Department of Engineering Science
ENGSCI 211 Mathematical Modelling 2 Course Book
Semester One 2022
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1 Basic Skills and Revision 8
1.1 BasicSkillsChecklist…………………………… 8
1.2 OrdinaryDifferentialEquations ……………………… 13 1.2.1 NotationandDefinitions……………………… 13 1.2.2 LinearSuperposition ……………………….. 16 1.2.3 InitialValueProblems ………………………. 17 1.2.4 AnalyticSolutionMethods…………………….. 18
1.3 VectorsandMatrices…………………………… 22 1.3.1 VectorDefinition …………………………. 22 1.3.2 VectorDotProduct………………………… 22 1.3.3 MatrixDefinition…………………………. 23 1.3.4 Addition,SubtractionandMultiplication . . . . . . . . . . . . . . . . . . 24 1.3.5 Determinant,Inverse,TransposeandSymmetry . . . . . . . . . . . . . . 25
1.3.6 VectorCrossProduct……………………….. 26
1.3.7 Diagonal,TriangularandIdentityMatrix. . . . . . . . . . . . . . . . . . 27 1.3.8 MultiplicationwithInverseandIdentityMatrix . . . . . . . . . . . . . . 27 1.3.9 Coordinate Representation and Transformation . . . . . . . . . . . . . . 29
1.4 SolvingaSystemofLinearEquations…………………… 33 1.4.1 MatrixForm …………………………… 33 1.4.2 MatrixInverse…………………………… 34 1.4.3 GaussianElimination……………………….. 34 1.4.4 LUFactorisation …………………………. 36 1.4.5 SolvingMultipleSystemsofLinearEquations . . . . . . . . . . . . . . . 39 1.4.6 EfficientCalculationofMatrixDeterminant . . . . . . . . . . . . . . . . 42
2 Ordinary Differential Equations 44
2.1 Introduction……………………………….. 44
2.2 Second-OrderLinearODEs ……………………….. 45 2.2.1 SolutionMethod …………………………. 45 2.2.2 ComplementaryFunction……………………… 46 2.2.3 ParticularIntegral ………………………… 46 2.2.4 GeneralSolution …………………………. 47 2.2.5 Resonance…………………………….. 52
2.2.6 CaseStudy:CarSuspensionSystem ……………….. 56
2.3 NumericalMethods……………………………. 73 2.3.1 IntroductiontoNumericalMethods………………… 73 2.3.2 FirstOrderODEs…………………………. 73 2.3.3 HigherOrderODEs………………………… 77
2.4 LaplaceTransforms……………………………. 83 2.4.1 IntroductiontoLaplaceTransforms………………… 83 2.4.2 LaplaceTransformofElementaryFunctions . . . . . . . . . . . . . . . . 83 2.4.3 SolvinganODEwithLaplaceTransform……………… 89 2.4.4 PartialFractions(Revision) ……………………. 91 2.4.5 StepFunctionandDiracDeltaFunction ……………… 97 2.4.6 UsefulTheorems ………………………….101
3 Multivariable Calculus 108
3.1 SpaceCurves ……………………………….108 3.1.1 ScalarandVectorFunctions…………………….108 3.1.2 ParametricEquations………………………..111 3.1.3 SpaceCurve…………………………….112 3.1.4 VelocityVector …………………………..115 3.1.5 DistanceandArcLength………………………116
3.1.6 Acceleration…………………………….118
3.1.7 WorkDone …………………………….121
3.2 DifferentialCalculus ……………………………124 3.2.1 PartialDifferentiation ……………………….124 3.2.2 MultivariableChainRule………………………127 3.2.3 StationaryPoints………………………….128 3.2.4 VectorCalculus…………………………..134
3.3 IntegralCalculus ……………………………..140 3.3.1 IntroductiontoIntegralCalculus ………………….140 3.3.2 DoubleIntegral …………………………..140 3.3.3 RegionofIntegration………………………..142 3.3.4 ChangetheOrderofIntegration…………………..144 3.3.5 ChangeofVariable …………………………152
4 Data Analysis and Machine Learning 162
5 Linear Algebra 250
5.1 IntroductiontoLinearAlgebra ………………………250
5.2 PLUFactorisation …………………………….251 5.2.1 PermutationMatrix ………………………..252 5.2.2 RowSwapsinGaussianElimination ………………..254
5.2.3 FindingaPLUFactorisation…………………….255
5.2.4 SolutionwithPLUFactorisation ………………….260
5.2.5 ApplicationsofPLUFactorisation………………….263
5.3 LinearIndependence ……………………………265
5.3.1 Linear Independence of a System of Linear Equations . . . . . . . . . . . 265
5.3.2 LinearIndependenceofVectors …………………..269
5.3.3 Summary of Linear Independence and Square Matrices . . . . . . . . . . 272
5.4 IterativeSolutionMethods…………………………273
5.4.1 JacobiMethod …………………………..273
5.4.2 Gauss-SeidelMethod………………………..274
