numerical methods phd thesis

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Masters Theses & Specialist Projects

Analysis and implementation of numerical methods for solving ordinary differential equations.

Muhammad Sohel Rana , Western Kentucky University Follow

Publication Date

Advisor(s) - committee chair.

Dr. Mark Robinson (Director), Dr. Ferhan Atici and Dr. Ngoc Nguyen

Degree Program

Department of Mathematics

Degree Type

Master of Science

Numerical methods to solve initial value problems of differential equations progressed quite a bit in the last century. We give a brief summary of how useful numerical methods are for ordinary differential equations of first and higher order. In this thesis both computational and theoretical discussion of the application of numerical methods on differential equations takes place. The thesis consists of an investigation of various categories of numerical methods for the solution of ordinary differential equations including the numerical solution of ordinary differential equations from a number of practical fields such as equations arising in population dynamics and astrophysics. It includes discussion what are the advantages and disadvantages of implicit methods over explicit methods, the accuracy and stability of methods and how the order of various methods can be approximated numerically. Also, semidiscretization of some partial differential equations and stiff systems which may arise from these semidiscretizations are examined.

• Disciplines

Numerical Analysis and Computation | Ordinary Differential Equations and Applied Dynamics | Partial Differential Equations

Recommended Citation

Rana, Muhammad Sohel, "Analysis and Implementation of Numerical Methods for Solving Ordinary Differential Equations" (2017). Masters Theses & Specialist Projects. Paper 2053. https://digitalcommons.wku.edu/theses/2053

Since November 27, 2017

Included in

Numerical Analysis and Computation Commons , Ordinary Differential Equations and Applied Dynamics Commons , Partial Differential Equations Commons

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Mathematical Modeling Doctor of Philosophy (Ph.D.) Degree

Request Info about graduate study Visit Apply

The mathematical modeling Ph.D. enables you to develop mathematical models to investigate, analyze, predict, and solve the behaviors of a range of fields from medicine, engineering, and business to physics and science.

STEM-OPT Visa Eligible

Overview for Mathematical Modeling Ph.D.

Mathematical modeling is the process of developing mathematical descriptions, or models, of real-world systems. These models can be linear or nonlinear, discrete or continuous, deterministic or stochastic, and static or dynamic, and they enable investigating, analyzing, and predicting the behavior of systems in a wide variety of fields. Through extensive study and research, graduates of the mathematical modeling Ph.D. will have the expertise not only to use the tools of mathematical modeling in various application settings, but also to contribute in creative and innovative ways to the solution of complex interdisciplinary problems and to communicate effectively with domain experts in various fields.

Plan of Study

The degree requires at least 60 credit hours of course work and research. The curriculum consists of three required core courses, three required concentration foundation courses, a course in scientific computing and high-performance computing (HPC), three elective courses focused on the student’s chosen research concentration, and a doctoral dissertation. Elective courses are available from within the School of Mathematics and Statistics as well as from other graduate programs at RIT, which can provide application-specific courses of interest for particular research projects. A minimum of 30 credits hours of course work is required. In addition to courses, at least 30 credit hours of research, including the Graduate Research Seminar, and an interdisciplinary internship outside of RIT are required.

Students develop a plan of study in consultation with an application domain advisory committee. This committee consists of the program director, one of the concentration leads, and an expert from an application domain related to the student’s research interest. The committee ensures that all students have a roadmap for completing their degree based on their background and research interests. The plan of study may be revised as needed. Learn more about our mathematical modeling doctoral students and view a selection of mathematical modeling seminars hosted by the department.

Qualifying Examinations

All students must pass two qualifying examinations to determine whether they have sufficient knowledge of modeling principles, mathematics, and computational methods to conduct doctoral research. Students must pass the examinations in order to continue in the Ph.D. program.

The first exam is based on the Numerical Analysis I (MATH-602) and Mathematical Modeling I, II (MATH-622, 722). The second exam is based on the student's concentration foundation courses and additional material deemed appropriate by the committee and consists of a short research project.

