16.3. Formation of Partial Differential Equations

PDEs can be formed by eliminating arbitrary constants or arbitrary functions from a given relation.

The formation of partial differential equations (PDEs) often starts with existing relationships that involve functions and variables. By modifying these relationships, particularly through the removal of arbitrary constants or functions, one can derive a PDE that captures the underlying dynamics of the problem. This process is fundamental in creating mathematical models for physical phenomena in engineering and science.

Think of a cooking recipe that calls for a specific amount of an ingredient (like salt). If you have a general method that involves 'add salt to taste,' you can remove the specific measurement and create your flexible recipe that accounts for personal preference. Similarly, in mathematics, by removing constants or functions, one creates a general formula that can apply to various scenarios.

A. By Eliminating Arbitrary Constants Let the relation involve constants a,b: Example: z = ax + by + ab Differentiate partially: • ∂z = a
∂x
• ∂z = b
∂y
Eliminate a, b to get a PDE.

This method demonstrates how to derive a PDE by eliminating arbitrary constants from an equation. Take the equation z = ax + by + ab, where 'a' and 'b' are constants. By differentiating z with respect to x and y, we obtain expressions for the derivatives. The next step is to manipulate these expressions to eliminate 'a' and 'b,' resulting in a relationship that only involves z along with its partial derivatives, thus forming a PDE.

Imagine you have a general budget formula for purchasing items, like a = price of item 1 and b = price of item 2. The constants in your budget formula could be adjusted based on sales or discounts. By differentiating your budget equation with respect to those items, you could generate a new budgeting rule that doesn't depend on fixed prices but rather on fluctuations in prices.

B. By Eliminating Arbitrary Functions Let the relation be: z = f(x² + y²) Differentiate partially and eliminate the arbitrary function f, or its derivatives f′, to form a PDE.

In this method, we consider a functional relationship involving an arbitrary function instead of constants. For instance, in the equation z = f(x² + y²), 'f' is not a constant but a function of the variable (x² + y²). By partially differentiating with respect to x and y, we can derive expressions that involve the derivatives of 'f.' The goal is to eliminate the function 'f' from the equations, resulting in a PDE that no longer contains 'f' but describes the relationships between z and its derivatives.

Consider a situation where the temperature distribution in a circular region depends on a complex function that indicates temperature at various points (like a profile of heat flow). Instead of using the temperature profile directly, we find ways to express how changes in position affect temperature without specifically relying on that complex function. This simplification helps us understand the general heat distribution without needing to evaluate the complex temperature function.

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