Lab Notes for Math 201: Differential Equations for Engineers (small format)

Mathematics

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For the pendulum case, the value of and fixes the system. If the state is truly fixed, then the system should theoretically be insensitive to the starting initial guess. For certain systems, the computation of initial conditions is sensitive to the starting guess.

To demonstrate, consider the following system. By using a default value of 1 as the starting guess, the algorithm is forced to change the specified initial condition. The code essentially switches to a different solution branch. This happens because the default starting guess for the state is 1.

During the Newton iterations, the system becomes singular because the second equation is identically satisfied, and as a result the iterations diverge. To avoid complete failure, the algorithm is forced to modify the specified condition of. To avoid this behavior, a "better" starting guess can be given. Using a starting guess of -1 , the iterations converge to the expected consistent initial condition. The QR decomposition method is based on the work of Shampine [S02]. This algorithm is intended for index-1 or index-0 systems. The algorithm makes use of the efficient QR decomposition technique to decouple the system variables and their derivatives.

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In doing so, at each iteration step the solution of one system is broken down into two smaller systems of equations. For an -variable DAE system, the QR algorithm needs to handle at most a matrix of size as opposed to a matrix of size if the collocation method is used. In most cases, the algorithm only needs to handle matrices much smaller than.

This makes the method quite efficient in handling very large systems. A brief description of the underlying algorithm is provided in this section. Given an index-reduced system of the form , the system can first be linearized about an initial guess and as. A QR decomposition is performed on the Jacobian matrix such that , where is an orthogonal matrix, is an upper triangular matrix and is a permutation matrix.

For DAEs, the matrix will not have a full row-rank. Depending on the rank of the matrix , the linearized system is then written as. Once the components are found, the transformed variables are converted back into the original form, using the permutation matrix, and the process is repeated until convergence. Given that this is a high-index DAE, an index reduction must be performed on the system. In the preceding example, the initial conditions are given such that they lead to a consistent set of starting values.

In the event that the initial conditions are not consistent, the algorithm attempts to find consistent starting values. By specifying the option "DefaultStartingValue" , the initial guess used for the iteration is changed. For inconsistent initial conditions , this may lead to a different consistent set. The BLT method is based on transforming an index-reduced system into a block lower triangular form and then solving subsets of the system iteratively.

The subsets themselves are solved using a Gauss — Newton method. This is sometimes advantageous because the index reduction algorithm that generates the index-reduced system tends to already have a block lower triangular form and it reduces the computational complexity by working with smaller subsystems with correspondingly smaller matrix computations.

The algorithm also checks for inconsistent initial conditions and attempts to correct them. To see the advantage of using the BLT method, it is important to recognize that the index reduction algorithm leads to an expanded system of equations that are index 1 or index 0. Consider the following chemical systems example that describes an exothermic reaction.

The original DAE consists of four equations, while the index-reduced system consists of eight equations. The index-reduced equations are as follows. Note that the expanded system now contains dummy derivatives. The expanded system can be analyzed further to see the structure of the system. This indicates that the expanded system can actually be solved sequentially. Therefore, rather than having one system of nine equations, a single equation can be solved sequentially.

This is the main principle behind the initialization using the BLT method. The BLT method is found to be especially useful and computationally efficient when dealing with large systems having a lower triangular form. For cases with inconsistent initial conditions, the BLT method can suitably modify the initial conditions. It is, however, important to note that the results that obtained from the BLT method and the QR method may be different. Consider the pendulum example with inconsistent initial conditions.

One of the main challenges in DAE system formulations and their solution is the fact that the starting initial conditions must be consistent. If NDSolve encounters initial conditions that it finds to be inconsistent, it will attempt to correct them.

To illustrate the points about hidden constraints and inconsistent initial conditions, consider the pendulum system 2. As a first case, consider when the initial conditions violate the algebraic constraint for the system. This is done by intentionally setting the initial condition for , so that it always violates the constraint.

It then attempts to find a consistent initial condition using the inconsistent value as a starting guess for its iterations. Sometimes, it may not be clear if the specified conditions might be consistent or not. It may seem that the initial conditions satisfy the constraints, but may not satisfy the hidden constraint. These initial conditions satisfy the equation but fail to satisfy the implicit constraint. This constraint is not obvious because it is hidden and is only found by index reduction.

