mathematics

Partial differential algebraic equation

Partial differential algebraic equation is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Partial differential algebraic equation rather than just read about it. In short: In mathematics a partial differential algebraic equation (PDAE) set is an incomplete system of partial differential equations that is closed with a set of algebraic equations. Definition A general PDAE is defined as: 0 = F ( x , y , ∂ y i ∂ x j , ∂ 2 y i ∂ x j ∂ x k , … , z ) , {\displaystyle 0=\mathbf {F} \left(\mathbf {x} ,\mathbf {y} ,{\frac {\partial y_{i}}{\partial x_{j}}},{\frac {\partial ^{2}y_{i}}{\partial x…

Key takeaways

Partial differential algebraic equation belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Partial differential algebraic equation to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Partial differential algebraic equation from memory before moving on to harder problems.

Reference excerpt

In mathematics a partial differential algebraic equation (PDAE) set is an incomplete system of partial differential equations that is closed with a set of algebraic equations.

Definition A general PDAE is defined as:

0 = F ( x , y , ∂ y i ∂ x j , ∂ 2 y i ∂ x j ∂ x k , … , z ) , {\displaystyle 0=\mathbf {F} \left(\mathbf {x} ,\mathbf {y} ,{\frac {\partial y_{i}}{\partial x_{j}}},{\frac {\partial ^{2}y_{i}}{\partial x_{j}\partial x_{k}}},\ldots ,\mathbf {z} \right),}

where:

F is a set of arbitrary functions; x is a set of independent variables; y is a set of dependent variables for which partial derivatives are defined; and z is a set of dependent variables for which no partial derivatives are defined. The relationship between a PDAE and a partial differential equation (PDE) is analogous to the relationship between an ordinary differential equation (ODE) and a differential algebraic equation (DAE). PDAEs of this general form are challenging to solve. Simplified forms are studied in more detail in the literature. Even as recently as 2000, the term "PDAE" has been handled as unfamiliar by those in related fields.

Solution methods Semi-discretization is a common method for solving PDAEs whose independent variables are those of time and space, and has been used for decades. This method involves removing the spatial variables using a discretization method, such as the finite volume method, and incorporating the resulting linear equations as part of the algebraic relations. This reduces the system to a DAE, for which conventional solution methods can be employed.

References

Worked examples

Example 1 — a first encounter with Partial differential algebraic equation

Start with the simplest possible case. Write down what Partial differential algebraic equation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Partial differential algebraic equation before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Partial differential algebraic equation ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Partial differential algebraic equation

In research: Partial differential algebraic equation appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Partial differential algebraic equation in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Partial differential algebraic equation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Multivariable calculus, Numerical analysis, Partial differential equations, so understanding it makes those chapters shorter.
In everyday life: Look for Partial differential algebraic equation outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Partial differential algebraic equation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Partial differential algebraic equation means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Partial differential algebraic equation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Partial differential algebraic equation in simple terms?

In mathematics a partial differential algebraic equation (PDAE) set is an incomplete system of partial differential equations that is closed with a set of algebraic equations. Definition A general PDAE is defined as: 0 = F ( x , y , ∂ y i ∂ x j , ∂ 2 y i ∂ x j ∂ x k , … , z ) , {\displaystyle 0=\ma…

Why does Partial differential algebraic equation matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Partial differential algebraic equation?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Partial differential algebraic equation.