Dissertation Research Advisor and Committee

A dissertation research advisor is selected from the program faculty based on the student's research interests, faculty research interest, and discussions with the program director. Once a student has chosen a dissertation advisor, the student, in consultation with the advisor, forms a dissertation committee consisting of at least four members, including the dissertation advisor. The committee includes the dissertation advisor, one other member of the mathematical modeling program faculty, and an external chair appointed by the dean of graduate education. The external chair must be a tenured member of the RIT faculty who is not a current member of the mathematical modeling program faculty. The fourth committee member must not be a member of the RIT faculty and may be a professional affiliated with industry or with another institution; the program director must approve this committee member.

The main duties of the dissertation committee are administering both the candidacy exam and final dissertation defense. In addition, the dissertation committee assists students in planning and conducting their dissertation research and provides guidance during the writing of the dissertation.

Admission to Candidacy

When a student has developed an in-depth understanding of their dissertation research topic, the dissertation committee administers an examination to determine if the student will be admitted to candidacy for the doctoral degree. The purpose of the examination is to ensure that the student has the necessary background knowledge, command of the problem, and intellectual maturity to carry out the specific doctoral-level research project. The examination may include a review of the literature, preliminary research results, and proposed research directions for the completed dissertation. Requirements for the candidacy exam include both a written dissertation proposal and the presentation of an oral defense of the proposal. This examination must be completed at least one year before the student can graduate.

Dissertation Defense and Final Examination

The dissertation defense and final examination may be scheduled after the dissertation has been written and distributed to the dissertation committee and the committee has consented to administer the final examination. Copies of the dissertation must be distributed to all members of the dissertation committee at least four weeks prior to the final examination. The dissertation defense consists of an oral presentation of the dissertation research, which is open to the public. This public presentation must be scheduled and publicly advertised at least four weeks prior to the examination. After the presentation, questions will be fielded from the attending audience and the final examination, which consists of a private questioning of the candidate by the dissertation committee, will ensue. After the questioning, the dissertation committee immediately deliberates and thereafter notifies the candidate and the mathematical modeling graduate director of the result of the examination.

All students in the program must spend at least two consecutive semesters (summer excluded) as resident full-time students to be eligible to receive the doctoral degree.

Maximum Time Limitations

University policy requires that doctoral programs be completed within seven years of the date of the student passing the qualifying exam. All candidates must maintain continuous enrollment during the research phase of the program. Such enrollment is not limited by the maximum number of research credits that apply to the degree.

National Labs Career Fair

Hosted by RIT’s Office of Career Services and Cooperative Education, the National Labs Career Fair is an annual event that brings representatives to campus from the United States’ federally funded research and development labs. These national labs focus on scientific discovery, clean energy development, national security, technology advancements, and more. Students are invited to attend the career fair to network with lab professionals, learn about opportunities, and interview for co-ops, internships, research positions, and full-time employment.

Students are also interested in: Applied and Computational Mathematics MS

The College of Science consistently receives research grant awards from organizations that include the National Science Foundation , National Institutes of Health , and NASA , which provide you with unique opportunities to conduct cutting-edge research with our faculty members.

Faculty in the School of Mathematics and Statistics conducts research on a broad variety of topics including:

• applied inverse problems and optimization
• applied statistics and data analytics
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• dynamical systems and fluid dynamics
• geometry, relativity, and gravitation
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Curriculum for 2023-2024 for Mathematical Modeling Ph.D.

Current Students: See Curriculum Requirements

Mathematical Modeling, Ph.D. degree, typical course sequence

Concentrations, applied inverse problems, biomedical mathematics, discrete mathematics, dynamical systems and fluid dynamics, geometry, relativity and gravitation, admissions and financial aid.

This program is available on-campus only.

Full-time study is 9+ semester credit hours. International students requiring a visa to study at the RIT Rochester campus must study full‑time.

Application Details

To be considered for admission to the Mathematical Modeling Ph.D. program, candidates must fulfill the following requirements:

• Complete an online graduate application .
• Submit copies of official transcript(s) (in English) of all previously completed undergraduate and graduate course work, including any transfer credit earned.
• Hold a baccalaureate degree (or US equivalent) from an accredited university or college.
• A recommended minimum cumulative GPA of 3.0 (or equivalent).
• Submit a current resume or curriculum vitae.
• Submit a statement of purpose for research which will allow the Admissions Committee to learn the most about you as a prospective researcher.
• Submit two letters of recommendation .
• Entrance exam requirements: None
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• Submit English language test scores (TOEFL, IELTS, PTE Academic), if required. Details are below.

