Generalization of Exp-function and other standard function methods
Transcript of Generalization of Exp-function and other standard function methods
Generalization of Exp-function and other standard function methods
Zenonas Navickas a, Liepa Bikulciene a, Minvydas Ragulskis b,*
aDepartment of Applied Mathematics, Kaunas University of Technology, Studentu 50-325, Kaunas LT-51368, LithuaniabResearch Group for Mathematical and Numerical Analysis of Dynamical Systems, Kaunas University of Technology, Studentu 50-222, Kaunas LT-51368, Lithuania
a r t i c l e i n f o
Keywords:
The Exp-function method
Multiplicative operator
H-rank
Standard functions
a b s t r a c t
Exp-function and other standard function methods for solving nonlinear differential equa-
tions are generalized in this paper. An analytical criterion determining if a solution can be
expressed in a form comprising standard functions is derived. New computational algo-
rithm for automatic identification of the structure of the solution is constructed. The algo-
rithm provides information if the solution can be expressed as a sum of standard functions,
a ratio of sums of standard functions, or even a more complex algebraic form involving
standard functions. Several examples are used to illustrate the proposed concept.
� 2010 Elsevier Inc. All rights reserved.
1. Introduction
The Exp-function method was proposed in 2006 by He and Wu [1] to seek solitary solutions, periodic solutions and com-
pacton-like solutions of nonlinear differential equations. It has been demonstrated that the Exp-function method, with the
help of symbolic computation, provides a powerful mathematical tool for solving high-dimensional nonlinear evolutions in
mathematical physics. The Exp-function method has been exploited for the determination of exact solutions of many non-
linear differential equations [2–5]; we give only a few of many references available.
Several alternative modifications of the Exp-function method have been developed. The tanh, extended tanh, improved
tanh and generalized tanh function methods [6–9], sech and rational Exp-function method [10], the modified simplest equa-
tion method [11] and many similar standard function methods are also successfully used for the construction of solutions of
nonlinear differential equations.
An analytical criterion determining if a solution of a differential equation can be expressed in an analytical form compris-
ing exponential functions is developed in [12]. The employment of this criterion does not only give an answer to the above-
stated question but gives the structure of the solution so that one does not have to guess what the form of the solution is. The
load of symbolic calculations is brought before the structure of the solution is identified. This is in contrary to the Exp-func-
tion type methods where the structure of the solution is first guessed, and then symbolic calculations are exploited for the
identification of parameters.
The object of this paper is to develop and generalize Exp-function type methods. We seek three objectives in this process:
(i) To investigate possibilities to express solutions of differential equations in forms comprising not only exponential,
hyperbolic tangent functions, but other standard functions (for example radical functions). The objective is to form
a class of standard functions which could be used to express solutions of a widest possible class of differential
equations.
0096-3003/$ - see front matter � 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2010.03.083
* Corresponding author.
E-mail addresses: [email protected] (Z. Navickas), [email protected] (L. Bikulciene), [email protected] (M. Ragulskis).
URL: http://www.personalas.ktu.lt/~mragul (M. Ragulskis).
Applied Mathematics and Computation 216 (2010) 2380–2393
Contents lists available at ScienceDirect
Applied Mathematics and Computation
journal homepage: www.elsevier .com/ locate /amc
(ii) To develop a convenient computational algorithm for the identification of the structure of the solution. These algo-
rithms must eliminate the need for guessing how the structure of the solution looks like. In this sense our approach
is an alternative (and much more general) method compared to Exp-function type methods where the structure of the
solution is first guessed, and only then symbolic computations are used to calculate appropriate coefficients. Our algo-
rithm identifies the structure of the solution automatically and provides information on how the solution is expressed
in a form comprising a standard function (if it is a sum of standard functions, a ratio of sums of standard functions, or
even a more complex form involving standard functions).
(iii) It is well known that eigenvalues and eigenvectors play a central role in analysis of linear differential equations. It
would be of high interest to find analogies of eigenvalues and eigenvectors for nonlinear differential equations and
dynamical systems. We propose an insight into nonlinear dynamical systems through the definition of the H-rank,
the Hankel characteristic equation and its solutions.
