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##  eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2022; 62(4): 729-736

Published online December 31, 2022

### 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

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 y=ayn+by+c,n>1, with constant coefficients a,b,c. The role of this class of equations in NLEEs is explained in the introduction below. The basic algorithm to compute the coefficients of the power series solutions of the class, emerged long ago and is further developed in this paper. Practical application for hitherto unknown solutions is exemplified.

Keywords: Abel equation, Bernoulli equation, Chini equation, JCPMiller algorithm, Riccati equation, series solution

### 1. Introduction

There is as yet no general solution f for the equation

f=afn+bf+c

where f is a function dependent on t, n>1 is a positive integer, and a, b, and c are given constants unequal to 0 in F of characteristic zero. The design of the equation is by D. Bernoulli  with c=0, n =2 , as studied by him to predict the effect of smallpox vaccination. If c0,n=2 then (1.1) is named the Riccati equation with constant coefficients , and n=3 is a special case of the classical Abel equation discussed on p. 24 in . The importance of this equation nowadays originates from the need in the physics of fluid dynamics to develop e.g. accurate weather prediction, neurological implants for deaf and blind people, or control fluids in nano capacities (for medicine administration through the brain-body blood barrier), etc. Fluid dynamics models have the format of partial differential equations (PDEs) of a high order, most often in the class of non-linear evolution Equations (NLEEs). Nature often has a complicated relationship between the phase speed of traveling waves (for instance tsunamis) and the parameters in the PDEs predicting them. The order of an NLEE becomes higher if more precision is required, e.g. dispersion effects are taken into account, but most often these model equations then become unsolvable. Kudryashov  designed the Simplest Equation Method' (SEM) to reduce the order of intractable partial differential equations (PDEs) by assuming a series solution such that it solves a lower order Riccati equation.

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 a, b, and c are constants. To this end, we devise an algorithmic i.e. a constructive, method to find series solutions of any power n>1, with illustrative applications from , , . There is as yet no general solution for this differential equation (1.1), if n>1 and a, b, c are general, i.e. non-constant.

### 2. Power Series Solutions

Recall the formal derivative of power series over a field F with characteristic zero with

f=A0+A1t+A2t2++Aktk+

an element of the power series ring F[[t]], where t is an indeterminate over F. Then is the formal derivative f' of f defined as

f:=A1+2A2t++(k+1)Ak+1tk+.

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 F be a field of characteristic zero and k=0Aktk a formal power series in F[[t]] with the indeterminate t. Then for each positive integer n,

( k=0Aktk)n= k=0Cktk,

where C0=(A0)n, and

Cm=1mA0 k=1 m(knm+k)AkCmk,  for all m1.

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 F be a field of characteristic zero and f= k=0Aktk be a power series solution for the differential equation (1.1)

f=afn+bf+c

where n>1 is a positive integer, and a, b, c, and A0 are given elements in F. If C0=(A0)n and m1,

Cm=1mA0 k=1 m(knm+k)AkCmk,

then

A1=aC0+bA0+c,

and

Am=1m(aCm1+bAm1),  for all m>1.

Proof. Assume that a, b, c, and f(0)=A0 are given elements in F and let f= k=0Aktk be a power series solution for the differential equation (1.1). It is clear that f= k=0(k+1)Ak+1tk. On the other hand, by Lemma 1, a power series raised to the power n is calculated with the formula

( k=0Aktk)n= k=0Cktk,

where C0=(A0)n and

Cm=1mA0 k=1 m(knm+k)AkCmk,  for all m1.

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

A1=aC0+bA0+c,Am=1m(aCm1+bAm1),  for all m>1.

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 f and f/y be continuous in some rectangle n<t<β and γ<y<δ containing the point (t0,y0). Then, in some interval t0h<t<t0+h contained in n<t<β there is a unique solution y=φ(t) of the initial value problem y=f(t,y) and y(t0)=y0.

Corollary 4. Let a,b, and c be arbitrary real constants with a0 and let n>1 be integer. Then, there is a unique solution y=φ(t) satisfying equation (1.1)

dydt=y=ayn+by+c

with y(t0)=y0 on an open interval of real numbers including x0.

Proof. Let us define f(t,y)=ayn+by+c. Then it is obvious that the functions f and f/y are continuous and so by Theorem 3, the solution is unique.

