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Dr Robert Van Gorder

(MSc, PhD)Generic Silhouette

Senior Lecturer
Director of Studies for 100-level Mathematics

Office: Science III, room 212
Tel +64 3 479 7783
Email robert.vangorder@otago.ac.nz

About

I obtained BS (2009) and PhD (2014) degrees in mathematics from the University of Central Florida (UCF), USA, where I held a Trustees Doctoral Fellowship (2009-2011) and then a National Science Foundation Graduate Research Fellowship (2011-2014). After completion of my PhD, I moved to the Mathematical Institute at the University of Oxford, UK, and took up the post of Research Fellow in Nonlinear Dynamics (2014-2015), followed by a Glasstone Research Fellowship in Science (2015-2018). In 2019, I joined the faculty of the Department of Mathematics and Statistics at the University of Otago as a Senior Lecturer.

Teaching responsibilities

Robert's teaching responsibilities include:

  • MATH 140 Fundamentals of Modern Mathematics 2
  • MATH 302 Complex Analysis
  • MATH4A1 Techniques in Applied Mathematics 1
  • MATH4A2 Techniques in Applied Mathematics 2

Research Interests

In my research I seek to better understand how physical phenomena can be described, predicted, and even modified using tools from applied mathematics. I specialize in the trifecta of mathematical modelling (writing down relations between physical quantities within the language of mathematics... this is where all of those equations come from!), analytical and asymptotic solution methods (deriving an exact or approximate solution to a mathematical problem using pen-and-paper approaches), and numerical simulations (using a computer program to solve a mathematical problem). I apply these methods to study physical phenomena, primarily those falling into one of the following areas:

Fluid dynamics

If you’re alive, you’re interacting with fluids: The air you breathe and the water you drink are examples of fluids. Despite the ubiquitous nature of fluids, there are still many questions we have about their behavior. My interests in fluids include better understanding the dynamics governing fundamental structures – such as vortices, bubbles, waves, and boundary layers – that can then be used as building blocks for more complex fluid flows. What forms can these structures take? Do they persist, or break apart over time? What happens if we attempt to manipulate or control these structures? These are the kinds of questions I am interested in.

Spatial instabilities and pattern formation

Diffusive instabilities – such as the Turing and Benjamin-Feir instabilities – have been proposed as mechanisms for the formation of patterns in many real-world systems, ranging from spot formation on the coats of big cats to optical turbulence in lasers. I am interested in understanding how these instability mechanisms extended to more generic non-autonomous or spatially heterogeneous systems, where they result in messier yet perhaps more realistic patterning. These heterogeneous systems arise from models of reaction-diffusion processes such as chemical reactions in the presence of thermal forcing, fluid flows, or evolving space domains, and are found in applications ranging from physics and chemistry to biology and epidemiology. Under what conditions can spatial or spatiotemporal patterns emerge from such systems? How might these patterns be controlled?

Quantum theory, quantum fluids, and nonlinear waves

My interests in theoretical physics include quantum mechanics and quantum field theory, with a particular focus on modelling problems in low-temperature physics and condensed matter physics as quantum fluids. Some of my work in this area has involved understanding the dynamics Bose-Einstein condensates, quantized vortex filaments in superfluid helium, and confined quantum systems. While these topics are farther removed from our daily experiences, they exhibit exotic and interesting behaviors, providing a number of fascinating scientific challenges. Nonlinear waves arise in these and other applications, and I am interested in understanding how nonlinear waves evolve under various conditions relevant to realistic experimental configurations, such as confined space domains or external forcing.

I am happy to supervise PhD and honours projects in any of these areas.

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Publications

Luckins, E. K., Oliver, J. M., Please, C. P., Sloman, B. M., & Van Gorder, R. A. (2022). Modelling and analysis of an endothermic reacting counter-current flow. Journal of Fluid Mechanics, 949, A21. doi: 10.1017/jfm.2022.702

Luckins, E. K., Oliver, J. M., Please, C. P., Sloman, B. M., Valderhaug, A. M., & Van Gorder, R. A. (2022). Modelling alternating current effects in a submerged arc furnace. IMA Journal of Applied Mathematics, 87, 492-520. doi: 10.1093/imamat/hxac012

Ritchie, J. S., Krause, A. L., & Van Gorder, R. A. (2022). Turing and wave instabilities in hyperbolic reaction–diffusion systems: The role of second-order time derivatives and cross-diffusion terms on pattern formation. Annals of Physics. Advance online publication. doi: 10.1016/j.aop.2022.169033

Van Gorder, R. A. (2022). Adiabatic soliton management: Controlling solitary wave motion while keeping the wave envelope unchanged. Physics Letters A. Advance online publication. doi: 10.1016/j.physleta.2022.128284