The collocation algorithm does not perform index reduction and operates on the original equations. Even in the absence of an index-reduced system, the collocation algorithm is able to detect an inconsistency in the initial conditions and attempts to rectify it. Consider a system of two equations with a discrete variable where a state change is made at. In the preceding example, initial conditions for and are given.

The initialization code automatically computes the consistent initial condition. At time , an event is triggered that says that the discrete variable should be changed from 1 to 2. At time , a reinitialization of the variables needs to take place. The variable is called a modified variable. The following strategy is adopted during reinitialization:. Keep all the state, discrete, and modified variables fixed. If a state variable has a derivative fixed or specified to be changed , then that variable can be changed.

If strategy 1 fails to reinitialize, then keep the state variables and modified variables fixed and allow other variables and their derivatives to change. If strategy 2 fails, then the modified variables and the state variables with derivatives in the DAE are kept fixed. All other variables and their derivatives can be changed. If strategy 3 fails, then only the modified variables are kept fixed and all the state variables and their derivatives are allowed to change. Just before the reinitialization, the values of , , and are as follows. Based on the outlined strategy, at time , using strategies 1 and 2, it is required that , , and do not change.

Given that , this clearly violates the algebraic equation. Strategies 1 and 2 therefore fail. Strategy 3 requires that variables with derivatives should not change. Therefore is kept fixed along with and is allowed to change. This modifies the value of as shown in the plot the blue line. WhenEvent allows for simultaneous and sequential evaluation of event actions, including state changes.

After each state change, even for the same event, the DAE is reinitialized using the preceding strategy, so if possible, it is more efficient to just give one rule for the state changes needed. The final state when there are multiple state changes done at the same event time may be different because the new values are used in each subsequent evaluation and setting and for DAEs these values may depend on the reinitialization.

Consider the previous problem with additional events. To see why the solution is different from the one preceding, it is necessary to follow the sequence of settings. First is set to 2 and the DAE is reinitialized before evaluating the other settings. This reinitialization changes the value of to 2 since that is how the strategy finds a solution consistent with the residual. Execution of the second action sets to 2 the current value of , but this is inconsistent with the residual, so since is an algebraic variable its value is set to 4.

Finally, setting to resets to 2, and to avoid modifying the newly set algebraic variable, the value of is reset to 1. One way to see the sequence of changes is to put in Print commands between the settings. So a different sequence of settings leads to a different solution. With a single rule there is only one initialization, so if all of the variables are set, the values given should be consistent, or reinitialization strategy may fail or not give the expected values.

Here , , and are fixed. This is a very restrictive constraint, and all the strategies fail. Setting a variable to its current value within a single rule in effect forces it to stay the same. Since stays the same, is forced to change. This is a test problem taken from Campbell and Moore [CM95]. The DAE describes the motion of a particle in 3D. This is an index-3 DAE with three position variables, three velocity variables, and one constraint.

The solution manifold is four dimensional, and the exact solution is given by. This is a simplified model of a dynamic process in petrochemical engineering. The details of this model can be found in Hairer et.

This is an index-2 system. This example is a simple constrained multibody system. The system consists of a crank of mass m1 , which is hinged on a fixed surface, a slider of mass m3 , and a connecting rod of mass m2. The slider is constrained to move only in the horizontal plane.

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Due to the masses of the individual components, the moments of inertia would have to be considered during the construction of the system. A schematic of the problem is shown following. The system can be completely defined using the two angles. The state of the slider-crank mechanism can be described completely by two angles and the distance of the slider from the origin. The equations of motion are derived by resolving the forces and applying Newton's law and. The crank and connecting rod have mass and therefore inertia. This is an extension of the simple slider crank mechanism.

A spring and damper system is attached between the crank and the connecting rod. This spring constrains the range of motion of the crank, while the damper slowly brings the system to a steady state. The schematic of the system is given following. The two-link planar robotic arm is a frictionless, constrained mechanical system.

The system consists of two links attached to each other through a joint, such that one end of the first link is held fixed, while the trajectory at the end of the second link is specified. The following figure shows a schematic of the system. The system can be completely defined using the angles between the links. Additional details of this system can be found in Ascher et al.