English Language Test Scores

International applicants whose native language is not English must submit one of the following official English language test scores. Some international applicants may be considered for an English test requirement waiver .

International students below the minimum requirement may be considered for conditional admission. Each program requires balanced sub-scores when determining an applicant’s need for additional English language courses.

How to Apply   Start or Manage Your Application

Cost and Financial Aid

An RIT graduate degree is an investment with lifelong returns. Ph.D. students typically receive full tuition and an RIT Graduate Assistantship that will consist of a research assistantship (stipend) or a teaching assistantship (salary).

Additional Information

Foundation courses.

Mathematical modeling encompasses a wide variety of scientific disciplines, and candidates from diverse backgrounds are encouraged to apply. If applicants have not taken the expected foundational course work, the program director may require the student to successfully complete foundational courses prior to matriculating into the Ph.D. program. Typical foundation course work includes calculus through multivariable and vector calculus, differential equations, linear algebra, probability and statistics, one course in computer programming, and at least one course in real analysis, numerical analysis, or upper-level discrete mathematics.

Numerical Methods in Scientific Computing

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Jos van Kan, Delft University of Technology

Guus Segal, Delft University of Technology

Fred Vermolen, University of Hasselt

Copyright Year: 2023

ISBN 13: 9789463667401

Publisher: TU Delft Open

Language: English

Formats Available

Conditions of use.

Table of Contents

• Review of some basic mathematical concept
• A crash course in PDE’s
• Finite difference methods
• Finite volume methods
• Minimization problems in physics
• The numerical solution of minimization problems
• The weak formulation and Galerkin’s method
• Extension of the FEM
• Solution of large systems of equations
• The heat- or diffusion equation
• The wave equation
• The transport equation
• Moving boundary problems

Ancillary Material

About the book.

This is a book about numerically solving partial differential equations occurring in technical and physical contexts and the authors have set themselves a more ambitious target than to just talk about the numerics. Their aim is to show the place of numerical solutions in the general modeling process and this must inevitably lead to considerations about modeling itself. Partial differential equations usually are a consequence of applying first principles to a technical or physical problem at hand. That means, that most of the time the physics also have to be taken into account especially for validation of the numerical solution obtained. This book aims especially at engineers and scientists who have ’real world’ problems. It will concern itself less with pesky mathematical detail. For the interested reader though, we have included sections on mathematical theory to provide the necessary mathematical background. Since this treatment had to be on the superficial side we have provided further reference to the literature where necessary.

About the Contributors

Jos van Kan (1944) graduated in 1968 from Delft University of Technology, Delft, Netherlands, in Numerical Analysis and was assistant professor at the Department of Mathematics of that institute until 2009. He wrote several articles on Numerical Fluid Mechanics (pressure correction methods) and has written a multigrid pressure solver for the Delft software package to solve the Navier-Stokes equations. He was teaching classes in Numerical Analysis from 1971 until 2009, and wrote several books on the subject. Currently he is a retired professor.

Guus Segal (1948) graduated in 1971 from Delft University of Technology, Delft, Netherlands, in Numerical Analysis and was part time assistant professor at the Department of Mathematics of that institute until 2013. He also worked in the consultancy and numerical software company SEPRA in The Hague, Netherlands. He wrote a number of articles on Finite Element Methods and several articles on curvilinear Finite Volume Methods and Numerical Fluid Mechanics. He has written a book on Finite Element methods and Navier-Stokes equations. He is the main developer of the finite element package SEPRAN. He was teaching classes in Numerical Analysis from 1973 until 2013.

Fred Vermolen (1969) graduated in 1993 from Delft University of Technology, Delft, Netherlands and defended his PhD thesis on numerical methods for moving boundary problems in 1998. He has written several contributions on Stefan problems, computational mechanics, mathematical analysis and uncertainty quantification with most of the applications in medicine. He has held an assistant and associate professorship in Numerical Analysis at the Delft University from 2000 until 2020. In 2020 he started his current position as a full professor in Computational Mathematics at the University of Hasselt in Belgium.