Van Gorder, R. A. (2022). Finite time blowup of incompressible flows surrounding compressible bubbles evolving under soft equations of state. Proceedings of the Royal Society A, 478(2262), 20220172. doi: 10.1098/rspa.2022.0172

Vajravelu, K., & Van Gorder, R. A. (2012). Nonlinear flow phenomena and homotopy analysis: Fluid flow and heat transfer. Heidelberg, Germany: Springer, 190p. doi: 10.1007/978-3-642-32102-3

Authored Book - Research

Van Gorder, R. A. (2014). Stability of auxiliary linear operator and convergence-control parameter in the homotopy analysis method. In S. Liao (Ed.), Advances in the homotopy analysis method. (pp. 123-180). World Scientific. doi: 10.1142/9789814551250_0004

Chapter in Book - Research

Luckins, E. K., Oliver, J. M., Please, C. P., Sloman, B. M., & Van Gorder, R. A. (2022). Modelling and analysis of an endothermic reacting counter-current flow. Journal of Fluid Mechanics, 949, A21. doi: 10.1017/jfm.2022.702

Journal - Research Article

Luckins, E. K., Oliver, J. M., Please, C. P., Sloman, B. M., Valderhaug, A. M., & Van Gorder, R. A. (2022). Modelling alternating current effects in a submerged arc furnace. IMA Journal of Applied Mathematics, 87, 492-520. doi: 10.1093/imamat/hxac012

Journal - Research Article

Ritchie, J. S., Krause, A. L., & Van Gorder, R. A. (2022). Turing and wave instabilities in hyperbolic reaction–diffusion systems: The role of second-order time derivatives and cross-diffusion terms on pattern formation. Annals of Physics. Advance online publication. doi: 10.1016/j.aop.2022.169033

Journal - Research Article

Van Gorder, R. A. (2022). Adiabatic soliton management: Controlling solitary wave motion while keeping the wave envelope unchanged. Physics Letters A. Advance online publication. doi: 10.1016/j.physleta.2022.128284

Journal - Research Article

Van Gorder, R. A. (2022). Finite time blowup of incompressible flows surrounding compressible bubbles evolving under soft equations of state. Proceedings of the Royal Society A, 478(2262), 20220172. doi: 10.1098/rspa.2022.0172

Journal - Research Article

Erban, R., & Van Gorder, R. A. (2021). On the sum of positive divisors functions. Research in Number Theory, 7, 25. doi: 10.1007/s40993-021-00240-6

Journal - Research Article

Luckins, E. K., Oliver, J. M., Please, C. P., Sloman, B. M., & van Gorder, R. A. (2021). Homogenised model for the electrical current distribution within a submerged arc furnace for silicon production. European Journal of Applied Mathematics. Advance online publication. doi: 10.1017/s0956792521000243

Journal - Research Article

Van Gorder, R. A. (2021). A theory of pattern formation for reaction–diffusion systems on temporal networks. Proceedings of the Royal Society A, 477, 20200753. doi: 10.1098/rspa.2020.0753

Journal - Research Article

Van Gorder, R. A. (2021). Pattern formation from spatially heterogeneous reaction-diffusion systems. Philosophical Transactions of the Royal Society A, 379, 20210001. doi: 10.1098/rsta.2021.0001

Journal - Research Article

Van Gorder, R. A. (2021). Time-varying Bose–Einstein condensates. Proceedings of the Royal Society A, 477, 20210443. doi: 10.1098/rspa.2021.0443

Journal - Research Article

Van Gorder, R. A., Kamilova, A., Birkeland, R. G., & Krause, A. L. (2021). Locating the baking isotherm in a Søderberg electrode: Analysis of a moving thermistor model. SIAM Journal on Applied Mathematics, 81(4), 1691-1716. doi: 10.1137/20M1314276

Journal - Research Article

Van Gorder, R. A., Klika, V., & Krause, A. L. (2021). Turing conditions for pattern forming systems on evolving manifolds. Journal of Mathematical Biology, 82, 4. doi: 10.1007/s00285-021-01552-y

Journal - Research Article

González-Fariña, R., Münch, A., Oliver, J. M., & van Gorder, R. A. (2020). Modeling microsilica particle formation and growth due to the combustion reaction of silicon monoxide with oxygen. SIAM Journal on Applied Mathematics, 80(2), 1003-1033. doi: 10.1137/19M1287080

Journal - Research Article

Krause, A. L., & Van Gorder, R. A. (2020). A non-local cross-diffusion model of population dynamics II: Exact, approximate, and numerical traveling waves in single- and multi-species populations. Bulletin of Mathematical Biology, 82, 113. doi: 10.1007/s11538-020-00787-y