2. Preliminary definitions
2.1. Functions and their extensions
Definition 1. A function f ðx; s0; . . . ; sn�1Þ, where
f :¼ f ðx; s0; . . . ; sn�1Þ ¼X
þ1
j¼0
pjðs0; . . . ; sn�1Þðx� cÞj
j!; jx� cj < Rf ð1Þ
is called an extended function if its domain is extended from its region of convergence jx� cj < Rf into a wider region
(the real axis or the complex plane) with the exception of a limited number of special singular points x1; . . . ; xmwhere limx!xi jf ðxÞj ¼ þ1; i ¼ 1;2; . . . ;m; where x; c 2 R; fs0; s1; . . . ; sn�1g is a finite set of parameters; n is the number of
parameters; s0; s1; . . . ; sn�1 2 C; complex functions pjðs0; . . . ; sn�1Þ are polynomials, ratios of polynomials or radicals of
polynomials.
Fx;s0 ;...;sn�1is denoted as a set of extended functions.
Definition 2. A function
yðxÞ ¼X
þ1
j¼0
qj
j!xj; ð2Þ
where qj; q0 ¼ 1; qj ¼Qj�1
j¼0ðaþ bkÞ; qj – 0; j ¼ 1;2; . . . are coefficients of the expansion, is called a standard function.
We will use several standard functions for the construction of generalized solutions of differential equations (we analyze
these solutions only on the real x-axis):
y1ðxÞ ¼X
þ1
j¼0
xj
j!; y1ðxÞ ¼ expðxÞ; x 2 R; qj ¼ 1;
y2ðxÞ ¼X
þ1
j¼0
xj; jxj < 1; y2ðxÞ ¼1
1� x; x – 1; qj ¼ j!;
y3ðxÞ ¼X
þ1
j¼0
ð2j� 1Þ!!j!
xj; ð�1Þ!! ¼ 1; 1!! ¼ 1; 3!! ¼ 1 � � �3; . . . ; jxj < 1
2; y3ðxÞ ¼
1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 2xp ; x <
1
2; qj ¼ ð2j� 1Þ!!:
ð3Þ
Necessary order derivatives of the described functions do exist and this fact will be not stressed further on.
2.2. Differential and multiplicative operators
Definition 3. The generalized differential operator is defined as [13]:
D :¼ Q0Ds0 þ � � � þ Qn�1Dsn�1; ð4Þ
where Q0; . . . ;Qn�1 2 Fs0 ;...;sn�1; Fs0 ;...;sn�1
is a set of extended functions which do not depend on x; Fs0 ;...;sn�1� Fx;s0 ;...;sn�1
; Dsj are
conventional linear differential operators (partial differentiation is performed in respect of a parameter sjÞ;D0
sj:¼ 1; j ¼ 0;1; . . . ;n� 1; 1 is an identical linear operator.
Z. Navickas et al. / Applied Mathematics and Computation 216 (2010) 2380–2393 2381
Usual properties of differentiation hold for the generalized differential operator [13]:
DX
m
j¼1
cjfj
!
¼X
m
j¼1
cj � Dfj;
Dnðfk � flÞ ¼X
n
j¼0
n
j
� �
Djfk
� �
Dn�jfl
� �
;
Dfm ¼ mf
m�1Df ;
Dfkfl¼ ðDfkÞfl � fkðDflÞ
f 2l;
� � �m; n ¼ 0;1; . . . ; cj 2 C; j ¼ 1;2; . . . ;m; k; l ¼ 1;2; . . . ;n; f ; fk; fl 2 Fx;s0 ;...;sn�1
:
ð5Þ
It can be noted that the generalized differential operator sometimes is referred to as a vector field [15].
Definition 4. The multiplicative operator [13] is a linear operator:
G :¼ GD :¼X
þ1
j¼0
xj
j!Dj: ð6Þ
It can be noted that the multiplicative operator is denoted as the exponential operator GD :¼ exD in [15,16].