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Mathematics PhD theses

A selection of Mathematics PhD thesis titles is listed below, some of which are available online:

2022   2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Melanie Kobras –  Low order models of storm track variability

Ed Clark –  Vectorial Variational Problems in L∞ and Applications to Data Assimilation

Katerina Christou – Modelling PDEs in Population Dynamics using Fixed and Moving Meshes  

Chiara Cecilia Maiocchi –  Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems

Samuel R Harrison – Stalactite Inspired Thin Film Flow

Elena Saggioro – Causal network approaches for the study of sub-seasonal to seasonal variability and predictability

Cathie A Wells – Reformulating aircraft routing algorithms to reduce fuel burn and thus CO 2 emissions  

Jennifer E. Israelsson –  The spatial statistical distribution for multiple rainfall intensities over Ghana

Giulia Carigi –  Ergodic properties and response theory for a stochastic two-layer model of geophysical fluid dynamics

André Macedo –  Local-global principles for norms

Tsz Yan Leung  –  Weather Predictability: Some Theoretical Considerations

Jehan Alswaihli –  Iteration of Inverse Problems and Data Assimilation Techniques for Neural Field Equations

Jemima M Tabeart –  On the treatment of correlated observation errors in data assimilation

Chris Davies –  Computer Simulation Studies of Dynamics and Self-Assembly Behaviour of Charged Polymer Systems

Birzhan Ayanbayev –  Some Problems in Vectorial Calculus of Variations in L∞

Penpark Sirimark –  Mathematical Modelling of Liquid Transport in Porous Materials at Low Levels of Saturation

Adam Barker –  Path Properties of Levy Processes

Hasen Mekki Öztürk –  Spectra of Indefinite Linear Operator Pencils

Carlo Cafaro –  Information gain that convective-scale models bring to probabilistic weather forecasts

Nicola Thorn –  The boundedness and spectral properties of multiplicative Toeplitz operators

James Jackaman  – Finite element methods as geometric structure preserving algorithms

Changqiong Wang - Applications of Monte Carlo Methods in Studying Polymer Dynamics

Jack Kirk - The molecular dynamics and rheology of polymer melts near the flat surface

Hussien Ali Hussien Abugirda - Linear and Nonlinear Non-Divergence Elliptic Systems of Partial Differential Equations

Andrew Gibbs - Numerical methods for high frequency scattering by multiple obstacles (PDF-2.63MB)

Mohammad Al Azah - Fast Evaluation of Special Functions by the Modified Trapezium Rule (PDF-913KB)

Katarzyna (Kasia) Kozlowska - Riemann-Hilbert Problems and their applications in mathematical physics (PDF-1.16MB)

Anna Watkins - A Moving Mesh Finite Element Method and its Application to Population Dynamics (PDF-2.46MB)

Niall Arthurs - An Investigation of Conservative Moving-Mesh Methods for Conservation Laws (PDF-1.1MB)

Samuel Groth - Numerical and asymptotic methods for scattering by penetrable obstacles (PDF-6.29MB)

Katherine E. Howes - Accounting for Model Error in Four-Dimensional Variational Data Assimilation (PDF-2.69MB)

Jian Zhu - Multiscale Computer Simulation Studies of Entangled Branched Polymers (PDF-1.69MB)

Tommy Liu - Stochastic Resonance for a Model with Two Pathways (PDF-11.4MB)

Matthew Paul Edgington - Mathematical modelling of bacterial chemotaxis signalling pathways (PDF-9.04MB)

Anne Reinarz - Sparse space-time boundary element methods for the heat equation (PDF-1.39MB)

Adam El-Said - Conditioning of the Weak-Constraint Variational Data Assimilation Problem for Numerical Weather Prediction (PDF-2.64MB)

Nicholas Bird - A Moving-Mesh Method for High Order Nonlinear Diffusion (PDF-1.30MB)

Charlotta Jasmine Howarth - New generation finite element methods for forward seismic modelling (PDF-5,52MB)

Aldo Rota - From the classical moment problem to the realizability problem on basic semi-algebraic sets of generalized functions (PDF-1.0MB)

