### Article

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 : nasehpour@gut.ac.ir and nasehpour@gmail.com

**Received**: June 17, 2021; **Revised**: May 13, 2022; **Accepted**: May 31, 2022

### Abstract

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

### 1. Introduction

There is as yet no general solution

where

An example of such use of a lower-order differential equation is in [9] 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 [23] which hit upon a Chini equation. Also, Chini’s method is still under study in applications [17] 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 [1], Formula 0.314 in [5], Theorem 1.6c in [7], [14] Ch. 17 (1st edn.), and Ch. 21 (2nd edn.). We bring it for reference.

**Lemma 1.** Let

where

Optimization of computation of power series expansions has been coined JCPMiller Algorithm in [6]. Later publications of the algorithm are in [7], [14], [22] though the algorithm has a long history. It was apparently known to Euler, according to Henrici [7], p. 65, but Wimp in [22] p. 2 refers to Lord Rayleigh [19] in 1910. Knopp [11], however, attributes it to Glaisher in 1875.

Now we prove the main theorem of this section:

**Theorem 2.** Let

where _{0}

then

and

_{0}

where

Finally, from the differential equation (1.1), we obtain

This completes the proof.

Let us recall the following Theorem 2.4.2 in [3]:

**Theorem 3.** (Existence and Uniqueness for First-Order Nonlinear Equations)

Let the functions

**Corollary 4.** Let

with _{0}

### 3. Automating Power Series Solutions

The first recurrence in Theorem 2 expresses _{m}_{m}

Maple's symbolic output of successive calls of Miller's algorithm _{0})

**Remark 5.** The proof of Theorem 2 also holds if the assumed solution is a formal Laurent series _{0} ≠ 0

**Remark 6.** The recurrences (2.2) and (2.3) of Theorem 1 are such that each _{m}_{0}

An algorithm - embedding previous tools - to find all symbolic solutions of the equation (2.4), whenever the constants _{0}_{0}, A_{1}, C_{1}_{k}_{k}

This algorithm has to be changed slightly if non-units solutions are needed, i.e. the assumed power series _{0}=0

### 4. Applications

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 _{1} ≠ 0

The Claurent algorithm gives coefficients of

### 5. Conclusion

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 _{1}_{1}

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 [12] 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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