Solving the PDEs in a mathematical model makes it possible to predict the outcome of an experiment and helps engineers and scientists understand the process or phenomenon that is described by that mathematical model.

We thus assume that PDEs can be used to describe the laws of physics. In this sense, x, y, z, and t are independent of temperature. However, a value for T, taken from the solution of the PDEs, does not automatically give us the position in time and space. In other words, T depends on x, y, z, and t. We can select a position in space and time and get a unique value for T by solving the partial differential equations. In such a case, temperature ( T) is called the dependent variable. The law for the conservation of energy in solids and fluids can be described as a PDE expressing the changes of temperature, T, in space. Here, t, x, y, and z are called the independent variables. Assuming that we know the solution at time t and all positions ( x, y, z), we can use a PDE that describes a system to numerically estimate the solution after a very small change in time and all positions. For example, a change of a system in time (t) and space ( x, y, and z) can be described with a PDE. More specifically, partial differential equations (PDEs) describe such changes in more than one independent variable. 1)ĭifferential equations describe the change of a system rather than its state over space and time. He further states: “There is only one precise way of presenting the laws, and that is by means of differential equations.” (Ref. He mentions that one possible approach to get a good idea of the behavior of a system is by looking at the solution of the differential equations that describe such a system for different circumstances. In “Lectures on Physics”, Richard Feynman discusses the analysis of physical problems. For example, the law for the conservation of energy can not only be applied to matter but also to related concepts such as electromagnetic radiation. The laws of physics define the rules, as we observe them, for the motion of matter and related concepts in space and time. Physics, PDEs, and Numerical Modeling The Laws of Physics, Mathematical Models, and PDEs

Balanced equation science phenomena