Sarah Lianne Cole - Truncation Error Estimates for Mesh Refinement in Lagrangian Hydrocodes (PDF-2.84MB)

Alexander J. F. Moodey - Instability and Regularization for Data Assimilation (PDF-1.32MB)

Dale Partridge - Numerical Modelling of Glaciers: Moving Meshes and Data Assimilation (PDF-3.19MB)

Joanne A. Waller - Using Observations at Different Spatial Scales in Data Assimilation for Environmental Prediction (PDF-6.75MB)

Faez Ali AL-Maamori - Theory and Examples of Generalised Prime Systems (PDF-503KB)

Mark Parsons - Mathematical Modelling of Evolving Networks

Natalie L.H. Lowery - Classification methods for an ill-posed reconstruction with an application to fuel cell monitoring

David Gilbert - Analysis of large-scale atmospheric flows

Peter Spence - Free and Moving Boundary Problems in Ion Beam Dynamics (PDF-5MB)

Timothy S. Palmer - Modelling a single polymer entanglement (PDF-5.02MB)

Mohamad Shukor Talib - Dynamics of Entangled Polymer Chain in a Grid of Obstacles (PDF-2.49MB)

Cassandra A.J. Moran - Wave scattering by harbours and offshore structures

Ashley Twigger - Boundary element methods for high frequency scattering

David A. Smith - Spectral theory of ordinary and partial linear differential operators on finite intervals (PDF-1.05MB)

Stephen A. Haben - Conditioning and Preconditioning of the Minimisation Problem in Variational Data Assimilation (PDF-3.51MB)

Jing Cao - Molecular dynamics study of polymer melts (PDF-3.98MB)

Bonhi Bhattacharya - Mathematical Modelling of Low Density Lipoprotein Metabolism. Intracellular Cholesterol Regulation (PDF-4.06MB)

Tamsin E. Lee - Modelling time-dependent partial differential equations using a moving mesh approach based on conservation (PDF-2.17MB)

Polly J. Smith - Joint state and parameter estimation using data assimilation with application to morphodynamic modelling (PDF-3Mb)

Corinna Burkard - Three-dimensional Scattering Problems with applications to Optical Security Devices (PDF-1.85Mb)

Laura M. Stewart - Correlated observation errors in data assimilation (PDF-4.07MB)

R.D. Giddings - Mesh Movement via Optimal Transportation (PDF-29.1MbB)

G.M. Baxter - 4D-Var for high resolution, nested models with a range of scales (PDF-1.06MB)

C. Spencer - A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table.

P. Jelfs - A C-property satisfying RKDG Scheme with Application to the Morphodynamic Equations (PDF-11.7MB)

L. Bennetts - Wave scattering by ice sheets of varying thickness

M. Preston - Boundary Integral Equations method for 3-D water waves

J. Percival - Displacement Assimilation for Ocean Models (PDF - 7.70MB)

D. Katz - The Application of PV-based Control Variable Transformations in Variational Data Assimilation (PDF- 1.75MB)

S. Pimentel - Estimation of the Diurnal Variability of sea surface temperatures using numerical modelling and the assimilation of satellite observations (PDF-5.9MB)

J.M. Morrell - A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian method for the numerical solution of the Euler equations (PDF-7.7MB)

L. Watkinson - Four dimensional variational data assimilation for Hamiltonian problems

M. Hunt - Unique extension of atomic functionals of JB*-Triples

D. Chilton - An alternative approach to the analysis of two-point boundary value problems for linear evolutionary PDEs and applications

T.H.A. Frame - Methods of targeting observations for the improvement of weather forecast skill

C. Hughes - On the topographical scattering and near-trapping of water waves

B.V. Wells - A moving mesh finite element method for the numerical solution of partial differential equations and systems

D.A. Bailey - A ghost fluid, finite volume continuous rezone/remap Eulerian method for time-dependent compressible Euler flows

M. Henderson - Extending the edge-colouring of graphs

K. Allen - The propagation of large scale sediment structures in closed channels

D. Cariolaro - The 1-Factorization problem and same related conjectures

A.C.P. Steptoe - Extreme functionals and Stone-Weierstrass theory of inner ideals in JB*-Triples