Journal - Research Article

Sloman, B. M., Please, C. P., & Van Gorder, R. A. (2020). Melting and dripping of a heated material with temperature-dependent viscosity in a thin vertical tube. Journal of Fluid Mechanics, 905, A16. doi: 10.1017/jfm.2020.727

Journal - Research Article

Taylor, N. P., Kim, H., Krause, A. L., & Van Gorder, R. A. (2020). A non-local cross-diffusion model of population dynamics I: Emergent spatial and spatiotemporal patterns. Bulletin of Mathematical Biology, 82, 112. doi: 10.1007/s11538-020-00786-z

Journal - Research Article

Van Gorder, R. A. (2020). Influence of temperature on Turing pattern formation. Proceedings of the Royal Society A, 476(2240), 20200356. doi: 10.1098/rspa.2020.0356

Journal - Research Article

Van Gorder, R. A. (2020). Perturbation theory for Bose-Einstein condensates on bounded space domains. Proceedings of the Royal Society A, 476, 20200674. doi: 10.1098/rspa.2020.0674

Journal - Research Article

Van Gorder, R. A. (2020). Turing and Benjamin–Feir instability mechanisms in non-autonomous systems. Proceedings of the Royal Society A, 476(2238). doi: 10.1098/rspa.2020.0003

Journal - Research Article

Van Gorder, R. A., Krause, A. L., Malomed, B. A., & Kaup, D. J. (2020). Unstaggered-staggered solitons on one- and two-dimensional two-component discrete nonlinear Schrödinger lattices. Communications in Nonlinear Science & Numerical Simulation, 85, 105244. doi: 10.1016/j.cnsns.2020.105244

Journal - Research Article

Xu, Y., Krause, A. L., & Van Gorder, R. A. (2020). Generalist predator dynamics under Kolmogorov versus non-Kolmogorov models. Journal of Theoretical Biology, 486, 110060. doi: 10.1016/j.jtbi.2019.110060

Journal - Research Article

Conti, F., & Van Gorder, R. A. (2019). The role of network structure and time delay in a metapopulation Wilson-Cowan model. Journal of Theoretical Biology, 477, 1-13. doi: 10.1016/j.jtbi.2019.05.010

Journal - Research Article

Fadai, N. T., Please, C. P., & Van Gorder, R. A. (2019). Modelling structural deformations in a roasting coffee bean. International Journal of Non-Linear Mechanics, 110, 123-130. doi: 10.1016/j.ijnonlinmec.2018.12.006

Journal - Research Article

Fussell, E. F., Krause, A. L., & Van Gorder, R. A. (2019). Hybrid approach to modeling spatial dynamics of systems with generalist predators. Journal of Theoretical Biology, 462, 26-47. doi: 10.1016/j.jtbi.2018.10.054

Journal - Research Article

Horner, C. L., & Van Gorder, R. A. (2019). Dynamics of quantized vortex filaments under a local induction approximation with second-order correction. Physics of Fluids, 31(6), 065103. doi: 10.1063/1.5091567

Journal - Research Article

Hua, T., & Van Gorder, R. A. (2019). Wave propagation and pattern formation in two-dimensional hexagonally-packed granular crystals under various configurations. Granular Matter, 21, 3. doi: 10.1007/s10035-018-0852-8

Journal - Research Article

Kiradjiev, K. B., Halvorsen, S. A., Van Gorder, R. A., & Howison, S. D. (2019). Maxwell-type models for the effective thermal conductivity of a porous material with radiative transfer in the voids. International Journal of Thermal Sciences, 145, 106009. doi: 10.1016/j.ijthermalsci.2019.106009

Journal - Research Article

Perkins, M., & Van Gorder, R. A. (2019). Closed-form calculation of infinite products of Glaisher-type related to Dirichlet series. Ramanujan Journal, 49, 371-389. doi: 10.1007/s11139-018-0037-4

Journal - Research Article

Sachak-Patwa, R., Fadai, N. T., & Van Gorder, R. A. (2019). Modeling multi-group dynamics of related viral videos with delay differential equations. Physica A, 521, 197-217. doi: 10.1016/j.physa.2019.01.052

Journal - Research Article

Van Gorder, R. A. (2019). Optimal homotopy analysis and control of error for implicitly defined fully nonlinear differential equations. Numerical Algorithms, 81, 181-196. doi: 10.1007/s11075-018-0540-0

Journal - Research Article

Van Gorder, R. A., Kim, H., & Krause, A. L. (2019). Diffusive instabilities and spatial patterning from the coupling of reaction-diffusion processes with Stokes flow in complex domains. Journal of Fluid Mechanics, 877, 759-823. doi: 10.1017/jfm.2019.620

Journal - Research Article

Van Gorder, R. A., Krause, A. L., & Kwiecinski, J. A. (2019). Amplitude death criteria for coupled complex Ginzburg–Landau systems. Nonlinear Dynamics, 19, 151-159. doi: 10.1007/s11071-019-04961-3

