System of nonhomogeneous difference equation

In summary, to solve the system z(t+1)=Az(t)+b where A is a 2x2 matrix and z(t+1), z(t), b are 2x1 matrices, one can use the formula z(t)=P(D^t)(P^-1)z(0) for the homogeneous solution, where D is the diagonal matrix of eigenvalues of A and P is the matrix of eigenvectors. To solve the nonhomogeneous solution at the steady state, one can add the two solutions z(t) for the homogeneous and nonhomogeneous equations. To obtain the particular solution, one can guess a constant vector and use the formula x_{p}=(I-A)^{-1}b. This
  • #1
smilieevah
1
0
How do you solve the system z(t+1)=Az(t)+b where A is a 2x2 matrix and z(t+1), z(t), b are 2x1 matricies?
I solved the homogeneous solution: z(t)=P(D^t)(P^-1)z(0) where D is the diagonal matrix of eigenvalues of A and P is the matrix of eigenvectors.
I tried to solve the nonhomogeneous solution at the steady state where z(t+1)=z(t). I'm not sure if this is the right method.
Then I added the two solutions z(t) for the homogeneous and the nonhomogeneous equations.
 
Physics news on Phys.org
  • #2
Guess a constant vector for the particular solution. This will give
[tex]x_{p}=(I-A)^{-1}b[/tex]
Then add it to the homogeneous solution to obtain the solution.
 

Related to System of nonhomogeneous difference equation

1. What is a system of nonhomogeneous difference equations?

A system of nonhomogeneous difference equations is a set of equations that describe the relationship between the values of one or more variables over discrete time intervals. These equations are "nonhomogeneous" because they include a constant term, unlike homogeneous difference equations where the constant term is equal to zero.

2. How is a system of nonhomogeneous difference equations different from a system of homogeneous difference equations?

In a system of nonhomogeneous difference equations, the equations have a constant term, while in a system of homogeneous difference equations, the constant term is equal to zero. This means that the solutions to nonhomogeneous difference equations will have an additional term, while the solutions to homogeneous difference equations will not.

3. What are some real-world applications of a system of nonhomogeneous difference equations?

System of nonhomogeneous difference equations have various applications in fields such as economics, biology, and physics. For example, it can be used to model population growth, stock market fluctuations, and the spread of diseases.

4. What are the methods for solving a system of nonhomogeneous difference equations?

There are various methods for solving a system of nonhomogeneous difference equations, including the method of undetermined coefficients, variation of parameters, and Laplace transforms. These methods involve finding the particular solution and the general solution to the equations.

5. How can a system of nonhomogeneous difference equations be solved numerically?

A system of nonhomogeneous difference equations can be solved numerically using methods such as Euler's method, Runge-Kutta method, or finite difference method. These methods involve approximating the solutions to the equations using a series of calculations and iterations.

Similar threads

I Fundamental matrix of a second order 2x2 system of ODEs
    • Differential Equations
Replies
2
Views
2K
A Determining whether the Dirichlet problem is nonhomogenous
    • Differential Equations
Replies
2
Views
2K
I On sub and super solutions: Teschl and others
    • Differential Equations
Replies
5
Views
702
I Undetermined coefficients - nonhomogenous system
    • Differential Equations
Replies
1
Views
1K
A Identifying and solving a pair of ODE's
    • Differential Equations
Replies
6
Views
1K
I Question about solving linear first order non-homogeneous ODEs
    • Differential Equations
Replies
3
Views
847
B System of differential equations Basic question
    • Differential Equations
Replies
3
Views
1K
A Determine PDE Boundary Condition via Analytical solution
    • Differential Equations
Replies
1
Views
2K
MHB Matrix Methods for Difference Problems II
    • Differential Equations
Replies
7
Views
2K
I Homogeneous Wave Equation and its Solutions
    • Differential Equations
Replies
3
Views
3K

Hot Threads

Recent Insights

Back
Top