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VisualPDE: Rapid Interactive Simulations of Partial Differential Equations

Walker, Benjamin J.; Townsend, Adam K.; Chudasama, Alexander K.; Krause, Andrew L.

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Benjamin J. Walker

Alexander K. Chudasama


Computing has revolutionised the study of complex nonlinear systems, both by allowing us to solve previously intractable models and through the ability to visualise solutions in different ways. Using ubiquitous computing infrastructure, we provide a means to go one step further in using computers to understand complex models through instantaneous and interactive exploration. This ubiquitous infrastructure has enormous potential in education, outreach and research. Here, we present VisualPDE, an online, interactive solver for a broad class of 1D and 2D partial differential equation (PDE) systems. Abstract dynamical systems concepts such as symmetry-breaking instabilities, subcritical bifurcations and the role of initial data in multistable nonlinear models become much more intuitive when you can play with these models yourself, and immediately answer questions about how the system responds to changes in parameters, initial conditions, boundary conditions or even spatiotemporal forcing. Importantly, VisualPDE is freely available, open source and highly customisable. We give several examples in teaching, research and knowledge exchange, providing high-level discussions of how it may be employed in different settings. This includes designing web-based course materials structured around interactive simulations, or easily crafting specific simulations that can be shared with students or collaborators via a simple URL. We envisage VisualPDE becoming an invaluable resource for teaching and research in mathematical biology and beyond. We also hope that it inspires other efforts to make mathematics more interactive and accessible.


Walker, B. J., Townsend, A. K., Chudasama, A. K., & Krause, A. L. (2023). VisualPDE: Rapid Interactive Simulations of Partial Differential Equations. Bulletin of Mathematical Biology, 85(11), Article 113.

Journal Article Type Article
Acceptance Date Sep 25, 2023
Online Publication Date Oct 12, 2023
Publication Date Nov 1, 2023
Deposit Date Oct 9, 2023
Publicly Available Date Oct 18, 2023
Journal Bulletin of Mathematical Biology
Print ISSN 0092-8240
Electronic ISSN 1522-9602
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 85
Issue 11
Article Number 113
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