By Dongming Wang

ISBN-10: 3764373687

ISBN-13: 9783764373689

This e-book provides the cutting-edge in tackling differential equations utilizing complicated tools and software program instruments of symbolic computation. It makes a speciality of the symbolic-computational elements of 3 varieties of primary difficulties in differential equations: remodeling the equations, fixing the equations, and learning the constitution and homes in their options. The 20 chapters are written via best specialists and are based into 3 parts.

The e-book is worthy interpreting for researchers and scholars engaged on this interdisciplinary topic yet can also function a important reference for everybody attracted to differential equations, symbolic computation, and their interaction.

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Additional info for Differential Equations with Symbolic Computation

Example text

Ideally, to obtain the common solutions for the intersections of the first five quadratic terms, we would like to compute their Gr¨ o¨bner basis with respect to a lexicographic ordering. Unfortunately, the computations are sufficiently difficult that such an approach does not work directly. However, we can compute a Gr¨ o¨bner basis with a total degree ordering, which is much more efficient computationally, but less useful for obtaining explicit equations for the zeros of the polynomials. If the ideal was zero dimensional, then we could use the FGLM algorithm to return efficiently to a lexicographic basis, but in our case, it turns out that there is a non-trivial one dimensional solution (also shared by the sixth quadratic term).

5. 4) are also isochronous center conditions. 5) holds. 5) and the recursive formulas in Appendix B, we have, after careful computations, τ5 = 2b21 , −(a03 b03 ) + 4 λ , τ10 = 2 −(a12 2 b03 ) − a03 b12 2 τ15 = , 8 −(a03 2 b03 2 ) τ20 = . 5), it is easy to get the following result. 6. 17) λ = a21 = b21 = a30 = b30 = a03 = b03 = 0. 3) are zero. 7, the origin is an isochronous center. 3) becomes dz 2 a12 w6 z 5 3 b12 w4 z 7 =z+ + , dT 5 5 dw 3 a12 w7 z 4 2 b12 w5 z 6 = −(w + + ). 19) becomes u (10 u − 11 a12 v + b12 u12 v 9 − 2 a12 b12 u11 v 10 + a12 2 b12 u10 v 11 ) du = = U, dT 10 (u − a12 v) dv v (−10 u + 9 a12 v + b12 u12 v 9 − 2 a12 b12 u11 v 10 + a12 2 b12 u10 v 11 ) = = −V.

We note that an immediate corollary of the work is that there are components of the center variety of the class of all cubic systems which have codimension 12. ˙ adek. The result for cubic systems was first shown by Zol ¸ However, the system he considers is different from ours. This is because, as noted in his paper, it is not possible to generate 11 limit cycles from his system by considering the linear terms only. The nice thing about the result here is that it depends on only the simplest arguments and a direct calculation.

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Differential Equations with Symbolic Computation by Dongming Wang

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