Engineering analysis

Control of Medical System Engineering

Third year

2016/2017

Dr Safanah Mudheher

Course Description:

• This course provides students with the mathematical foundation for the System Engineering Program.

• It will address the fundamentals of mathematical tools that are part of the System Engineering Program.

• The topics covered are Higher Order Ordinary Differential Equations, Matrix Algebra, Curve-Fits, Interpolation, Eigenvalue and Eigenvector Problems, Partial Differential Equations.

Class Participation

• There will be discussion boards for students to discuss among themselves different aspects of the course, and I will participate in the discussions when it is appropriate.

• I encourage people to work together on homework assignments. Use the discussion board to post your questions and to read the responses from your classmates.

• Exams are on an individual basis. Any questions on exams should be directed to me. I will add information to the exams if there many students are experiencing particular problems.

Policy on late/missing assignments

• It is important that students maintain the proper pace in this course. For that reason, homework and other assignments are expected to be submitted on time. Homework will be accepted up to one week late with a 50% penalty.

• Work submitted after one week will not be accepted. Exception may be made for extraordinary circumstances, but these are on an individual basis and must be approved by me in advance of the due date for the homework. Examples of circumstances include illness and family emergencies.

Assessment / Grading

• Homework

• Exams

Part-1 Ordinary Differential Eqautions Review First-Order ODEs

• Understanding the basics of ODEs requires solving problems by hand.

• The process of setting up a model, solving it mathematically, and interpreting the result in physical or other terms is called mathematical modeling or, briefly, modeling.

• a model is very often an equation containing derivatives of an unknown function. Such a model is called a differential equation.

Some applications of differential equations

Geometric Meaning of y ‘= f (x, y) Direction Fields, Euler’s Method

Linear ODEs. Bernoulli Equation. Population Dynamics

• A first-order ODE is said to be linear if it can be brought into the form

• by algebra, and nonlinear if it cannot be brought into this form.

• p and r may be any given functions of x.

• If in an application the independent variable is time, we write t instead of x.

We multiply (1) by F(x), obtaining

Homogeneous Linear ODEs of Second Order

Homogeneous Linear ODEs with Constant Coefficients

the solution of the first-order linear ODE with a constant coefficient k

Thank you

Homogeneous Linear Higher Order Constant Coefficient Equations

Engineering Analysis

2016/2017

Lecture 2

Undetermined Coefficients: Particular Integrals

• Like the nonhomogeneous second order constant coefficient differential equation, a particular integral yp(x) of the nonhomogeneous linear higher order constant coefficient differential equation

• (53)

is a solution of the equation that does not contain arbitrary constants, so

• The complementary function yc(x) associated with (53) is the general solution of the homogeneous form of the equation

• It follows from the definitions of yc(x) and yp(x) and the linearity of the equation that the general solution y(x) of (53) can be written

Example: Find the general solution of

Example:

Example:

Theorem:

Example:

Engineering analysis

Control of Medical System Engineering

Third year

2015/2016

Dr Safanah Mudheher

Exact, Least-Squares, and Cubic Spline Curve-Fits

• Engineers conduct experiments and collect data in the laboratories. To make use of the collected data, these data often need to be fitted with some particularly selected curves.

• For example, one may want to find a parabolic equation y = c 1+ c2x + c3x^2.

• which passes three given points (x i,y i ) for i = 1,2,3.

• This is a problem of exact curvefit.

• In case that we may want express this straight line by the equation y = c 1+ c2x for the stress and strain data collected for a stretching test of a metal bar in the elastic range, then the question of how to determine the two coefficients c1 and c2 is a matter of deciding on which criterion to adopt.

• The Least-Squares method is one of the criteria which is most popularly used. The two cases cited are the consideration of adopting the two and three lowest polynomial terms, x(0), x(1), and x(2),and linearly combining them.

EXACT CURVE FIT

let us consider the problem of finding a parabolic equation

y = c 1+ c2x + c3x^2 which passes three given points

GENERALIZED LEAST-SQUARES CURVEFIT

Fitting a Straight Line

example

Exercises

• Given five points (1,1), (2,3), (3,2), (4,5), and (5,4), calculate the coefficients c1 and c2 in the linear equation y = c1 + c2x which fits the five points by the least-squares method.

• 2. For a given set of data (1,–2), (2,0), (3,1), and (4,3), two equations have been suggested to fit these points. They are Y = X–2 and Y = (-X^2 + 7X–10)/2. Based on the least-squares criterion, which equation should be chosen to provide a better fit? Explain why?

