Given the task of finding an example of differential equations in the world of applied mathematics posed an interesting task. The decision to to choose welding was made in an effort to connect complicated, high level mathematics, with a still complicated (but comparatively simple) process. While there are many different kinds of welding that can be done, one type in particular was chosen as it is the most exemplary process for discussing the application of differential equations.

Gas Tungsten Arc Welding (GTAW) – commonly known in the industry as ‘tig’, an acronym for ‘Tungsten Inert Gas’ – is used in the manufacturing of metal structures for many fields of expertise. It is perhaps most commonly known for being used in areas such as aerospace technology and the medical device industry, in addition to many of the simpler constructions used in modern day-to-day life.


An analytical solution for heat flow during the welding process is based on conduction heat transfer for predicting the shape of the weld pool for two and three-dimensional welds. The Fourier PDE (partial differential equation) of heat conduction was used in conjunction with the moving coordinate system to develop solutions for the point and line heat sources, which allowed for the analysis of the welding process, and included considerations for many of the parameters key to welding: current, voltage, welding speed, and weld geometry, among others.

Fourier Transform

Heat Transfer

How It Works:


Many factors can affect the quality and durability of a weld. These can be both environmental and material factors. Higher grade machines offer the user the ability to control the weld with high precision through the use of complicated settings that can impact the most minute details. This allows each individual user to fine tune the machine to best fit their own welding style. Because of this – and the physical and mental dexterity required for high quality GTAW process welds – there is no one equation that can define how a weld is mathematically made. Instead, the welding machine uses a series of interchanging differential equations that constantly adapt to the input by the user.

An example of this is given: a foot pedal is used to dictate the amount of amperage output by the machine at any given time, which effects the travel speed, and the weld penetration. The travel speed is also affected by the travel angle, which in turn affects the heat affected zone (HAZ), which can impact the weld penetration.


Professor Holly Ashton

Scott Mannering

Professor P Ravinder Reddy