D'Math University | Education & Specialist
PhD Differential Equations
A research-intensive doctoral programme at the frontier of modern differential equations theory. Combining deep analysis — functional analysis, Sobolev spaces, and geometric methods — with applications to physics, biology, and engineering, graduates become leading contributors to one of mathematics' most productive and applied research areas.
Doctoral
3–5 Years
Research-Intensive
Specialist Track
5
Research Specialisations
£72k+
Average Graduate Salary
50+
Partner Universities
3–5
Year Programme
Programme Overview
Programme Overview
- Advanced doctoral research focused on the theory and applications of differential equations
- Year 1: Intensive coursework in functional analysis, Sobolev spaces, and advanced PDE theory
- Year 2 onward: Original supervised research with world-leading differential equations experts
- Research areas include nonlinear PDEs, dynamical systems, geometric analysis, and stochastic PDEs
- Active collaborations with applied sciences — fluid dynamics, mathematical biology, and quantum theory
- Publications in journals such as Communications in PDEs, JMPA, and Inventiones expected
- International conference participation and collaboration supported throughout
Entry Requirements
- MSc in Mathematics with strong analysis component (first-class or high merit preferred)
- Deep knowledge of real analysis, functional analysis, and measure theory
- Prior exposure to PDEs at MSc or advanced undergraduate level essential
- Research proposal identifying specific equations or problems of interest
- Programming experience (MATLAB, Python, FEniCS) advantageous for computational work
- Two academic references attesting to analytical ability and research potential
- English proficiency: IELTS 7.0+ or equivalent
Research Areas & Coursework
Functional Analysis & Sobolev Spaces
Banach and Hilbert spaces, distributions, Sobolev embedding theorems, and their role in PDE existence theory.
Nonlinear PDEs
Variational methods, viscosity solutions, regularity theory, and maximum principles for nonlinear elliptic and parabolic equations.
Dynamical Systems
Stability theory, bifurcation analysis, chaos, strange attractors, and invariant manifold theory for ODEs and PDEs.
Hyperbolic Conservation Laws
Characteristics, shock waves, Rankine-Hugoniot conditions, entropy solutions, and Glimm's theorem.
Stochastic PDEs
SPDEs driven by space-time white noise, mild solutions, Itô integrals in infinite dimensions, and applications.
Geometric Analysis
Riemannian geometry, harmonic maps, geometric flows including Ricci flow, and curvature equations on manifolds.
Numerical PDE Methods
Finite element analysis, error estimates, adaptive methods, and high-performance implementation for research problems.
Doctoral Thesis
Original research making a substantial theoretical or applied contribution to differential equations, assessed by international experts.
Course Catalogue
Click any course to view its objective and learning outcomes.
DEQ 701 Research Methods +
Objective
To prepare doctoral candidates for ODE/PDE research.
Learning Outcomes
- Apply rigorous research design.
- Conduct interdisciplinary collaboration.
- Apply advanced analytical methods.
- Critique published research.
- Write proposals.
DEQ 702 Functional Analysis +
Objective
To master infinite-dimensional analysis for DE research.
Learning Outcomes
- Apply Banach and Hilbert space theory.
- Use spectral theory of operators.
- Apply Sobolev spaces.
- Use weak topologies.
- Apply distribution theory.
DEQ 703 Linear PDE Theory +
Objective
To study linear PDEs rigorously.
Learning Outcomes
- Apply elliptic regularity theory.
- Use parabolic and hyperbolic PDE theory.
- Apply semigroup theory.
- Use distributional solutions.
- Apply spectral methods.
Interactive Activity — Direction Field & Solution Curves
For dy/dx = f(x,y), each arrow shows the slope at that point. Click anywhere to drop a starting point and trace the solution curve.
dy/dx =
Click anywhere to trace a solution curve.
DEQ 704 Nonlinear PDE Theory +
Objective
To research nonlinear PDEs.
Learning Outcomes
- Apply variational methods.