We will enumerate several properties of the multiplicative operator:
GX
m
j¼1
cjfj ¼X
m
j¼1
cjGfj;
Gf ðc; s0; . . . ; sn�1Þ ¼ f ðGc;Gs0; . . . ;Gsn�1Þ;
Gfkðc; s0; . . . ; sn�1Þflðc; s0; . . . ; sn�1Þ
¼ fkðGc;Gs0; . . . ;Gsn�1ÞflðGc;Gs0; . . . ;Gsn�1Þ
;
ð7Þ
where cj 2 C; j ¼ 1;2; . . . ;m; f ðx; s0; . . . ; sn�1Þ; f kðx; s0; . . . ; sn�1Þ; f lðx; s0; . . . ; sn�1Þ 2 Fx;s0 ;...;sn�1.
For example,
exDccn ¼ ðxþ cÞn; exDc f ðc; s0; . . . ; sn�1Þ ¼ f ðc þ x; s0; . . . ; sn�1Þ: ð8Þ
2.3. Operator method for solving ordinary differential equations
2.3.1. First order ordinary differential equations, the initial value problem
The exact solution of a differential equation
y0x ¼ P1ðx; yÞ; yðx0; sÞ ¼ s; x0 2 R; y 2 Fx;s0 ð9Þ
reads [13]:
y ¼X
þ1
j¼0
ðx� x0Þjj!
ððDc þ P1ðc; sÞDsÞjsÞ�
�
�
�
�
c¼x0
; ð10Þ
where the superscript denotes a full derivative and the subscript denotes the variable of differentiation. It can be noted that
the solution is expressed in a form of an infinite order polynomial.
2.3.2. nth order ordinary differential equations, the initial value problem
The exact solution of an initial value problem
yðnÞx ¼ Pn x; y; y0x; . . . ; yðn�1Þx
�
;
yðjÞx ðx; s0; s1; . . . ; sn�1Þ�
�
x¼x0¼ sj; x0 2 R; Pnðx; s0; s1; . . . ; sn�1Þ 2 Fx;s0 ;...;sn�1
ð11Þ
reads:
y ¼X
þ1
j¼0
ðx� x0Þjj!
Djs0
� �
�
�
�
�
�
c¼x0
; ð12Þ
where
D :¼ Dc þ s1Ds0 þ s2Ds1 þ � � � þ sn�1Dsn�2þ Pnðc; s0; s1; . . . ; sn�1ÞDsn�1
: ð13Þ
2382 Z. Navickas et al. / Applied Mathematics and Computation 216 (2010) 2380–2393
Example 1. Let Dnþ1s0 ¼ 0, but Dns0 – 0. Then, Dnþks0 ¼ 0 for k ¼ 1;2; . . ., and yðx; s0; s1; . . . ; sn�1Þ ¼Pn
j¼0ðx�x0Þj
j! Djs0
� �
jc¼x0;
meaning that the function yðx; s0; s1; . . . ; sn�1Þ is an nth order polynomial of x. It can be noted that fcDnsjc 2 Cg#KerD.
2.3.3. Systems of first order ordinary differential equations, the initial value problem
The exact solution of a system of differential equations
z0x ¼ Rðx; z;u; vÞ; zðx0; s0; s1; s2Þ ¼ s0;
u0x ¼ Pðx; z;u; vÞ;uðx0; s0; s1; s2Þ ¼ s1;
v0x ¼ Qðx; z;u; vÞ;vðx0; s0; s1; s2Þ ¼ s2;
8
>
<
>
:
ð14Þ
reads:
z ¼X
þ1
j¼0
ðx� x0Þjj!
Djs0
� �
�
�
�
�
�
c¼x0
;
u ¼X
þ1
j¼0
ðx� x0Þjj!
Djs1
� �
�
�
�
�
�
c¼x0
;
v ¼X
þ1
j¼0
ðx� x0Þjj!
Djs2
� �
�
�
�
�
�
c¼x0
;
ð15Þ
where D ¼ Dc þ Rðc; s0; s1; s2ÞDs0 þ Pðc; s0; s1; s2ÞDs1 þ Qðc; s0; s1; s2ÞDs2 .