D.E. Brown - Preconditioners for inhomogeneous anisotropic problems with spherical geometry in ocean modelling

S.J. Fletcher - High Order Balance Conditions using Hamiltonian Dynamics for Numerical Weather Prediction

C. Johnson - Information Content of Observations in Variational Data Assimilation

M.A. Wakefield - Bounds on Quantities of Physical Interest

M. Johnson - Some problems on graphs and designs

A.C. Lemos - Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts

R.K. Lashley - Automatic Generation of Accurate Advection Schemes on Structured Grids and their Application to Meteorological Problems

J.V. Morgan - Numerical Methods for Macroscopic Traffic Models

M.A. Wlasak - The Examination of Balanced and Unbalanced Flow using Potential Vorticity in Atmospheric Modelling

M. Martin - Data Assimilation in Ocean circulation models with systematic errors

K.W. Blake - Moving Mesh Methods for Non-Linear Parabolic Partial Differential Equations

J. Hudson - Numerical Techniques for Morphodynamic Modelling

A.S. Lawless - Development of linear models for data assimilation in numerical weather prediction .

C.J.Smith - The semi lagrangian method in atmospheric modelling

T.C. Johnson - Implicit Numerical Schemes for Transcritical Shallow Water Flow

M.J. Hoyle - Some Approximations to Water Wave Motion over Topography.

P. Samuels - An Account of Research into an Area of Analytical Fluid Mechnaics. Volume II. Some mathematical Proofs of Property u of the Weak End of Shocks.

M.J. Martin - Data Assimulation in Ocean Circulation with Systematic Errors

P. Sims - Interface Tracking using Lagrangian Eulerian Methods.

P. Macabe - The Mathematical Analysis of a Class of Singular Reaction-Diffusion Systems.

B. Sheppard - On Generalisations of the Stone-Weisstrass Theorem to Jordan Structures.

S. Leary - Least Squares Methods with Adjustable Nodes for Steady Hyperbolic PDEs.

I. Sciriha - On Some Aspects of Graph Spectra.

P.A. Burton - Convergence of flux limiter schemes for hyperbolic conservation laws with source terms.

J.F. Goodwin - Developing a practical approach to water wave scattering problems.

N.R.T. Biggs - Integral equation embedding methods in wave-diffraction methods.

L.P. Gibson - Bifurcation analysis of eigenstructure assignment control in a simple nonlinear aircraft model.

A.K. Griffith - Data assimilation for numerical weather prediction using control theory. .

J. Bryans - Denotational semantic models for real-time LOTOS.

I. MacDonald - Analysis and computation of steady open channel flow .

A. Morton - Higher order Godunov IMPES compositional modelling of oil reservoirs.

S.M. Allen - Extended edge-colourings of graphs.

M.E. Hubbard - Multidimensional upwinding and grid adaptation for conservation laws.

C.J. Chikunji - On the classification of finite rings.

S.J.G. Bell - Numerical techniques for smooth transformation and regularisation of time-varying linear descriptor systems.

D.J. Staziker - Water wave scattering by undulating bed topography .

K.J. Neylon - Non-symmetric methods in the modelling of contaminant transport in porous media. .

D.M. Littleboy - Numerical techniques for eigenstructure assignment by output feedback in aircraft applications .

M.P. Dainton - Numerical methods for the solution of systems of uncertain differential equations with application in numerical modelling of oil recovery from underground reservoirs .

M.H. Mawson - The shallow-water semi-geostrophic equations on the sphere. .

S.M. Stringer - The use of robust observers in the simulation of gas supply networks .

S.L. Wakelin - Variational principles and the finite element method for channel flows. .

E.M. Dicks - Higher order Godunov black-oil simulations for compressible flow in porous media .

C.P. Reeves - Moving finite elements and overturning solutions .

A.J. Malcolm - Data dependent triangular grid generation. .

