MATH 250 Linear Algebra and Differential Equations for Engineers

Tuesdays: 16:30 – 19:20 C204

Fridays: 13:30 – 16:20 C203

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Course OutcomesUpon completing this course students should be ableto:1. Fundamentals of Matrix algebra2. Produce solutions to various algebraic equations.2. The students will demonstrate their ability to use

tools from differential equations in providing exact and qualitative solutions for problems arising in physics and other scientific and engineering applications.

3. The students will be able to choose the appropriate techniques from calculus and geometry to generate

exact and qualitative solutions of differential equations.

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4. The students will be able to solve problems in ordinary differential equations, dynamical systems and a number of applications to scientific and engineering problems.

5. The students will recall techniques to solve second degree non-homogeneous and

homogeneous linear differential equations.6. The students will recall Laplace Transform

techniques to solve differential equations.7. The students will recall numerical techniques to

solve differential equations.8. The students will recall algorithms to develop

mathematical models for engineering problems.3

Course Description

• Lectures will consist of theories, problem solving and techniques presented with programs being written and run (in groups and individually) in order to demonstrate the introduced material.

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Syllabus• Lect1: Orientation and Introduction• Lect1: Matrices and Determinants Systems of Linear

Equations. Matrices and Matrix Operations. Inverses of Matrices.

• Lect2: Special Matrices and Additional Properties of Matrices. Determinants. Further Properties of Determinants. Proofs of Theorems on Determinants.

• Lect3: Vector Spaces. Subspaces and Spanning Sets. Linear Independence and Bases. Dimension; Nullspace, Rowspace, and Column Space.

• Lect4: Linear Transformations, Eigenvalues, and Eigenvectors Linear Transformations. The Algebra of Linear Transformations; Differential Operators and Differential Equations. Matrices for Linear Transformations.5

• Lect 5: Eigenvectors and Eigenvalues of Matrices. Similar Matrices, Diagonalization, and Jordan Canonical Form.Eigenvectors and Eigenvalues of Linear Transformations

• Lect 6: First Order Ordinary Differential Equations Introduction to Differential Equations. Separable Differential Equations. Exact Differential Equations. Linear Differential Equations.

• Lect7: More Techniques for Solving First Order Differential Equations. Modeling With Differential Equations. Reduction of Order. The Theory of First Order Differential Equations. Numerical Solutions of Ordinary Differential Equations.

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• Lect 8: Linear Differential Equations The Theory of Higher Order Linear Differential Equations. Homogenous Constant Coefficient Linear Differential Equations. The Method of Undetermined Coefficients. The Method of Variation of Parameters. Some Applications of Higher Order Differential Equations

• Lect 9: Systems of Differential Equations The Theory of Systems of Linear Differential Equations. Homogenous Systems with Constant Coefficients: The Diagonalizable Case.

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• Lect 10: Homogenous Systems with Constant Coefficients: The Nondiagonalizable Case. Nonhomogenous Linear Systems. Nonhomogenous Linear Systems

• Lect 11: Converting Differential Equations to First Order Systems. Applications Involving Systems of Linear Differential Equations. 2x2 Systems of Nonlinear Differential Equations.

• Lect 12: Laplace Transform Definition and Properties of the Laplace Transform. Solving Constant Coefficient Linear Initial Value Problems with Laplace Transforms

• Lect 13: Power Series Solutions to Linear Differential Equations Introduction to Power Series Solutions. Series Solutions for Second Order Linear Differential equations.

• Lect 14: Euler Type Equations. Series Solutions Near a Regular Singular Point..

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References:• Differential Equations and Linear Algebra, 3/E Authors: C. Henry Edwards & David E. Penney

Publisher: Pearson • Linear Algebra, an applied first course, Kolman

& Hill 8th edition, Pearson

• MATLAB: An Engineer’s Guide to MATLAB Authors: Magrab, Azarm, Balachandran,

Duncan, Herold, Walsh Publisher: Pearson• Prerequisites:

• MATH 153 9

Grading Policy:

• Homework: 5% • Midterm Examinations: 40%• Quizes: 20%• Final Examination: 35%

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MATHEMATICAL MODELING

Principles

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Why Modeling?

• Fundamental and quantitative way to understand and analyze complex systems and phenomena

• Complement to Theory and Experiments, and often Intergate them

• Becoming widespread in: Computational Physics, Chemistry, Mechanics, Materials, …, Biology

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Modeling

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Mathematical Modeling? Mathematical modeling seeks to gain an

understanding of science through the use of mathematical models on computers.