- Use viscosity solutions.
- Apply fixed-point methods.
- Use compactness methods.
- Discuss blow-up phenomena.
Interactive Activity — Direction Field & Solution Curves
For dy/dx = f(x,y), each arrow shows the slope at that point. Click anywhere to drop a starting point and trace the solution curve.
dy/dx =
Click anywhere to trace a solution curve.
Interactive Activity — 1D Wave Equation
Solve ∂²u/∂t² = c² ∂²u/∂x² on a string fixed at both ends. Pick an initial profile and watch waves propagate, reflect and superpose.
Initial:
c =
5.0
DEQ 705 Dynamical Systems +
Objective
To research nonlinear dynamics.
Learning Outcomes
- Apply infinite-dimensional dynamics.
- Use bifurcation theory.
- Apply chaos theory.
- Use Hamiltonian dynamics.
- Apply ergodic theory.
DEQ 706 Doctoral Seminar +
Objective
To engage with current DE research.
Learning Outcomes
- Present and critique papers.
- Engage with international research.
- Participate in peer review.
- Build a network.
- Develop presentation skills.
DEQ 707 Teaching Practicum +
Objective
To develop teaching skills.
Learning Outcomes
- Plan and deliver lectures.
- Design assessments.
- Apply pedagogical theory.
- Mentor undergraduates.
- Engage in curriculum design.
Interactive Activity — Truth Table Builder
Type a logical expression using p, q, r and operators (AND, OR, NOT). The truth table generates instantly.
Operators:
AND OR NOT XOR -> <->
DEQ 708 PhD Thesis I +
Objective
To produce original research.
Learning Outcomes
- Identify an original research question.
- Conduct literature review.
- Develop methodology.
- Produce preliminary results.
- Present at conferences.
DEQ 709 PhD Thesis II +
Objective
To advance the doctoral research.
Learning Outcomes
- Develop original methodology.
- Generate substantial findings.
- Publish peer-reviewed papers.
- Develop thesis structure.
- Defend methodology.
DEQ 710 PhD Thesis III +
Objective
To consolidate research into a thesis.
Learning Outcomes
- Write 80,000-100,000 word thesis.
- Synthesise multiple contributions.
- Defend viva voce.
- Publish multiple articles.
- Contribute to the field.
Career Pathways
Academic Mathematician
Establish a research career in analysis and PDEs at a leading mathematics department, with postdoctoral positions worldwide.
Research Institute Scientist
Work at institutes such as the Clay Mathematics Institute, MSRI, or Fields Institute on focused mathematical research programmes.
Fluid Dynamics Researcher
Apply Navier-Stokes and related equations to problems in aerospace, ocean modelling, and environmental science.
Physics-Informed AI Researcher
Develop physics-informed neural networks and scientific machine learning models grounded in PDE theory.
Biomedical Modeller
Use differential equations to model tumour growth, neural activity, drug diffusion, and physiological systems.
Quantum Computing Theorist
Apply operator theory and spectral methods from differential equations to quantum algorithms and quantum field theory.
Top Global Universities
Courant Institute (NYU)
University of Cambridge
ETH Zürich
University of Oxford
MIT
Caltech
University of Chicago
Imperial College London
Princeton University
University of Bonn
Why D'Math University
STEP 01
Deep Theoretical Rigour
Differential equations at the doctoral level requires mastery of modern analysis. Our curriculum goes to the full depth of the subject — no shortcuts.
STEP 02
World-Class Supervisors
Supervision by faculty with active research in nonlinear analysis, geometric flows, and stochastic equations — publishing in top mathematics journals.
STEP 03
Interdisciplinary Impact
DEs research connects directly to physics, biology, engineering, and AI — our doctoral candidates work across disciplines with real-world collaborators.
STEP 04
Global Research Network
Annual conference attendance support, international collaboration grants, and visiting researcher exchanges at partner institutions worldwide.
Doctoral applications reviewed year-round — contact us to discuss your research interests.