• 3. During a tensile-strength test of a metallic material the following data (Xi,Yi) for i = 1,2,…,7 where X and Y represent strain (extension per unit length) and stress (force per unit area), respectively, have been collected:

Example

exercises

•Thank you

Series Solutions of

Differential Equations

A General Approach to Power Series Solutions of Homogeneous Equations

Example

Legendre’s equation • An important application of the power series method of solution is to

the Legendre differential equation

Similar to previous example

• As the solutions y1(x) and y2(x) are not proportional, they must be linearly independent solutions of the Legendre equation (10). We leave as an exercise the task of showing that each series is convergent in the interval −1 < x < 1, so the general solution (12) has this same interval of convergence.

• Examination of the recurrence relation (11) shows that if α = n is a nonnegative integer, the terms an+2 = an+4 = an+6 = · · · all vanish. Thus, if α = n is even, the series y1(x) will reduce to a polynomial of degree n in even powers of x, whereas if α = n is odd the series y2(x) will reduce to a polynomial of degree n in odd powers of x.

Legendre polynomials

• Results (15a, b) provide a general definition for a Legendre polynomial of any order, though when only a few low order polynomials are required it is often more convenient to generate them by means of the following recurrence relation that

Singular Points of Linear Differential Equations

• Previously, the power series method was used to find a solution of a homogeneous variable coefficient differential equation of the form

• Expressed differently, when (19) is written in the standard form

• the power series method can be applied to develop a solution about any point x0 at which the functions P(x) and Q(x) are analytic.

• Points regular and singular where P(x) and Q(x) are analytic are called regular points of the differential equation, and points where at least one is not analytic are called singular points.

• Equation (20) will be said to have a regular singular point at x0 if the functions

• are analytic at x0, and so have Taylor series expansions about x0. If at least one of these functions is not analytic at x0, the point will be said to be an irregular singular point.

Eigenvalues, Eigenvectors

Control and System Engineering Dept./Branch of

Control of Medical Engineering Systems

Third year

2016/2017

We can write (12) as

For n=2

Example

Systems of ODEs as Models in Engineering Applications

• Mixing Problem Involving Two Tanks

• A mixing problem involving a single tank is modeled by a single ODE.

• The model will be a system of two first-order ODEs.

• Tank and in Fig. contain initially 100 gal of water each. In the water is pure, whereas 150 lb of fertilizer are dissolved in . By circulating liquid at a rate of and stirring (to keep the mixture uniform) the amounts of fertilizer in and in change with time t.

• How long should we let the liquid circulate so that T1 will contain at least half as much fertilizer as there will be left in T2?

Electrical Network

• Find the currents I1 and I2 in the network. Assume all currents and charges to be zero at t=0, the instant when the switch is closed.

• Solution. Step 1. Setting up the mathematical model. The model of this network is obtained from

Kirchhoff’s Voltage Law,

• Let I1(t) and I2(t) be the currents

COMPLEX ANALYSIS Analytic Functions

Control of Medical System Engineering/

Third year

Control & System Engineering Dept.

2016-2017

Complex Functions and Mappings

• A typical example of a complex function is the nth degree polynomial

where the coefficients a0, a1, . . . , an are complex numbers and z = x + iy is an arbitrary complex variable.

• an arbitrary complex function w = f (z) can be introduced by considering two complex planes, one the z-plane containing the points z = x + iy and the other the w-plane containing the points w = u + iv, as shown in Fig. 13.1

To develop this idea further, let a set of points D in the z-plane be such

that to each point z in D there corresponds a unique complex number w belonging

to another set of points in the w-plane.

Then the set D is said to be mapped onto the set by a single-valued function of

the complex variable z.

A point w0 in the w-plane corresponding to a point z0 in the z-plane is called the image of z0.

• The term single-valued is used because, by hypothesis, each point of D corresponds to one and only one point of Ω , and the name mapping is used because an arbitrary curve in D will correspond (be mapped) to a corresponding curve in Ω , with each point of the curve in the image of a point in D.

• The relationship between the points in D and the corresponding points in Ω is shown by the usual functional notation

• Set Dis called the domain of definition of the complex function f (z), and set is called its range.

• A set G will be said to be connected if every pair of points in G can be joined by an unbroken path with the property that every point of the path also belongs to G. Here, the path may be either a curve or a set of straight line segments joined end to end.

