.

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General Engineering: The upward velocity of a rocket is given as a function of time. To find the distance covered by the rocket from lift off requires integration.

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Industrial Engineering: Are there as many tissues as advertised in the toilet paper roll? To find the probability of the number of tissues to be less than advertised, one needs to use integrate a function.

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Mechanical Engineering: To find how much a steel shaft will contract so that it can be shrink fit into a hollow hub, we need to use integration of a thermal expansion vs. temperature curve.

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PHYSICAL PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Chemical Engineering: Controlling sodium chloride waste while making soap requires the solution of an ordinary differential equation.

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Civil Engineering: When is it safe to go in the lake after a pollutant has exceeded its limit requires the solution of an ordinary differential equation.

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Computer Engineering: Sizing a rectifier-based power supply requires the solution of an ordinary differential equation.

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Electrical Engineering: Sizing a rectifier-based power supply requires the solution of an ordinary differential equation.

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General Engineering: How much time will it take a heated ball to cool down requires the solution of an ordinary differential equation.

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Industrial Engineering: Speed Control of DC motors requires the solution of an ordinary differential equation.

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Mechanical Engineering: Finding how much time it would take for a cylinder to cool down for shrink fitting requires the solution of an ordinary differential equation.

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PHYSICAL PROBLEMS OF DIFFERENTIATION

Chemical Engineering: How quickly does the slick spread requires requires differentiation.

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Civil Engineering : Finding stress concentration around a hole in a plate requires requires differentiation.

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Computer Engineering: Detecting edges of images requires differentiation.

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Electrical Engineering: Avoiding arcing in switches requires differentiation.

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General Engineering: Finding acceleration of a rocket from velocity requires differentiation.

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Industrial Engineering: Finding acceleration of a rocket from velocity requires differentiation.

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Mechanical Engineering: How does thermal expansion coefficient vary at different temperatures requires differentiation.

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