The equations of motion are derived by resolving the forces and applying Newton's law. The differential equations are derived for the angles. The Cartesian coordinates and are therefore represented in terms of the angle. The following system describes a chemical process in which two species, FLB and ZHU, are mixed while is constantly added to the system. The chemical names are fictitious. Details of the system and the dynamics associated with it can be found in [N93]. The reaction equations follow.

Associated with each reaction, there are rates of reaction, which are specified. The algebraic constraint comes from the last reaction, which requires FLB and ZHU and its mixture to maintain a constant relationship. The car-axis model is a simple multibody system that is designed to simulate the behavior of a car axis on a bumpy road. A schematic of the system is shown following. The model consists of two wheels: The bumpy surface is modeled as a sinusoid. The left and right wheels transfer their motion to the chassis of the car through two massless springs as shown following.

The chassis of the car is modeled as a bar of mass M. Additional details of this model can be found in Mazzia et. This is an index-3 system. The left tire at is always on a flat surface, while the right tire periodically goes over a bump. The distance between the wheels must always remain fixed and the car chassis must always have a fixed length. The following is a synthetic example for solving a parabolic and an elliptic PDE that are coupled together in a nonlinear fashion. Spatial discretization of the PDEs results in a system that contains a significant number of algebraic variables.

The equations are set up such that the analytic solution exists for the system. The details of the coupled system are given following. This example is a model of the multispecies food web [B86]. Unlike the typical predator-prey problem, there is a spatial component of this problem.

The predator-prey relationships are defined over a 2D spatial domain via a series of diffusion parameters. The general coupled PDE is given by. In this system, the first species are the prey and the next species are the predators. The specific case of is considered here. This means that there is one predator and one prey species. A uniform distribution of the predators is specified initially. This is leads to a inconsistent initial condition that is corrected by the built-in algorithms. The Andrews squeezer mechanism is a multibody system containing seven different links. The dimensions of the various links and their specifications are given following.

The system can be completely defined using seven angles. The angles are labeled q 1 — q 7 in the following system. Additionally, there are six algebraic constraints associated with the link dimensions and connections. This leads to a dimensional, index-3 system. Details of the system can be found in Hairer and Wanner [HW96].

The dimensions associated with the various links are shown below.

The seven links are labeled K 1 — K 7. The notation for the parameters used in the model and their description can be found in Hairer and Wanner [HW96]. The motion of the system can be defined using seven angles. The angles are defined as here. The equations of motion can be derived using Hamiltonian principles resulting in a system of equations of the form where is a mass matrix, is a vector of forces, is a vector of constraints, and are Lagrange multipliers.

Please try again later. If you continue to experience a problem or if you have any questions, please contact us. Enable JavaScript to interact with content and submit forms on Wolfram websites. A system of DAEs with three equations, but only one differential term. Solve the system of DAEs starting with a specified initial condition for and with. Solve the system of DAEs starting with. Automatic try to solve the system symbolically in the form ; if a solution cannot be found or takes too long , try simplifying using the "MassMatrix" or "Residual" methods; this is the default "Solve" solve the system symbolically in the form if possible "MassMatrix" reduce the system to the form if possible "Residual" subtract right-hand sides of equations from left-hand sides to form a residual function.

NDSolve automatically tries to put the system in an explicit form.

Generate a system of the form using the option "MassMatrix". Generate a system of the form using the option "Residual". Show correspondence between working and specified variables.

Use the suboption option "TimeConstraint" to control the amount of time in seconds NDSolve spends on obtaining an explicit expression. Solve a system where analytic solutions can only be found for some of the dependent variables. Solve an ODE that contains constant parameters as part of the system. This defines the equation, invariant constraint, and starting condition for a simulation of the motion of a pendulum.

NDSolve does not solve this system directly, as indicated by the message. Using the "Projection" time integration method gets a solution that satisfies the invariant at the time steps. Plot the invariant at the time steps; it is satisfied nearly to machine precision. Define system variables, equations, and initial conditions. Solve the system in the form using the option "MassMatrix". Define equations and a complete set of initial values for the constrained pendulum system. None no index reduction default Automatic choose the method automatically "Pantelides" graph-based Pantelides algorithm "StructuralMatrix" structural matrix-based algorithm.