Example 2. The generalized differential operator for a system
z0x ¼ 1; zðx0; c; sÞ ¼ c;
u0x ¼ u
z;uðx0; c; sÞ ¼ s;
(
ð16Þ
reads:D ¼ Dc þ scDs. Thus,D
0c ¼ c; Dc ¼ 1; D0s ¼ s; Ds ¼ sc; butD2c ¼ D2s ¼ 0. Thus, theexpressionon the solution ðz;uÞ reads:
z ¼ c þ ðx� x0Þ;
u ¼ sþ s
cðx� x0Þ:
ð17Þ
It can be noted that the system (16) is equivalent to the differential equation xy0 ¼ y which general solution is
y ¼ ax; a ¼ s
c2 C
� �
: ð18Þ
2.4. The H-rank of a sequence
A Hankel matrix [14] for the sequence ðpj; j ¼ 0;1;2; . . .Þ where pj 2 Fx;s0 ;...;sn�1or pj 2 C is defined as:
Hn :¼
p0 p1 � � � pn�1
p1 p2 � � � pn
� � �pn�1 pn � � � p2n�2
2
6
6
6
4
3
7
7
7
5
; n ¼ 1;2; . . . : ð19Þ
Definition 5. A sequence ðpj; j 2 Z0Þ has an H-rank Hrðpj; j 2 Z0Þ ¼ m; m 2 Z0;m < þ1 if the sequence of determinants of
Hankel matrixes has the following form: ðd1; d2; . . . ; dm; 0;0; . . .Þ; where dm – 0; dmþ1 ¼ dmþ2 ¼ � � � ¼ 0.
Definition 6. The length of a sequence ðpj; j 2 Z0Þ is Lðpj; j 2 Z0Þ ¼ m; m 2 Z0, if the sequence has the form
ðp0; p1; . . . ; pm;0;0; . . .Þ; where pm – 0; pmþ1 ¼ pmþ2 ¼ � � � ¼ 0.
Corollary 1. If Lðpj; j 2 Z0Þ ¼ m; m < þ1, then Hrðpj; j 2 Z0Þ ¼ m.
Definition 7. The characteristic Hankel equation for a sequence ðpj; j 2 Z0Þ which H-rank is equal to m is defined as:
det
p0 p1 � � � pm
p1 p2 � � � pmþ1
� � �pm�1 pm � � � p2m�1
1 q � � � qm
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
¼ 0: ð20Þ
Z. Navickas et al. / Applied Mathematics and Computation 216 (2010) 2380–2393 2383
Expansion of the determinant in Eq. (20) yields an mth order algebraic equation for determination of roots of the character-
istic equation:
qm þ Am�1qm�1 þ � � � þ A1qþ A0 ¼ 0: ð21Þ
Corollary 2. The following statement holds true [14]:Let Hrðpj; j 2 Z0Þ ¼ m and the multiplicity of roots q1;q2; . . . ;ql of the char-
acteristic equation (Eq. (21)) is accordingly m1;m2; . . . ;ml;Pl
r¼1mr ¼ m. Then,
pj ¼X
l
r¼1
X
mr�1
k¼0
lrk
j
k
� �
qj�kr ; ð22Þ
where lrk;qr 2 C.
The opposite statement also holds true. If Eq. (22) holds, then
Hrðpj; j 2 Z0Þ ¼ m1 þm2 þ � � � þml: ð23Þ
Definition 8. A sequence is entitled as an algebraic sequence if elements of that sequence ðpj; j 2 Z0Þ are defined by Eq. (22).
We assume that lrk
jk
� �
qj�kr ¼ 0 if
jk
� �
¼ 0 what is true when 0 6 j < k, wherejk
� �
¼ j!k!ðj�kÞ!.
Corollary 3. In case when all roots of the characteristic equation are different, Eq. (22) obtains a more simple form:
pj ¼X
m
r¼1
lrqjr: ð24Þ
It can be noted that coefficients lrk (or just lrÞ can be found solving the linear algebraic system of equations (q1;q2; . . . ;ql
are already determined):
X
l
r¼1
X
mr�1
k¼0
j
k
� �
qj�kr lrk ¼ pj; j ¼ 0;1; . . . ;m� 1: ð25Þ
This system of equations always has the only solution [14].
It can be also noted that one can use Eq. (22) or Eq. (24) to calculate all elements of the series ðpj; j 2 Z0Þ starting from p2m
if Hrðpj; j 2 Z0Þ ¼ m and the first 2m elements of that series are known.