5.4.3 MatrixFormofIterativeMethods………………….276
5.4.4 Convergence and Divergence of Iterative Methods . . . . . . . . . . . . . 277
5.5 EigenvaluesandEigenvectors ……………………….283
5.5.1 DefinitionofEigenvaluesandEigenvectors . . . . . . . . . . . . . . . . . 283
5.5.2 FindingEigenvaluesandEigenvectors………………..284
5.5.3 DiagonalFactorisationofaSquareMatrix . . . . . . . . . . . . . . . . . 287
5.5.4 DiagonalFactorisationofaSymmetricMatrix . . . . . . . . . . . . . . . 289
5.5.5 SystemofLinearFirst-OrderODEs…………………291
6 Fourier Series 294
6.1 Introduction………………………………..294
6.2 MathematicalConcepts…………………………..295
6.2.1 PiecewiseFunctions…………………………295
6.2.2 PeriodicFunctions …………………………297
6.2.3 EvenandOddFunctions………………………299
6.2.4 ProductsofEvenandOddFunctions………………..302
6.2.5 Half-RangeIntegrationofEvenandOddFunctions . . . . . . . . . . . . 305
6.3 FourierSeries ……………………………….308
6.4 PartialSums………………………………..320
6.5 FourierSeriesofNon-PeriodicFunctions ………………….323
6.6 ConvergenceatDiscontinuities……………………….331
.1 DerivationofFourierCoefficients(Non-Examinable). . . . . . . . . . . . . . . . 335
.2 ComplexFourierSeries(Non-Examinable) …………………339
Basic Skills and Revision
1.1 Basic Skills Checklist
This course has a number of different prerequisite courses. Currently, these are ENGSCI 111, or a B+ grade or higher in MATHS 108, or MATHS 110, or MATHS 150, or MATHS 153, or a B+ grade or higher in MATHS 120 and 130.
If your prerequisite course was not ENGSCI 111, then you may be unfamiliar with a few topics that will appear in ENGSCI 211. This includes:
Solution of ordinary differential equations (ODEs).
Solution of a system of linear equations using LU factorisation.
If unfamiliar with either, relevant material to self-teach these are provided later in this revision chapter.
In order to succeed in this course, you will need a good working understanding of a variety of basic skills covered in your prerequisite course(s). These will be assessed through an online quiz in the first week of semester.
A full list of the examinable topics in the basic skills quiz is given below. The Book of Practice Problems contains a range of practice questions on each of these topics.
1.1. BASIC SKILLS CHECKLIST
Basic algebra
Simplify added or subtracted algebraic fractions. Simplify multiplied or divided algebraic fractions. Expand algebraic expressions.
Factorise algebraic expressions.
Use log rules to simplify log expressions.
Use power rules to simplify expressions, including converting between power/surd form.
Polynomial and Rational Functions
Find the roots of a quadratic, including using the quadratic formula. Division of polynomials.
Determine a partial fraction expansion.
Determine coefficients in a partial fraction expansion.
Other Functions
Determine a valid domain and range of a function.
Calculate the result of putting a value into a function of a function.
Evaluate sin, cos, e and ln for ‘nice’ values, without a calculator.
Graph of basic functions e.g. line, quadratic, cubic, hyperbola, circle, sin, cos, tan.
Use trigonometric identities to re-write a trigonometric expression.
Solve simple trigonometric equations.
Complex Numbers
Addition and subtraction. Multiplication.
Division.
Modulus.
Argument. Conjugate.
Differentiation
Find and classify stationary points of a function. Use the product rule.
Use the quotient rule.
Use the chain rule.
Implicit differentiation.
Integration
Integration of basic functions e.g. powers, trig. expressions, logarithms, exponentials. Definite integration i.e. substituting in integration limits.
Integration by substitution.
Integration by parts.
Integration of simple fractions, requiring the use of ln.