Solve using automatically chosen methods of index reduction. Use index reduction and solve with the differentiated equations. Compare how well the constraint is satisfied for the two solutions. Get the structural incidence matrix for the system , for the variables ,. Get the structural incidence matrix for the system , with the second equation differentiated for the variables ,. Get the structural incidence matrix for the constrained pendulum system with respect to the highest-order derivatives that appear. This tests the literal presence of the symbols x and y in each of the equations.

Plot quantities that should be invariant, based on the originally specified equations. Solve the system using the structural matrix method for index reduction. Attempt to solve the system using the Pantelides algorithm for index reduction. Solve the system using the structural matrix algorithm for index reduction. Solve the system as an index-0 system using the Pantelides algorithm. Plot the result of the system as obtained using "StructuralMatrix". Solve the system as an index-0 system using the structural matrix method.

This solves the system using the index reduction with dummy derivatives built into NDSolve. Plot quantities that should be invariant based on the originally specified equations. Try to integrate the system with manually introduced dummy derivatives replacing and. Using dummy derivatives is the default so no option setting is needed. Solve the ODEs using the "Projection" method. This solves the system using the index reduction with projection built into NDSolve. This gets the energy for a solution at the time steps and plots the energy for the dummy derivative and projection solutions.

Automatic determine the method to use automatically "Collocation" use a collocation method; this algorithm can be used for initialization of high-index systems "QR" use a QR decomposition-based algorithm for index-1 systems "BLT" use a Block Lower Triangular ordering BLT approach for index-1 systems.

Obtain a consistent set of initial conditions for the pendulum DAE. The default setting for the method "CollocationPoints" finds the correct initialization. Do the initialization using constant and linear basis functions. Try solving the initialization problem with a range of collocation points. Plotting the results as a function of the collocation points gives an indication of how the solution converges.

Red , Green , and Blue correspond to the variables x 1 , x 2 , x 3 , respectively. Define variables and equations for an index-6 linear system of DAEs. Find the exact solution for this case using DSolve. The General Mathematics Option is recommended for students who wish to pursue mathematics either by itself or in combination with a related field e.

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The plan of work toward the doctor of philosophy degree will include a comprehensive examination, a qualifying examination, and an advanced topic examination. A language exam may be required at the discretion of the thesis committee. The content of the advanced topic examination is determined by a department committee. A general examination, the doctoral dissertation and its defense complete the work for the Ph. Each candidate's plan of work must be approved by a special committee of the department.

A language examination may be required at the discretion of the thesis committee. Intensive review of fundamental concepts in mathematics utilized in calculus, including functions and graphs, exponentials and logarithms, and trigonometry. This course is for students who need to take MATH 51 or 21, but who require remediation in precalculus. The credits for this course do not count toward graduation, but do count toward GPA and current credit count. Consent of department required. Meaning, content, and methods of mathematical thought illustrated by topics that may be chosen from number theory, abstract algebra, combinatorics, finite or nonEuclidean geometries, game theory, mathematical logic, set theory, topology.

Systems of linear equations, matrices, introduction to linear programming. Sets, counting methods, probability, random variables, introduction to Markov chains. A first course in the basic concepts and methods of statistics with illustrations from the social, behavioral, and biological sciences. Descriptive statistics; frequency distributions, mean and standard deviation, two-way tables, correlation and regression; random sampling, rules of probability, probability distributions and parameters, parameter estimation, confidence intervals, hypothesis testing, statistical significance.

Functions and graphs; limits and continuity; derivative, differential, and applications; indefinite and definite integrals; trigonometric, logarithmic, exponential, and hyperbolic functions. Applications of integration; techniques of integration; separable differential equations; infinite sequences and series; Taylor's Theorem and other approximations; curves and vectors in the plane.

Vectors in space; partial derivatives; Lagrange multipliers; multiple integrals; vector analysis; line integrals; Green's Theorem, Gauss's Theorem. Matrices, vectors, vector spaces and mathematical systems, special kinds of matrices, elementary matrix transformations, systems of linear equations, convex sets, introduction to linear programming. The derivative and applications to extrema, approximation, and related rates. Exponential and logarithm functions, growth and decay. Trigonometric functions and related derivatives and integrals.