Mathematical modeling involves teamwork

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Mathematical Modeling

Complements, but does not replace, theory and experimentation in scientific research.

Experiment

Computation

Theory

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Mathematical Modeling• Is often used in place of experiments when

experiments are too large, too expensive, too dangerous, or too time consuming.

• Can be useful in “what if” studies.• Is a modern tool for scientific investigation.

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Mathematical Modeling Process

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Example: Industry

• First jetliner to be digitally designed, "pre-assembled" on computer, eliminating need for costly, full-scale mockup.

• Computational modeling improved the quality of work and reduced changes, errors, and rework.

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Example: Climate Modeling

• 3-D shaded relief representation of a portion of PA using color to show max daily temperatures.

• Displaying multiple data sets at once helps users quickly explore and analyze their data.

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Real World Problem

Identify Real-World Problem:– Perform background research,

focus on a workable problem. – Conduct investigations (Labs),

if appropriate.– Learn the use of a computational tool:

Matlab, Mathematica, Excel, Java. Understand current activity and predict

future behavior.

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Example: Falling Rock

Determine the motion of a rock dropped from height, H, above the ground with initial velocity, V.

A discrete model: Find the position and velocity of the rock above the ground at the equally spaced times, t0, t1, t2, …; e.g. t0 = 0 sec., t1 = 1 sec., t2 = 2 sec., etc.

|______|______|____________|______ t0 t1 t2 … tn

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Working Model

Simplify Working Model: Identify and select factors to describe important aspects of Real World Problem; deter- mine those factors that can be neglected. – State simplifying assumptions. – Determine governing principles, physical laws.– Identify model variables and inter-relationships.

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Example: Falling Rock

• Governing principles: d = v*t and v = a*t.

• Simplifying assumptions: – Gravity is the only force acting on the body.– Flat earth.– No drag (air resistance).– Model variables are H,V, g; t, x, and v– Rock’s position and velocity above the

ground will be modeled at discrete times (t0, t1, t2, …) until rock hits the ground.

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Mathematical Model Represent Mathematical

Model: Express the Working Model in mathematical terms;

write down mathematical equa- tions whose solution describes the Working Model.

In general, the success of a mathematical model depends on how easy it is to use

and how accurately it predicts.

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Example: Falling Rock v0 v1 v2 … vn x0 x1 x2 … xn |______|______|____________|_____ t0 t1 t2 … tn

t0 = 0; x0 = H; v0 = V

t1= t0 + Δt

x1= x0 + (v0*Δt) v1= v0 - (g*Δt)

t2= t1 + Δtx2= x1 + (v1*Δt) v2= v1 - (g*Δt) …

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Computational Model

Translate Computational Model: Change Mathema- tical Model into a form suit- able for computational solution.

• Existence of unique solution

• Choice of the numerical method

• Choice of the algorithm

• Software

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Computational Model

Translate Computational Model: Change Mathema- tical Model into a form suit- able for computational solution.

Computational models include software such as Matlab, Excel, or Mathematica, or languages such as Fortran, C, C++, or Java.

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Example: Falling RockPseudo CodeInput

V, initial velocity; H, initial heightg, acceleration due to gravityΔt, time step; imax, maximum number of steps

Outputti, t-value at time step ixi, height at time tivi, velocity at time ti

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Example: Falling Rock

InitializeSet ti = t0 = 0; vi = v0 = V; xi = x0 = Hprint ti, xi, vi

Time stepping: i = 1, imaxSet ti = ti + ΔtSet xi = xi + vi*ΔtSet vi = vi - g*Δtprint ti, xi, viif (xi <= 0), Set xi = 0; quit

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Results/Conclusions

Simulate Results/Con- clusions: Run “Computational Model” to obtain Results; draw Conclusions.– Verify your computer program; use check

cases; explore ranges of validity. – Graphs, charts, and other visualization

tools are useful in summarizing results and drawing conclusions.

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Falling Rock: Model

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Real World Problem Interpret Conclusions:

Compare with Real World Problem behavior.

– If model results do not “agree” with physical reality or experimental data, reexamine the Working Model (relax assumptions) and repeat modeling steps.

– Often, the modeling process proceeds through several iterations until model is“acceptable”.

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Example: Falling Rock• To create a more realistic model of a

falling rock, some of the simplifying assumptions could be dropped; e.g., incor-porate drag - depends on shape of the rock, is proportional to velocity.

• Improve discrete model: – Approximate velocities in the midpoint of time

intervals instead of the beginning.– Reduce the size of Δt.