Integration using tables e.g. integrals which use inverse trigonometric functions. Page 10
1.1. BASIC SKILLS CHECKLIST
1.1. BASIC SKILLS CHECKLIST
Ordinary Differential Equations
Classification: order, linearity and homogeneity, constant or variable coefficients. General solution using direct integration: y′ = f(t)
General solution using separation of variables: y′ = f(y)g(t)
General solution using exponential substitution: y′′ + b y′ + c y = 0
Particular solution of an ODE using initial conditions.
Magnitude and normalisation of a vector.
Vector dot product and the cosine rule.
Vector cross product and the sine rule.
Vector representation using the standard unit vectors: i, j, k
Multiplication of a vector or matrix.
Transpose of a vector or matrix.
Determinant of a 2 × 2 matrix.
Determinant of a 3 × 3 matrix using co-factor expansion.
Inverse of a 2 × 2 matrix.
Solve a system of linear equations using the methods of: matrix inverse, Gaussian elimi- nation, LU factorisation.
Probability and Statistics
Evaluate the mean, median and mode of a sample. Page 11
Determine mean and variance of a uniform, exponential or normal probability distribu- tion.
Find cumulative distribution function associated with a probability distribution function.
Determine the probability of a specified event from the associated probability distribution.
Determine the expected value of a random variable whose distribution is given.
Determine the expected value of a function of a random variable whose distribution is given.
1.1. BASIC SKILLS CHECKLIST
1.2. ORDINARY DIFFERENTIAL EQUATIONS
1.2 Ordinary Differential Equations 1.2.1 Notation and Definitions
Consider the simple ODE:
The dependent variable of this ODE, y, will change as a function of the independent variable, t. In solving this ODE, we find the dependent variable as a function of the independent variable, y (t). There are three properties of an ODE that are important and closely related to the physical behaviour of the system they are modelling. These are the order, linearity, and homogeneity.
Order The order of an ODE is equal to the order of the highest derivative term. This property is related to the complexity or number of degrees of freedom of the physical system being modelled by the differential equation. For example, a mathematical model of the oscillation of a large building under earthquake loading can be an ODE of higher order, perhaps 100 to 1000.
Linearity A linear ODE is one that has constant coefficients, or coefficients that depend only on the independent variable, in front of each derivative term. For example:
a0(t)y+a1(t)dy +a2(t)d2y +…+an(t)dny =f(t) dt dt2 dtn
would be linear. In general, linear ODEs are easier to solve mathematically than nonlinear ODEs. Most real-world systems are nonlinear and therefore difficult to solve analytically, though can often be approximated by a linear ODE. For example, the nonlinear ODE for a swinging pendulum is:
d2θ + ω2 sin (θ) = 0 dt2
For small angles only, this can be approximated by the linear ODE: d2θ + ω2θ = 0
which is easier to solve.
1.2. ORDINARY DIFFERENTIAL EQUATIONS
Homogeneity A homogeneous ODE is one in which all terms in the differential equation in- volve the dependent variable. Physically, a system governed by a homogeneous ODE will exhibit a free (or unforced) behaviour i.e. not subject to any external forcing. In a nonhomogeneous ODE, the term(s) not involving the dependent variable correspond to forcing terms.
Example 1.2.1 ……………………………………………………………….. Find the order, linearity and homogeneity of the ODE:
d2y + dy = 0 dt2 dt
This ODE is second-order, linear, and homogeneous. …………………………………………………………………………………
Example 1.2.2 ……………………………………………………………….. Find the order, linearity and homogeneity of the ODE:
yd3y = dy dt3 dt
This ODE is third-order, nonlinear, and homogeneous. …………………………………………………………………………………
Example 1.2.3 ……………………………………………………………….. Find the order, linearity and homogeneity of the ODE:
dy + y2 = 0 dt
This ODE is first-order, nonlinear and homogeneous. …………………………………………………………………………………
Example 1.2.4 ……………………………………………………………….. Find the order, linearity and homogeneity of the ODE:
dy + t2 = 0 dt
This ODE is first-order, linear and nonhomogeneous. …………………………………………………………………………………
1.2. ORDINARY DIFFERENTIAL EQUATIONS
Example 1.2.5 ……………………………………………………………….. Find the order, linearity and homogeneity of the ODE:
y dy + t2 = 0 dt
This ODE is first-order, nonlinear and nonhomogeneous. …………………………………………………………………………………
Example 1.2.6 ……………………………………………………………….. Find the order, linearity and homogeneity of the ODE:
tdy + t3y = 0 dt
This ODE is first-order, linear and homogeneous. …………………………………………………………………………………
Example 1.2.7 ……………………………………………………………….. Find the order, linearity and homogeneity of the ODE:
dP = λP dt
This ODE, commonly used in modelling population dynamics, is first-order, linear and homo- geneous.
