Solving autonomous linear systems of differential/difference equations

[jozko.slaninka]

I would like to ask if anybody knows something about the methods of solving infinite linear autonomous systems of first-order differential (or possibly difference) equations.

There is a well-known method for solving finite-dimensional systems based on the computation of eigenvalues of the system matrix. I wonder if something similar can be done also for infinite-dimensional systems. Perhaps there is a method based on spectral theory...

I am mainly looking for references to literature. I have found a reference to a Russian book:

K.G. Valeev, O.A. Zhautykov, "Infinite systems of differential equations". (this is an English translation of the title)

However, I am quite unable to find this book in local libraries, nor to find out what matters are dealt with in it. If anyone knows this book, I would be grateful for any alternative references dealing with similar matters. As well as for any other references.

 

[Mandelbroth]

I would like to ask if anybody knows something about the methods of solving infinite linear autonomous systems of first-order differential (or possibly difference) equations.

There is a well-known method for solving finite-dimensional systems based on the computation of eigenvalues of the system matrix. I wonder if something similar can be done also for infinite-dimensional systems. Perhaps there is a method based on spectral theory...

I am mainly looking for references to literature. I have found a reference to a Russian book:

K.G. Valeev, O.A. Zhautykov, "Infinite systems of differential equations". (this is an English translation of the title)

However, I am quite unable to find this book in local libraries, nor to find out what matters are dealt with in it. If anyone knows this book, I would be grateful for any alternative references dealing with similar matters. As well as for any other references.

I've found, after cursory inspection...

• "On Infinite Systems of Linear Autonomous and Nonautonomous Stochastic Equations" by T. S. Rybnikova
• "Solution of an Infinite System of Differential Equations of the Analytic Type" by F. R. Moulton
• http://www.jstor.org/discover/10.2307/1989496?uid=3739840&uid=2129&uid=2&uid=70&uid=4&uid=3739256&sid=21102191980473 by William T. Reid
• Stack Exchange is typically my second stop for math research after Wikipedia.
• Or, of course, you could just go ask the magic fairies of internet land. :tongue:

The book Numerical-Analytical Methods in the Theory of Boundary-Value Problems by N. Nikolai Iosifovich Ronto and A. Anatolii Mikhailovich Samoilenko references K.G. Valeev and O.A. Zhautykov's work on infinite systems. That may be a good stop if you can't find their book firsthand.

 

[jozko.slaninka]

I've found, after cursory inspection...

• "On Infinite Systems of Linear Autonomous and Nonautonomous Stochastic Equations" by T. S. Rybnikova
• "Solution of an Infinite System of Differential Equations of the Analytic Type" by F. R. Moulton
• http://www.jstor.org/discover/10.2307/1989496?uid=3739840&uid=2129&uid=2&uid=70&uid=4&uid=3739256&sid=21102191980473 by William T. Reid
• Stack Exchange is typically my second stop for math research after Wikipedia.
• Or, of course, you could just go ask the magic fairies of internet land. :tongue:

The book Numerical-Analytical Methods in the Theory of Boundary-Value Problems by N. Nikolai Iosifovich Ronto and A. Anatolii Mikhailovich Samoilenko references K.G. Valeev and O.A. Zhautykov's work on infinite systems. That may be a good stop if you can't find their book firsthand.

Thanks a lot!

 

[Mandelbroth]

Thanks a lot!

You're most certainly welcome. Math is interesting!

 

[blue_raver22]

I am familiar with the methods for solving finite-dimensional systems of differential or difference equations, but I am not as well versed in the methods for solving infinite-dimensional systems. However, I do know that there are methods based on spectral theory that can be used for solving such systems. These methods involve studying the eigenvalues and eigenvectors of the system matrix, similar to the approach used for finite-dimensional systems.

One potential reference for this topic is the book "Infinite-Dimensional Linear Systems Theory" by João P. Hespanha. This book covers various methods for solving infinite-dimensional linear systems, including spectral theory and operator semigroups.

Another useful resource is the paper "Solving Infinite-Dimensional Linear Systems Using Spectral Theory" by Michal Kocvara and Jan Vlcek. This paper discusses different approaches for solving infinite-dimensional systems, including spectral methods, and provides numerical examples.

Lastly, I would also recommend checking out relevant journal articles and conference proceedings in the field of differential equations and spectral theory for more specific and current information on solving autonomous linear systems of differential or difference equations.