PhD Mathematics Thesis Writing in Mathematics with ThesisLikho
A PhD Mathematics Thesis in Applied Mathematics is a big project that takes several years to complete. It is a lot of research, analysis, modelling, computation and academic investigation. This thesis is not like an academic project. It must show a research problem, a well-established research gap, the right mathematical methodology, rigorous analysis and an original contribution to knowledge.
Applied Mathematics is a field that connects theories with problems from science, engineering, economics, technology, biology and other disciplines. A doctoral scholar may work with modelling, differential equations, numerical methods, optimization, statistics, computational mathematics or other specialized areas.
Doing mathematical research is only one part of the PhD journey. The research must also be presented in an organized thesis that clearly communicates the problem, methodology, mathematical analysis, results, contribution, limitations and future research possibilities.
ThesisLikho provides guidance for scholars working on Applied Mathematics research and supports the development organization editing and formatting of their PhD thesis.
Introduction
What Is Applied Mathematics
Applied Mathematics is about using concepts and techniques to understand and solve problems outside purely theoretical mathematics.
It can involve:
Mathematical modelling
Differential equations
Numerical analysis
Optimization
Operations research
Mathematical physics
Computational mathematics
Mathematical biology
Financial mathematics
Engineering mathematics
The specific research area determines the methods and analytical techniques used in the thesis.
A typical Applied Mathematics research pathway is:
Theoretical Problem
↓
Mathematical Formulation
↓
Research Methodology
↓
Mathematical Analysis
↓
Numerical or Computational Analysis
↓
Results
↓
Validation
↓
Original Contribution
Why Applied Mathematics Is Important for PhD Research
Applied Mathematics provides frameworks for investigating complex systems.
For example mathematical methods can be used to study:
Population dynamics
Disease transmission
Financial systems
Fluid flow
Heat transfer
Environmental processes
Engineering systems
Optimization problems
Energy systems
Computational problems
The purpose of research is not simply to apply an existing mathematical formula. The researcher should identify a problem and develop or extend mathematical knowledge in that area.
Selecting a PhD Mathematics Research Topic
Choosing a research topic is one of the most important stages of a PhD.
A strong topic should be:
Researchable
Mathematically meaningful
Supported by sufficient literature
of producing an original contribution
Appropriate for the scholars mathematical background
Feasible within the available resources
For example Applied Mathematics is too broad as a topic.
It can be narrowed into:
Applied Mathematics
↓
Mathematical Modelling
↓
Differential Equation Modelling
↓
Nonlinear Mathematical Model
↓
Specific Research Problem
This creates a research direction.
Identifying the Research Problem
A PhD thesis should begin with a defined research problem.
The problem may emerge from:
An mathematical question
A limitation in an existing model
A method that requires improvement
An application that has not been adequately studied
Inadequate mathematical analysis
Computational limitations
Conflicting findings in studies
The research problem should eventually lead to clearly defined objectives and research questions.
Research Gap in Applied Mathematics
The research gap explains what is missing from existing research.
A researcher may identify gaps involving:
Mathematical Gap
An existing problem has not been adequately analyzed.
Methodological Gap
Available mathematical or numerical methods have limitations.
Computational Gap
Existing methods may require computational resources.
Application Gap
A mathematical approach has not been sufficiently investigated in an application.
Validation Gap
Existing models may have limited validation.
The gap should be established through a review of relevant research.
Literature Review for a PhD Mathematics Thesis
A literature review is more than a collection of summaries.
A strong Applied Mathematics literature review should examine:
Existing mathematical models
Mathematical theories
Analytical techniques
Numerical methods
Computational approaches
Previous research findings
Limitations
Research gaps
The review should establish how the proposed research relates to existing knowledge.
A useful structure is:
Existing Research
↓
Strengths
↓
Limitations
↓
Research Gap
↓
Proposed Research
Mathematical Modelling in Applied Mathematics
Mathematical modelling is frequently used in Applied Mathematics.
A model transforms a system into form.
The process may involve:
1. Defining the problem
2. Identifying variables
3. Identifying parameters
4. Establishing assumptions
5. Formulating equations
6. Solving the model
7. Analyzing results
8. Validating the model
Depending on the research area the model may contain equations, differential equations, integral equations, optimization formulations or other mathematical structures.
Mathematical Methodology
The methodology chapter should clearly explain how the research problem will be investigated.
It may include:
Mathematical formulation
methods
Numerical methods
Computational algorithms
Simulation procedures
Parameter estimation
Error analysis
Convergence analysis
Stability analysis
Validation procedures
The selected methodology should be directly connected to the research objectives.
Analytical Methods
Some Applied Mathematics problems can be investigated using approaches.
Depending on the research area methods may include:
Separation of variables
Integral transforms
Fourier analysis
Laplace transforms
Perturbation techniques
analysis
Eigenvalue methods
Variational methods
The thesis should explain why a particular analytical method is appropriate.
Numerical Methods
When analytical solutions are difficult or unavailable numerical techniques can provide solutions.