Journal - Research Article

Zhou, Z., & Van Gorder, R. A. (2019). Turing instability and colony formation in spatially extended Rosenzweig-MacArthur predator-prey models with allochthonous resources. Bulletin of Mathematical Biology, 81, 5009-5053. doi: 10.1007/s11538-019-00667-0

Journal - Research Article

Baines, L. W. S., & Van Gorder, R. A. (2018). Soliton wave-speed management: Slowing, stopping, or reversing a solitary wave. Physical Review A, 97, 063814. doi: 10.1103/PhysRevA.97.063814

Journal - Research Article

Bakhtiari, P., Abbasbandy, S., & Van Gorder, R. A. (2018). Reproducing kernel method for the numerical solution of the 1D Swift–Hohenberg equation. Applied Mathematics & Computation, 339, 132-143. doi: 10.1016/j.amc.2018.07.006

Journal - Research Article

Bakhtiari, P., Abbasbandy, S., & Van Gorder, R. A. (2018). Solving the Dym initial value problem in reproducing kernel space. Numerical Algorithms, 78(2), 405-421. doi: 10.1007/s11075-017-0381-2

Journal - Research Article

Bhatt, A., & Van Gorder, R. A. (2018). Chaos in a non-autonomous nonlinear system describing asymmetric water wheels. Nonlinear Dynamics, 93(4), 1977-1988. doi: 10.1007/s11071-018-4301-3

Journal - Research Article

Eide, R. M., Krause, A. L., Fadai, N. T., & Van Gorder, R. A. (2018). Predator-prey-subsidy population dynamics on stepping-stone domains with dispersal delays. Journal of Theoretical Biology, 451, 19-34. doi: 10.1016/j.jtbi.2018.04.038

Journal - Research Article

Fadai, N. T., Please, C. P., & Van Gorder, R. A. (2018). Asymptotic analysis of a multiphase drying model motivated by coffee bean roasting. SIAM Journal on Applied Mathematics, 78(1), 418-436. doi: 10.1137/16M1095500

Journal - Research Article

Flora Meng, X., Van Gorder, R. A., & Porter, M. A. (2018). Opinion formation and distribution in a bounded-confidence model on various networks. Physical Review E: Statistical, Nonlinear, & Soft Matter Physics, 97(2), 022312. doi: 10.1103/PhysRevE.97.022312

Journal - Research Article

Jansen, J. E., & Van Gorder, R. A. (2018). Dynamics from a predator-prey-quarry-resource-scavenger model. Theoretical Ecology, 11(1), 19-38. doi: 10.1007/s12080-017-0346-z

Journal - Research Article

Kekić, A., & Van Gorder, R. A. (2018). Wave propagation across interfaces induced by different interaction exponents in ordered and disordered Hertz-like granular chains. Physica D, 384-385, 18-33. doi: 10.1016/j.physd.2018.07.007

Journal - Research Article

Kennedy, D. T., & Van Gorder, R. A. (2018). Motion of open vortex-current filaments under the Biot-Savart model. Journal of Fluid Mechanics, 836, 532-559. doi: 10.1017/jfm.2017.826

Journal - Research Article

Kleanthous, A., Hua, T., Manai, A., Yawar, K., & Van Gorder, R. A. (2018). Dynamics from a mathematical model of a two-state gas laser. Physica A, 497, 26-40. doi: 10.1016/j.physa.2017.12.110

Journal - Research Article

Krause, A. L., Beliaev, D., Van Gorder, R. A., & Waters, S. L. (2018). Bifurcations and dynamics emergent from lattice and continuum models of bioactive porous media. International Journal of Bifurcation & Chaos, 28(11), 1830037. doi: 10.1142/S0218127418300379

Journal - Research Article

Krause, A. L., Burton, A. M., Fadai, N. T., & Van Gorder, R. A. (2018). Emergent structures in reaction-advection-diffusion systems on a sphere. Physical Review E: Statistical, Nonlinear, & Soft Matter Physics, 97(4), 042215. doi: 10.1103/PhysRevE.97.042215

Journal - Research Article

Krause, A. L., Kurowski, L., Yawar, K., & Van Gorder, R. A. (2018). Stochastic epidemic metapopulation models on networks: SIS dynamics and control strategies. Journal of Theoretical Biology, 449, 35-52. doi: 10.1016/j.jtbi.2018.04.023

Journal - Research Article

Kwiecinski, J. A., & Van Gorder, R. A. (2018). Dynamics of nearly parallel interacting vortex filaments. Journal of Fluid Mechanics, 835, 575-623. doi: 10.1017/jfm.2017.792

Journal - Research Article

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