• Graduate Catalog >
• Norm Asbjornson College of Engineering >
• Department of Electrical and Computer Engineering >

Ph.D. in Electrical Engineering

Ph.D. students earn at least 60 post-baccalaureate credits, including at least 18 credits of dissertation work. In progressing toward this degree, the student must pass the following examinations:

• A written departmental Graduate Study Qualifying Examination administered to all M.S. and Ph.D. students in their first year of work.
• A comprehensive examination to be taken within two years of the qualifying examination and after completing two-thirds of their total coursework.
• A final oral examination and defense of a dissertation based on the student's research.

There is no foreign language requirement for the degree.

Degree requirements--60 credits total:

• ≤ 28 credits  Doctoral Thesis ( EELE 690 ) , with a minimum 18 credits.
• 2 credits  Scientific Communication and Proposal Development ( ENGR 650 ) , taken just before the comprehensive examination
• ≥ 3 credits Advanced Math, Numerical Methods, or Statistics (committee approved)
• ≥  27  advisor-approved credits with all graded credits having earned a B or better.
• ≤ 9 credits at 4xx level

All credits must also meet the following conditions:

• Regardless of how many credits are applied from a previously earned master’s degree, the Ph.D. program of study must include at least 9 credits of major courses taken

 at Montana State University (ECE-specific exception granted by Grad School).

• All Ph.D. credits no more than ten (10) years old at time of graduation (this limit does not apply to courses counted from a previously earned master’s degree).
• ≤ 6 credits  Independent Study  (EELE 592).
• ≤ 3 credits pass/fail, excluding dissertation.
• ≤ 9 credits challenged.
• No credits of 488, 489, 490, 492, 494, 498, or 589 are allowed.

Montana State University

P.O. Box 172220

Bozeman, MT 59717-2220

Telephone: (406) 994-6650

Fax: (406) 994-1972

Email: [email protected]

Location: 101 Montana Hall

Antoni Campeau

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Some students are also worried that hiring professional writers and editors is something like an academic crime. In reality, it is not. Just make sure that you use the received papers smartly and never write your name on them. Use them in the same manner that you use books, journals, and encyclopedias for your papers. They can serve as samples, sources of ideas, and guidelines.

So, you have a writing assignment and a request, "Please, write my essay for me." We have a team of authors and editors with profound skills and knowledge in all fields of study, who know how to conduct research, collect data, analyze information, and express it in a clear way. Let's do it!

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Customer Reviews

Write My Essay Service - Working to Help You

Do you want to have more free time for personal development and fun? Or are you confused with your professor's directions? Whatever your reason for coming to us is, you are welcome! We are a legitimate professional writing service with student-friendly prices and with an aim to help you achieve academic excellence. To get an A on your next assignment simply place an order or contact our 24/7 support team.

Write my essay for me frequently asked questions

How Our Essay Service Works

Do my essay with us and meet all your requirements.

We give maximum priority to customer satisfaction and thus, we are completely dedicated to catering to your requirements related to the essay. The given topic can be effectively unfolded by our experts but at the same time, you may have some exclusive things to be included in your writing too. Keeping that in mind, we take both your ideas and our data together to make a brilliant draft for you, which is sure to get you good grades.

Bennie Hawra

Customer Reviews

For expository writing, our writers investigate a given idea, evaluate its various evidence, set forth interesting arguments by expounding on the idea, and that too concisely and clearly. Our online essay writing service has the eligibility to write marvelous expository essays for you.

Total Price

Need a personal essay writer try essaybot which is your professional essay typer..

• EssayBot is an essay writing assistant powered by Artificial Intelligence (AI).
• Given the title and prompt, EssayBot helps you find inspirational sources, suggest and paraphrase sentences, as well as generate and complete sentences using AI.
• If your essay will run through a plagiarism checker (such as Turnitin), don’t worry. EssayBot paraphrases for you and erases plagiarism concerns.
• EssayBot now includes a citation finder that generates citations matching with your essay.

Progressive delivery is highly recommended for your order. This additional service allows tracking the writing process of big orders as the paper will be sent to you for approval in parts/drafts* before the final deadline.

What is more, it guarantees:

• 30 days of free revision;
• A top writer and the best editor;
• A personal order manager.

* You can read more about this service here or please contact our Support team for more details.

It is a special offer that now costs only +15% to your order sum!

Would you like to order Progressive delivery for your paper?

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