• A neighborhood of a point z0 in G is defined as all the points of a set contained strictly inside a circle of arbitrarily small radius with its center at z0. A point z0 is called an interior point of Gif a neighborhood of z0 only contains points of G.

• If a neighborhood of z0 contains no points of G, the point z0 is called an exterior point of G. When any neighborhood of z0 contains both interior and exterior points of G, the point z0 is called a boundary point of G.

• A set G that contains no boundary points is called an open set.

• If every boundary and connectivity point of set G belongs to G, then G said to be closed.

• The name domain is given to an open connected set, while the more general term region is used to describe a connected set of points that may contain none, some, or all of its boundary points.

• A typical open connected set G is the disc |z| < 1 in the z-plane.

• The set is connected because every point in G can be joined to every other point in G by a curve lying entirely inside G,

• a neighborhood of z0 can always be found that only contains points of G.

• This becomes a closed set if the relation |z| < 1 is replaced by |z| ≤ 1, because then the boundary of G formed by the circle |z| = 1 belongs to the set.

• These ideas are illustrated in Fig. 13.2.

• The complex function (2) can be written in its cartesian form as

• where u(x, y) and v(x, y) are real functions of the real variables x and y denoted by

Example

Example

Example

Example

• Setting z= x + iy in w = 2z+ 1 gives w = u + iv = 2x + 1 + 2iy, so u = 2x + 1 and v = 2y. The top boundary of the area in the z-plane in Fig. 13.5 is −1 < x < 1, y = 2, so using these results in the mapping shows the image of this boundary in the w-plane to be given by u = 2x + 1, with −1 < x < 1, and v = 4.

• A repetition of this form of argument applied to the other three sides of the rectangle establishes that the image in the w-plane of the rectangle in the z-plane is the one illustrated on the right of Fig. 13.5.

• A general point (x, y) inside the rectangle in the z- plane maps to the point (2x + 1, 2y) in the w-plane with−1 < x < 1,−2 < y < 2, and this point is seen to lie inside the rectangular boundary in the w-plane.

• Consequently, all points inside the rectangle in the z-plane map to points inside the image rectangle in the w-plane.

• Inspection of Fig. 13.5 shows that the geometrical effect of this mapping is first to scale the rectangle in the z-plane uniformly by a factor 2 in both the x and y directions, and then to shift the origin parallel to the real axis.

exercises

Scalar and Vector Fields, Limits,

Continuity, and Differentiability Control of Medical System Engineering/

Third year

Control & System Engineering Dept.

2016-2017

• A scalar function F(x, y, z) defined over some region of space D is a function that assigns to each point P0 in D with coordinates (x0, y0, z0) the number F(P0) = F(x0, y0, z0).

• The set of all numbers F(P) for all points P in D are said to form a scalar field over D. If P has position vector r, we can write the scalar field F(x, y, z) in the form F(P) = F(r) to emphasize the fact that a scalar value F(r) is associated with the position vector r in D.

• In physical problems P is usually a point in space, and in addition to depending on P, the function F often also depends on the time t, so then F(P, t) = F(x, y, z, t) and in this case we can write F(P, t) = F(r, t).

• A typical example of a time dependent scalar field is provided by the temperature distribution throughout a block of metal heated in such a way that the temperatures on its sides vary with time.

• vector field defined by a vector fields function F(x, y, z) over some region of space D that assigns to each point P0 in D with coordinates (x0, y0, z0) the vector F(P0) = F(x0, y0, z0) with its tail at P0.

• Functions of this type are called either vector functions or vector-valued functions, and if P has position vector r we can write F(P) = F(r) to emphasize the fact that in this case a vector F(P) is associated with each position vector r in D.

• Like scalar fields, vector fields over D often depend on both position and the time t, so then F = F(x, y, z, t), and in this case we can write F(P, t) = F(r, t).

• An example of a time dependent vector field is provided by the fluid velocity vector in the unsteady flow of water around a bridge support column, because there the velocity depends on both the position vector r in the water and the time t at which the velocity is observed.

• In general, in terms of the unit vectors i, j, and k, a time-dependent vector-valued function can be defined by setting

example

In order to perform calculus on vectors it is necessary to introduce the idea of

a vector as a function. The simplest example of this kind is a vector F(t) of a single real variable t, which in terms of cartesian coordinates can be written

example

example

Example

Example

example

Example

Example

Exercises