Partial derivatives and extrema. Multiple integrals and applications. Meets three hours per week. Limits and continuity; exponential, logarithmic and trigonometric functions; derivatives; extrema; approximations; indefinite and definite integrals. Applications with emphasis on business and economics. A first course in logical theory, introducing the notions of logical consequence and proof, as well as related concepts such as consistency and contigency.

Formal systems taught may include: Elements of statistics and probability with emphasis on biological applications. Statistical analysis of experimental and observational data. An introduction to the discipline of mathematics for students considering a major in mathematics. Provides an introduction to rigorous mathematical reasoning, including basic proof techniques e.

Study of a topic in mathematics under individual supervision. Intended for students with specific interests in areas not covered in the listed courses.

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Consent of department chair required. Practice in solving problems from mathematical contests using a variety of techniques. Permission of instructor required.

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Course may be repeated. Preparation for the first actuarial exam — probability. Problems in calculus and probability with insurance applications. Preparation for the second actuarial exam - financial mathematics. Mathematics of interest and investments, interest rate measurement, present value, annuities, loan repayment schemes, bond valuation, introduction to derivative securities. Practice in solving problems from past exams.

Linear differential equations and applications; matrices and systems of linear equations; vector spaces; eigenvalues and application to linear systems of differential equations. Functions of a complex variable; calculus of residues; contour integration; applications to conformal mapping and Laplace transforms. Consent of instructor required. Discussion of geometry as an axiomatic system. History of and equivalent versions of Euclid's fifth postulate.

NonEuclidean geometries based upon negation of the fifth postulate: During the second session of each week, Penn students will teach the hands-on activity to a small group of UCHS students. The Penn students will also have an opportunity to develop their own activity and to implement it with the UCHS students. Small class sizes permit an informal, discussion-type atmosphere, and often the entire class works together on a given problem.

Homework is intended to be thought-provoking, rather than skill-sharpening. This is a course about mathematical reasoning and the media. Embedded in many stories one finds in the media are mathematical questions as well as implicit mathematical models for how the world behaves. We will discuss ways to recognize such questions and models, and how to think about them from a mathematical perspective. A key part of the course will be about what constitutes a mathematical proof, and what passes for proof in various media contexts.

The course will cover a variety of topics in logic, probability and statistics as well as how these subjects can be used and abused. Introduction to Fourier series and Fourier transforms. Computation of solutions, modeling using PDE's, geometric intuition, and qualitative understanding of the evolution of systems according to the type of partial differential operator.

Demetrius at MIT based on the second law of thermodynamics. The essential idea is that life exists because if increases overall entropy quicker than other process es. The course will include an introduction to thermodynamics as well as to the information theory difinition of entropy.

Eventually I want to apply this to other systems. For instance, I think the theory explains why political candidates who create chaos, tweet and require more bits of information to describe each day are favored by the second law. Demetrius makes a distinction between robust high entropy environments and precarious low entropy environments which is a very interesting in many different systems.

Typically the problems considered will come from mathematics, chemistry, biology, and materials science but sometimes they will also come from economics, finance, and social sciences. The research problems in the course vary from year to year. Mathematics majors are advised that MATH cannot be taken to satisfy the major requirements. This course will introduce students to some of the most widely used algorithms and illustrate how they are actually used. Applications will include the use of matrix computations to computer graphics, use of the discrete Fourier transform and related techniques in digital signal processing, the analysis of systems of linear differential equations, and singular value deompositions with application to a principal component analysis.

The ideas and tools provided by this course will be useful to students who intend to tackle higher level courses in digital signal processing, computer vision, robotics, and computer graphics. MATH is a masters level version of this course. Linear algebra, eigenvalues and eigenvectors of matrices, groups, rings and fields. After a review of vector calculus and a section on tensor algebra, we study manifolds and their intrinsic geometry, including metrics, connections, geodesics, and the Riemann curvature tensor. Introduction to the topology of metric spaces with an emphasis on higher dimensional Euclidean spaces.

The contraction mapping principle. Inverse and implicit function theorems.

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