Kyungpook Mathematical Journal 2022; 62(4): 729-736
Published online December 31, 2022
Copyright © Kyungpook Mathematical Journal.
Series Solution of High Order Abel, Bernoulli, Chini and Riccati Equations
Henk Koppelaar∗, Peyman Nasehpour
Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Delft, The Netherlands
e-mail : Koppelaar.Henk@gmail.com
Department of Engineering Science, Golpayegan University of Technology, Golpayegan, Iran
e-mail : firstname.lastname@example.org and email@example.com
Received: June 17, 2021; Revised: May 13, 2022; Accepted: May 31, 2022
To help solving intractable nonlinear evolution equations (NLEEs) of waves in the field of fluid dynamics we develop an algorithm to find new high order solutions of the class of Abel, Bernoulli, Chini and Riccati equations of the form
Keywords: Abel equation, Bernoulli equation, Chini equation, JCPMiller algorithm, Riccati equation, series solution
There is as yet no general solution
An example of such use of a lower-order differential equation is in  for the solution of otherwise unsolvable double dispersion equations, i.e. the Sharma-Tasso-Olver (STO) equation. To glimpse other examples of the application of the work reported in this paper is the Benjamin-Ono PDEs in  which hit upon a Chini equation. Also, Chini’s method is still under study in applications  to Gross-Pitaevskii PDEs.
In general, the field of fluid dynamics progresses if the SEM becomes wider applicable by raising the order of the SEM. This paper gives an automated technique, such that high order solutions become feasible for the SEM.
The main result of this paper is to aid application fields with a computer-assisted method for symbolic differential solutions of (1.1), if
2. Power Series Solutions
Recall the formal derivative of power series over a field
an element of the power series ring
The following lemma is known for a long time (see Formula 6.361 in , Formula 0.314 in , Theorem 1.6c in ,  Ch. 17 (1st edn.), and Ch. 21 (2nd edn.). We bring it for reference.
Lemma 1. Let
Optimization of computation of power series expansions has been coined JCPMiller Algorithm in . Later publications of the algorithm are in , ,  though the algorithm has a long history. It was apparently known to Euler, according to Henrici , p. 65, but Wimp in  p. 2 refers to Lord Rayleigh  in 1910. Knopp , however, attributes it to Glaisher in 1875.
Now we prove the main theorem of this section:
Theorem 2. Let
Finally, from the differential equation (1.1), we obtain
This completes the proof.
Let us recall the following Theorem 2.4.2 in :
Theorem 3. (Existence and Uniqueness for First-Order Nonlinear Equations)
Let the functions
Corollary 4. Let
3. Automating Power Series Solutions
The first recurrence in Theorem 2 expresses
Maple's symbolic output of successive calls of Miller's algorithm
Remark 5. The proof of Theorem 2 also holds if the assumed solution is a formal Laurent series
Remark 6. The recurrences (2.2) and (2.3) of Theorem 1 are such that each
An algorithm - embedding previous tools - to find all symbolic solutions of the equation (2.4), whenever the constants
This algorithm has to be changed slightly if non-units solutions are needed, i.e. the assumed power series
Below we give applications by showing outputs of our ABCR algorithm.
Example 7. The solution of Bernoulli's equation
where C is a constant depending on
For the particular case
to the result below, by executing ABCR(1, 1, 0, 7, 2, 1): by Maple. This gives the coefficients of the series solution by
Example 8. All test cases in the table below, taken from the literature, give corect results:
Example 9. Solution of an hitherto unknown Chini equation with
Example 10. Execution of the Laurent version of the algorithm
on a truncated non-unit series for a fourth order equation, for example with
The Claurent algorithm gives coefficients of
An advantage of ourmethod for equations with constant coefficients is the result of a series expansion without effort to analyse whether it has periodic solutions, or not. Our method grants solutions of all types. The series of solutions are obtained by the property discovered by Euler: a particular solution
For the moltenslag problem in metallurgy - a special case of the celebrated NLEEs in fluid flow dynamics - no general solution for the slag temperature
Also, by our method, we generate the needed expansion of the solution of Kudryashov's  SEM (Simplest Equation Method), to simplify otherwise intractable partial differential equations (PDEs) in the fluid dynamics field.
The technique is general, easy to implement via the two algorithms above, and yields exact expansion results.
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