Solutions of first-order matrix differential equations

In summary: Then, using the QR decomposition, we can write \frac{d}{dt}(B\psi) = Q\left ( \frac{dB}{dt}B^{-1}\right )B^{-1} + R\left ( \frac{dB}{dt}B^{-1}\right ) which is of the form Q\left (B\right )B^{-1} + R\left (B\right )
  • #1
Haorong Wu
413
89
Homework Statement
How to solve the following matrix differential equation, ##[A(t)+B(t) \partial_t]\left | \psi \right >=0 ##, where ##A(t)## and ##B(t)## are ##n\times n## matrices and ##\left | \psi \right >## is a ##n##-vector.
Relevant Equations
None
Hello, there. I am trying to solve the differential equation, ##[A(t)+B(t) \partial_t]\left | \psi \right >=0 ##. However, ##A(t)## and ##B(t)## can not be simultaneous diagonalized. I do not know is there any method that can apprixmately solve the equation.

I suppose I could write the equation as ##\partial_t \left | \psi \right >=-B^{-1}(t) A(t)\left | \psi \right > ## with general solutions being ##\left | \psi \right >=\exp \left ( \int_0^t -B^{-1}(t') A(t')dt'\right ) \left | c\right > ## and ##\left | c\right > ## is a constant vector. Then a first-order approximation may be ##\left | \psi \right >=\left (I+ \int_0^t -B^{-1}(t') A(t')dt'\right ) \left | c\right > ##.

I am not familiar with matrix differential equations. Does this method have any restrictions or problems? Or there may be other better approximation solutions? Any references would be greatly appreciated.

Thanks!
 
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  • #2
Haorong Wu said:
Homework Statement:: How to solve the following matrix differential equation, ##[A(t)+B(t) \partial_t]\left | \psi \right >=0 ##, where ##A(t)## and ##B(t)## are ##n\times n## matrices and ##\left | \psi \right >## is a ##n##-vector.
Relevant Equations:: None

Hello, there. I am trying to solve the differential equation, ##[A(t)+B(t) \partial_t]\left | \psi \right >=0 ##. However, ##A(t)## and ##B(t)## can not be simultaneous diagonalized. I do not know is there any method that can apprixmately solve the equation.

I suppose I could write the equation as ##\partial_t \left | \psi \right >=-B^{-1}(t) A(t)\left | \psi \right > ## with general solutions being ##\left | \psi \right >=\exp \left ( \int_0^t -B^{-1}(t') A(t')dt'\right ) \left | c\right > ## and ##\left | c\right > ## is a constant vector. Then a first-order approximation may be ##\left | \psi \right >=\left (I+ \int_0^t -B^{-1}(t') A(t')dt'\right ) \left | c\right > ##.

I am not familiar with matrix differential equations. Does this method have any restrictions or problems? Or there may be other better approximation solutions? Any references would be greatly appreciated.

Thanks!
I don't understand what you did. Starting from as ##\partial_t \left | \psi \right >=-B^{-1}(t) A(t)\left | \psi \right > ##, wouldn't you get ##\left | \psi \right >= \left ( \int_0^t -B^{-1}(t') A(t')dt'\right ) ##? You might be mixing up the concept of an integrating factor with the solution of a differential equation. It's been nearly 25 years since I worked on this stuff, so I could be mistaken.

Nit: Also, you have used the symbol ##\partial_t##. Since the matrices and vector are functions of a single variable t, the ordinary derivative would be more suitable, IMO.
 
  • #3
Thanks, @Mark44. There is a ##\left | \psi \right >## in the rhs, so it is like the equation as ##y'=\alpha y##, which solution is exponential functions. Also, thanks for the suggestions about ##\partial_t##, but I use ##\partial_t## for partial and ordinary derivatives if no confusion could occur.
 
  • #4
Haorong Wu said:
Thanks, @Mark44. There is a ##\left | \psi \right >## in the rhs, so it is like the equation as ##y'=\alpha y##, which solution is exponential functions.
OK, I understand. I'm not so familiar with the notation ##|\psi>##, as that's probably more of a physics notation rather than one used in mathematics.
 
  • #5
Mark44 said:
OK, I understand. I'm not so familiar with the notation ##|\psi>##, as that's probably more of a physics notation rather than one used in mathematics.
Sorry for the confusion. It can be treated as a vector.
 
  • #6
Assuming [itex]B[/itex] is invertible, we can rewrite the ODE as [tex]
\frac{d}{dt}(B\psi) + \left[ AB^{-1} - \frac{dB}{dt}B^{-1}\right](B\psi) = 0[/tex] which is of the standard form [tex]
\frac{du}{dt} + C(t)u = 0.[/tex] But in general there is no closed form solution unless [itex]C[/itex] commutes with [itex]\int_0^t C(s)\,ds[/itex], when the solution is [tex]u(t) = \exp\left(\int_0^t C(s)\,ds\right)u(0).[/tex]
 

Related to Solutions of first-order matrix differential equations

1. What is a first-order matrix differential equation?

A first-order matrix differential equation is an equation that involves a matrix, its derivatives, and one or more independent variables. It can be written in the form of X'(t) = A(t)X(t) + B(t), where X(t) is the vector of unknown functions, A(t) is the matrix of coefficients, and B(t) is the vector of known functions.

2. What is the general solution of a first-order matrix differential equation?

The general solution of a first-order matrix differential equation is a set of functions that satisfies the equation for all values of the independent variable. It can be expressed in the form of X(t) = e∫A(t)dt[C + ∫e-∫A(t)dtB(t)dt], where C is a vector of arbitrary constants.

3. How do you solve a first-order matrix differential equation using the matrix exponential?

The matrix exponential method involves finding the eigenvalues and eigenvectors of the matrix A(t) and using them to construct the matrix exponential e∫A(t)dt. The general solution can then be expressed as X(t) = e∫A(t)dt[C + ∫e-∫A(t)dtB(t)dt], where C is a vector of arbitrary constants.

4. Can a first-order matrix differential equation have multiple solutions?

Yes, a first-order matrix differential equation can have multiple solutions. This is because the general solution contains an arbitrary constant vector C, which can take on different values for different initial conditions. In other words, the solution to the equation is not unique and can vary depending on the initial conditions.

5. How are first-order matrix differential equations used in real-world applications?

First-order matrix differential equations are commonly used in physics, engineering, and other scientific fields to model and analyze systems that involve multiple variables and their rates of change. They are particularly useful in studying systems that can be described using matrices, such as electrical circuits, chemical reactions, and population dynamics.

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