### 3. Automating Power Series Solutions

The first recurrence in Theorem 2 expresses Cm,m=0,1,2, in terms of a, b, c, and Am. The Am are external for this algorithm, i.e. 'global' in Maple's terminology. Rewriting this by a recursive program in the mathematical symbolic programming language Maple, we obtain the listing below of Miller's algorithm with the shortcut name C. Maple's symbolic output of successive calls of Miller's algorithm C(m,4,A0) for m=0,1,,5 is

C(0,4,A0)=A04C(1,4,A0)=4A03A1C(2,4,A0)=4A03A2+6A02A12C(3,4,A0)=4A03A3+12A02A1A2+4A0A13C(4,4,A0)=4A03A4+12A02A1A3+6A02A22+12A0A12A2+A14C(5,4,A0)=4A03A5+12A02A1A4+12A02A2A3+12A0A12A3+12A0A1A22+4A13A2

Remark 5. The proof of Theorem 2 also holds if the assumed solution is a formal Laurent series k=qAktk, where q ≠ 0 is an integer number and may or may not be negative. A polynomial or power series over a field with a nonzero constant term, i.e. A0 ≠ 0 is named unit' in . Note that the formal Laurent series over a field form a field again. The algorithm CLaurent is the Laurent extension of Miller's algorithm C at p. 55 in , made available in Remark 6 hereafter.

Remark 6. The recurrences (2.2) and (2.3) of Theorem 1 are such that each Am can be expressed in terms of a, b, c, and A0, for each m1. Both A and C do necessarily execute in this order: A0C0A1C1Am1Cm1Am. The full computation is given in Theorem 2. The first two steps are A0C0 and subsequently C0A1. Thereafter follow Am=(aCm1+bAm1)/m,for all m>1.

An algorithm - embedding previous tools - to find all symbolic solutions of the equation (2.4), whenever the constants a, b, c, and A0 are given, is such that if these constants are numerically substituted, the algorithm solves correctly Bernoulli, Riccati, Abel as well as Chini's equations . Because of this generality, we name our algorithm ABCR in alphabetic order of the names of the original scholars Abel, Bernoulli, Chini and Riccati. The body of the algorithm ABCR first sets the initial values A0, A1, C1 and thereafter it alternates - as said - between updating Ck and Ak until the stop condition is satisfied. This algorithm has to be changed slightly if non-units solutions are needed, i.e. the assumed power series f has A0=0. Then the call to Miller's C algorithm needs to replaced by a call to the CLaurent algorithm, as follows. ### 4. Applications

Below we give applications by showing outputs of our ABCR algorithm.

Example 7. The solution of Bernoulli's equation c=0 in (2.4) is known in general

(1Ce(n1)t1)1n1,

where C is a constant depending on a,b and a boundary condition y(0).

For the particular case y(t)=(y(t))2+y(t),y(0)=1, let n=2,a=1,b=1,c=0 in (1.1), and collate the known solution

12et1=12(1+coth(t212ln(2)))

to the result below, by executing ABCR(1, 1, 0, 7, 2, 1): by Maple. This gives the coefficients of the series solution by seq(Aktk,k=07);. The output is the exact sequence of coefficients of expansion of the known solution.

1,2t,3t2,133t3,254t4,54160t5,1561120t6,472932520t7,

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 n=9, a=1, b=1, c=1 in (1.1):

1,3t,152t2,612t3,10618t4,2391740t5,13310548t6,220129571680t7,

Example 10. Execution of the Laurent version of the algorithm

seq(CLaurent(m,4,1,A1),m=07)

on a truncated non-unit series for a fourth order equation, for example with q=1, hence A1 ≠ 0, and a1t+a2t2++a7t7.

The Claurent algorithm gives coefficients of Cm,m=07, as follows.

0,0,0,0,A14,4A13A2,4A13A3+6A12A22,4A13A4+12A12A2A3+4A1A23.

### 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 f1 generates a series of solutions u via the substitution f = u + 1/f1.

For the moltenslag problem in metallurgy - a special case of the celebrated NLEEs in fluid flow dynamics - no general solution for the slag temperature y(t) flowing out of the furnace was found thus far, see . Our method solves this problem in full generality. The slag outflow temperature is fully known and predictable by our symbolic output. Our result is new to this metallurgy application field.

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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