Common numerical methods include:
Euler methods
Runge–Kutta methods
Finite difference methods
Finite element methods
Finite volume methods
Spectral methods
Iterative algorithms
Numerical research should generally consider accuracy, convergence, stability and computational efficiency where appropriate.
Computational Mathematics
Computational techniques can support research by allowing scholars to conduct simulations and numerical experiments.
Computational work may involve:
Algorithm implementation
Parameter variation
Numerical simulations
Error calculation
Convergence testing
Comparative analysis
Visualization of results
A thesis should clearly document the methodology so that the research process is understandable and reproducible.
Error Analysis
Numerical results are generally approximations.
Therefore error analysis can be a part of Applied Mathematics research.
For an approximation uh and a reference solution u, an absolute error can be expressed as:
E = |u. Uh|
A relative error may be represented as:
Er = |u| / |u. Uh|
where appropriate.
The exact error measure should be selected according to the problem.
Convergence Analysis
Convergence analysis is about seeing if a numerical solution gets closer to the solution when we make the computational parameters smaller.
For example a scholar may look at results for sizes like h, 2h 4h where h is a discretization parameter.
The thesis should explain what we see in the convergence behavior and how it helps us meet our research goals.
Stability Analysis
Stability is about seeing if errors stay under control when things change.
Depending on what area we're researching we may look at stability for:
Differential equations
Numerical schemes
Iterative algorithms
Dynamical systems
We need to choose the right mathematical framework for stability based on the problem we are trying to solve.
Results and Interpretation
The results chapter should show what we actually found out from our investigation.
Our results may include:
derivations
Theoretical findings
Numerical solutions
Error values
Convergence results
Stability results
Simulation outputs
Comparative tables
Graphs
We should not just show tables and graphs we need to explain what they mean.
Each major result should be explained in relation to our research goals.
Comparing Results with Methods
If we are proposing a new method, model or mathematical approach comparing it to existing methods can make our research stronger.
A comparison may look at:
Method Accuracy Error Convergence Computational Cost
Existing Method A Result Result Result Result
Existing Method B Result Result Result Result
Proposed Method Result Result Result Result
The numbers we use should be based on our research.
The point is to show the bad things about our proposed approach and what it adds to the field.
Original Contribution to Applied Mathematics
A PhD thesis must clearly say what it adds to the field of applied mathematics.
This contribution may be:
New Mathematical Model
A model for a problem that has not been studied much before.
Improved Numerical Method
A numerical method that works better under conditions.
New Analytical Result
A new mathematical relationship, theorem, solution or property.
Improved Computational Framework
A efficient way of doing computations.
New Application
Using an established method to solve a new and meaningful problem.
New Mathematical Analysis
Helping us understand how a mathematical system behaves.
Our contribution should be supported by evidence and comparisons to what other people have done.
Recommended PhD Mathematics Thesis Structure
A typical applied mathematics thesis may have the following chapters.
Chapter 1. This chapter can include:
Background information
Why we are doing this research
What problem we are trying to solve
Why this problem is
What we want to achieve
What questions we are trying to answer
What areas we will cover
Why this research is significant
What is new about our research
How the thesis is organized
Chapter 2. Literature Review
This chapter may talk about:
mathematical theories
Existing models
Existing methods
Numerical approaches
Previous research
What is missing in current research
What we can do to fill this gap
Chapter 3. Mathematical Formulation and Methodology
This chapter may contain:
assumptions we made
What variables and parameters we used
The equations that govern our problem
How we analyzed the problem mathematically
What numerical methods we used
What algorithms and computational procedures we followed
Chapter 4. Mathematical Analysis
Depending on the topic this may include:
analysis
Existence and uniqueness of solutions
Stability and convergence
Error analysis
What mathematical properties we found
Chapter 5. Computational / Numerical Results
This chapter may include:
How we designed our experiments
How we implemented our methods
What we found from our simulations
Tables and graphs of our results
How we analyzed our errors
Comparing our results to others
Chapter 6. Discussion
The discussion should explain:
What we found out
Why our results are important
How our results compare to previous research
What our results mean
What the limitations of our research are
What we contributed to the field
Chapter 7. Conclusion and Future Work
This chapter may include:
A summary of what we found
What our research added to the field
Our conclusions
What we could not do
What we suggest for research
The actual number and order of chapters should follow the guidelines of the university.
Writing Thesis Chapters
Each chapter should have a clear purpose.
For example:
What is the problem we are trying to solve?
Literature Review: What do we already know about this problem?
Methodology: How did we investigate this problem?
Results: What did we discover?
Discussion: What do our findings mean?
What did our research contribute to the field?
This structure helps the reader follow our research.
Thesis Editing and Proofreading
Mathematics theses need to be edited for language and technical consistency.
Important things to check include:
Mathematical Consistency:
Our equations
Variables
Symbols
Definitions
Mathematical notation
Academic Writing:
Grammar
Sentence structure
Clarity
Technical terms
transitions
References:
Citation consistency
Bibliography
Referencing style
Tables and Figures:
Numbering
Captions
Labels
Cross-references
Formatting:
Chapter headings
Page numbering
Margins
Spacing
University requirements
ThesisLikho Support for PhD Mathematics Research
ThesisLikho provides help for scholars working on their PhD Mathematics theses.