…………………………………………………………………………………
Example 1.2.8 ……………………………………………………………….. Find the order, linearity and homogeneity of the ODE:
dT =−α(T−Tair) dt
This ODE, also known as Newton’s law of cooling, is first-order, linear and nonhomogeneous. …………………………………………………………………………………
1.2.2 Linear Superposition
1.2. ORDINARY DIFFERENTIAL EQUATIONS
If a linear homogeneous ODE has two solutions, then a linear combination of these is also a solution. For example, if:
y1 =f(t) y2 = g (t)
are both solutions to a linear ODE, then: y = a y1 + b y2
is also a solution (where a and b are constants). This is a property called linear superposition.
Example 1.2.9 ……………………………………………………………….. Consider the ODE:
d2y + y = 0 dt2
This second-order, linear and homogeneous ODE has two solutions: y1 =sin(t)
y2 =cos(t)
You can verify this yourself by substituting these solutions into the ODE. As the ODE is linear and homogeneous, we know that any linear combination of the two solutions is also a solution. For example:
y = 2sin(t) + 3cos(t)
is also a solution. This can be easily verified through substitution.
…………………………………………………………………………………
1.2. ORDINARY DIFFERENTIAL EQUATIONS
1.2.3 Initial Value Problems
To fully describe a time-dependent process using an ODE, it is necessary to supply information on the initial conditions of the system. The number of initial conditions required is equal to the order of the ODE. Collectively, the ODE and its initial conditions are known as an initial value problem (IVP).
Example 1.2.10 ………………………………………………………………. An object falling under gravity can be modelled using the IVP:
dv =g, v(0)=v0 dt
The ODE is first-order, linear, nonhomogeneous and requires one initial condition to be solved. …………………………………………………………………………………
Example 1.2.11 ………………………………………………………………. Newton’s law of cooling is given by the IVP:
dT=−α(T−Tair) , T(0)=T0 dt
The ODE is first-order, linear, nonhomogeneous and requires one initial condition to be solved. …………………………………………………………………………………
Example 1.2.12 ………………………………………………………………. The amplitude of an oscillating pendulum is given by the IVP:
d2θ+ω2sin(θ)=0 , θ(0)=θ0 , dθ(0)=0 dt2 dt
The ODE is second-order, nonlinear, homogeneous and requires two initial conditions to be
solved. These are the initial amplitude, θ (0), and the initial rate of change of amplitude with
respect to time, dθ (0), which we assume to be zero i.e. pendulum starts from rest. dt
…………………………………………………………………………………
1.2. ORDINARY DIFFERENTIAL EQUATIONS
1.2.4 Analytic Solution Methods
ENGSCI 111 covered several analytic solution methods for different categories of ODE:
Direct integration.
Separation of variables.
Integrating factor (non-examinable in ENGSCI 211). Exponential substitution.
We will consider two distinct types of solution:
General solution – this satisfies the differential equation, without consideration of the initial condition(s). It will therefore include unknown coefficient(s).
Particular solution or total solution or full solution – this satisfies both the differential equation and the initial conditions. We can think of this as a specific instance of the general solution that satisfies the initial conditions.
Direct Integration This method works for ODEs of the general form: dny = f (t)
where n is some positive integer. The right-hand side can be integrated with respect to the
independent variable n times to obtain a general solution.
Separation of Variables This method works for first-order homogeneous ODEs of the gen- eral form:
dy =g(t)h(y) dt
where the right-hand side has been factored into a function of the independent variable multi- plied by a function of the dependent variable. To solve this, we perform separation of variables:
dy =g(t)dt h(y)
The left-hand side can now be integrated with respect to y, whereas the right-hand side can be integrated with respect to t.
1.2. ORDINARY DIFFERENTIAL EQUATIONS
Integrating Factor (Non-Examinable in ENGSCI 211) This method can work for first- order linear nonhomogeneous ODEs of the general form:
dy + g (t) y = f (t) dt
This solution method was covered in ENGSCI 111, but is not used in ENGSCI 211.
Exponential Substitution This method works for linear homogeneous ODEs with constant coefficients of the general form:
a y + a dy + . . . + dny = 0 1 2 dt dtn
ODEs of this form are very common in Engineering problems. They can be solved by assuming an exponential trial solution of the form:
which, through substitution into the ODE, can be used to find a polynomial equation for λ, also known as the characteristic equation. Once this polynomial equation has been solved for λ, a general solution to the O
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