Inhomogeneous recurrence relation

In summary, an inhomogeneous recurrence relation is a mathematical equation that describes how a sequence of numbers changes over time. Unlike a homogeneous recurrence relation, which only includes terms involving the previous values in the sequence, an inhomogeneous recurrence relation also includes an external term or constant. This external term can represent any external factors that may affect the sequence. Solving an inhomogeneous recurrence relation involves finding a particular solution and then adding it to the general solution of the corresponding homogeneous recurrence relation. This type of relation is commonly used in fields such as physics, economics, and biology to model dynamic systems.
  • #1
andrew1
20
0
Hi all,

Could someone please explain to me the process involved in converting an inhomogeneous recurrence to a homogeneous recurrence, I'm completely confused as to how it works.Thanks
 
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  • #2
andrew said:
Hi all,

Could someone please explain to me the process involved in converting an inhomogeneous recurrence to a homogeneous recurrence, I'm completely confused as to how it works.Thanks

For semplicity we suppose that we have linear second order difference equations. A homogeneous difference equation is written as...

$\displaystyle a_{n+2} + c_{1}\ a_{n+1}+ c_{0}\ a_{n} = 0\ (1)$

An inhomogeneous difference equation is written as...

$\displaystyle a_{n+2} + c_{1}\ a_{n+1} + c_{0}\ a_{n} = b_{n}\ (2)$

Kind regards

$\chi$ $\sigma$
 
  • #3
chisigma said:
For semplicity we suppose that we have linear second order difference equations. A homogeneous difference equation is written as...

$\displaystyle a_{n+2} + c_{1}\ a_{n+1}+ c_{0}\ a_{n} = 0\ (1)$

An inhomogeneous difference equation is written as...

$\displaystyle a_{n+2} + c_{1}\ a_{n+1} + c_{0}\ a_{n} = b_{n}\ (2)$

Kind regards

$\chi$ $\sigma$

Could you possibly provide an example, this would help me understand it a bit better.
 
  • #4
andrew said:
Could you possibly provide an example, this would help me understand it a bit better.

An example of linear homogeneous second order difference equation is here...

http://mathhelpboards.com/discrete-mathematics-set-theory-logic-15/recursive-sequences-finding-their-expressions-10478.html#post48615

Kind regards

$\chi$ $\sigma$
 
  • #5
chisigma said:
An example of linear homogeneous second order difference equation is here...

http://mathhelpboards.com/discrete-mathematics-set-theory-logic-15/recursive-sequences-finding-their-expressions-10478.html#post48615

Kind regards

$\chi$ $\sigma$

Sorry, I meant an example of an inhomogeneous recurrence relation, I understand how to solve a homogeneous recurrence relation, but converting an inhomogeneous recurrence is where I am struggling.
 
  • #6
andrew said:
Sorry, I meant an example of an inhomogeneous recurrence relation, I understand how to solve a homogeneous recurrence relation, but converting an inhomogeneous recurrence is where I am struggling.

A general procedure to attack inhomogeneous difference equation is illustrated here...

http://mathhelpboards.com/discrete-mathematics-set-theory-logic-15/difference-equation-tutorial-draft-part-ii-860.html#post4671

Kind regards

$\chi$ $\sigma$
 

FAQ: Inhomogeneous recurrence relation

What is an inhomogeneous recurrence relation?

An inhomogeneous recurrence relation is a mathematical equation that describes the relationship between successive terms in a sequence, where the terms on the right side of the equation are not all constants. This means that the equation contains terms that are dependent on previous terms in the sequence, as well as terms that are not related to the previous terms.

How is an inhomogeneous recurrence relation different from a homogeneous recurrence relation?

A homogeneous recurrence relation only contains terms that are dependent on previous terms in the sequence, while an inhomogeneous recurrence relation contains additional terms that are not related to the previous terms. Inhomogeneous recurrence relations are often more complex and difficult to solve than homogeneous ones.

What are some real-world examples of inhomogeneous recurrence relations?

Inhomogeneous recurrence relations can be found in various fields such as physics, economics, and biology. For example, in physics, the motion of a pendulum can be described by an inhomogeneous recurrence relation, where the acceleration term is dependent on the previous position and velocity of the pendulum, as well as external forces such as friction.

How are inhomogeneous recurrence relations solved?

The general solution for an inhomogeneous recurrence relation is a combination of the particular solution and the complementary solution. The particular solution is a specific solution to the inhomogeneous equation, while the complementary solution is a solution to the corresponding homogeneous equation. The particular solution can be found using methods such as variation of parameters or undetermined coefficients.

What are the applications of inhomogeneous recurrence relations?

Inhomogeneous recurrence relations are widely used in various fields of science and engineering to model and solve problems related to sequences and systems. They can be used to describe complex phenomena, such as population growth, chemical reactions, and signal processing. Inhomogeneous recurrence relations also have practical applications in computing, where they are used to create efficient algorithms for tasks such as sorting and searching.

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