This help can include:
Topic and Research Planning:
Refining our research topic
Identifying the problem
Setting our research objectives
Developing our research questions
Finding the gap in research
Mathematical Research Support:
Mathematical modelling
Planning our methodology
methods
Numerical methods
Computational analysis
Organizing our results
Thesis Chapter Support:
Introduction
Literature review
Methodology
Mathematical analysis
Results
Discussion
Conclusion
Editing and Formatting:
Academic editing
Proofreading
Referencing
Thesis formatting
Making sure our figures and tables are consistent
Checking if our thesis is ready for submission
The scholars own mathematical work, research, results and academic contribution are the main focus of the thesis.
How to Prepare for a PhD Mathematics Viva
A Mathematics PhD viva may ask about the research contribution and the mathematical reasoning behind it.
A scholar should be able to explain:
1. Why we chose this research problem.
2. What gap, in research we are trying to fill.
3. Why we used the methodology we chose.
4. What assumptions we made.
5. How we formulated our equations.
6. Why we used the analytical method we chose.
7. How we made sure our results are valid.
We need to be ready to talk about our research and its significance.
8. What do the numbers tell us about the results?
9. What thing does the research bring to the table?
10. What are the things that the research cannot do?
11. How does the research compare to studies that are already out there?
12. What new research can come from the work that is done here?
The person doing the research should know why they made every decision about how to do the research.
Common Problems People Face When Writing a PhD Thesis in Mathematics
1. The Research Is Too Big
If the topic is too broad and covers areas of math it can be hard to handle.
2. The Gap In Research Is Not Clear
If the literature review does not clearly show what is missing it can be hard to show what the research adds.
3. The Math Symbols Are Not Used Consistently
If the symbols or definitions change from one chapter to another it can cause confusion.
4. The Results Are Not Explained Well
The numbers should be explained in a way not just shown.
5. The Results Are Not Checked
The results should be checked using the mathematical, numerical or computational methods.
6. It Is Not Clear What Is New
The thesis should clearly say what is new and how it helps the existing research.
How to Do a PhD Mathematics Thesis
The whole process can be broken down into these steps:
Choose a Topic
↓
Find a Problem
↓
Review the Literature
↓
Find the Gap in Research
↓
Set the Research Goals
↓
Create the Math Plan
↓
Decide on the Method
↓
Do the Math Work
↓
Do the Numerical or Computational Work
↓
Get the Results
↓
Check the Results
↓
Talk About the Results
↓
Say What Is
↓
Write the Thesis
↓
Edit the Thesis
↓
Make the Thesis Look Good
↓
Get Ready to Submit
↓
Get Ready for the Viva
PhD Mathematics Thesis Checklist
Before turning in the thesis the researcher should check:
The Research
The problem is clearly stated
The gap in research is found
The goals are set
The research questions are answered
What is new is clearly said
The Math Work
The equations are correct
The variables are defined
The parameters are explained
The method is justified
The math work is done
The Numerical or Computational Work
The algorithms are documented
The computational settings are reported
The errors are checked
The results are validated
The Quality of the Thesis
The chapters make sense
The tables and figures are numbered
The equations are numbered
The references are checked
The grammar is correct
The thesis looks good
The university rules are followed
The viva is prepared
Frequently Asked Questions
What is a PhD Mathematics thesis in Applied Mathematics?
It is a document that shows the research done on a math problem using the methods and shows what is new.
What areas of Applied Mathematics are commonly studied for a PhD?
Common areas include math modeling, differential equations, numerical analysis, optimization, operations research, math biology, computational math and math physics.
Does every Applied Mathematics thesis need to have analysis?
No some research is about the theory or math while other studies use a lot of numerical and computational methods.
What should be in a Mathematics thesis methodology?
It depends on the topic. It may include creating the math plan using analytical methods, numerical schemes, computational algorithms and checking the errors.
How can a Mathematics thesis show what is new?
It can show what is new by having a model math result, numerical method, computational approach, application or by adding to existing research in a meaningful way.
Conclusion
A good PhD Mathematics Thesis in Applied Mathematics needs to have a connection between the research problem the math plan, the method, the math work, the results, the check and what is new.
The research journey can be summed up like this:
Research Problem → Research Gap → Math Plan → Method → Math Work → Numerical or Computational Work → Results → Check → Talk About → What Is New → Thesis → Viva
A good thesis should clearly say what was studied why it matters how it was done what math findings were obtained, how they were checked and how the research helps the existing math knowledge.
With some help researchers can make their math research into a thesis that is easy to understand, review, edit make look good and prepare for submission.
ThesisLikho helps researchers, with planning, math methods creating chapters presenting results, editing, making look good and preparing for submission.
Take the Next Step with ThesisLikho
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Working on a PhD Mathematics thesis in Applied Mathematics? Connect with ThesisLikho for structured support in research organization, mathematical analysis, thesis chapter development, editing, formatting, and submission preparation.