Example 3. Lets consider a sequence ðp0; p1; p2; 0;0; . . .Þ where p2 – 0 and pj ¼ 0; j ¼ 3;4; . . .. Then, detH3 ¼ �p32 – 0; but
detHj ¼ 0 for j ¼ 4;5; . . .. Thus, Hrðp0; p1; p2;0; 0; . . .Þ ¼ 3. It is clear that
p0 p1 p2 0
p1 p2 0 0
p2 0 0 0
1 q q2 q3
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
¼ 0; ð26Þ
what yields q3p3 ¼ 0; q1 ¼ q2 ¼ q3 ¼ 0. Therefore, finally:
pj ¼ p00j þ p1
j
1
� �
0j�1 þ p2
j
2
� �
0j�2; j ¼ 0;1;2; . . . ; where 00
:¼ 1; 01 ¼ 02 ¼ � � � ¼ 0: ð27Þ
2.5. Changing the independent variable of a differential equation
The independent variable of a differential equation can be changed in order to produce a more convenient form of the
exact solution of a differential equation.
Let an invertible function is given by following equalities:
x :¼ uðzÞ; z ¼ wðxÞ: ð28Þ
Then following expressions hold:
z0x ¼ w0x ¼
1
u0z
;
z00xx ¼ � 1
u0z
� 2u00
zz � z0x ¼ � u00zz
u0z
� 3;
� � � :
ð29Þ
2384 Z. Navickas et al. / Applied Mathematics and Computation 216 (2010) 2380–2393
Also, for every function
y ¼ yðxÞ ¼ yðuðzÞÞ :¼ xðzÞ ¼ xðwðxÞÞ ¼ x ð30Þ
following equalities hold:
y0x ¼ x0z
�
�
z¼wðxÞ � w0x ¼
1
u0z
�x0z
�
�
z¼wðxÞ;
y00xx ¼ x00zz
�
�
z¼wðxÞ � w0x
� 2 þx0z
�
�
z¼wðxÞ � w00xx ¼
1
u0z
� 2�x00
zz
�
�
z¼wðxÞ �u00
zz
u0z
� 3�x0
z
�
�
z¼wðxÞ;
� � � :
ð31Þ
Thus,
y0x�
�
x¼uðzÞ ¼1
u0z
�x0z;
y00xx�
�
x¼uðzÞ ¼1
u0z
� 2�x00
zz �u00
zz
u0z
� 3�x0
z;
� � � :
ð32Þ
Merging Eqs. (29)–(32) into one equality and using symbol s to identify the change of variable in the differential equation
yields:
s x; y; y0x; y00xx; . . .
�
:¼ xjx¼uðzÞ; yjx¼uðzÞ; y0xjx¼uðzÞ; y
00xxjx¼uðzÞ; � � �
� �
; ð33Þ
or
s x; y; y0x; y00xx; . . .
�
:¼ uðzÞ;x;1
u0z
�x0z;
1
u0z
� 2�x00
zz �u00
zz
u0z
� 3�x0
z; � � � !
: ð34Þ
For example, the image of a second order differential equation (Eq. (11) at n ¼ 2) takes the following form after the variable
change x ¼ uðzÞ:
1
u0z
� 2x00
zz �u00
zz
u0z
� 3x0
z ¼ P2 uðzÞ;x;1
u0z
x0z
� �
: ð35Þ
Example 4. Several typical variable changes are listed below:
(i) Exponential variable change expðaxÞ ¼ z:
s x; y; y0x; y00xx; . . .
�
:¼ 1
aln z;x;azx0
z;a2 z2x00
zz þ zx0z
�
; . . .
� �
: ð36Þ
(ii) Logarithmic variable change a ln x ¼ z:
s x; y; y0x; y00xx; . . .
�
:¼ expz
a
� �
;x; exp � z
a
� �
x0z; exp �2z
a
� �
x00zz �x0
z
�
; . . .
� �
: ð37Þ
(iii) Symmetric variable change ax¼ z:
s x; y; y0x; y00xx; . . .
�
:¼ az;x;� z2
ax0
z;2z3
a2x0
z þz4
a2x00
zz; . . .
� �
: ð38Þ
where a is a parameter of the variable change; a 2 R;a– 0.
3. Main theorems
We will prove one of the fundamental properties of the generalized differential operator. This property can be used to
express a solution of a differential equation in a form comprising standard functions.
Z. Navickas et al. / Applied Mathematics and Computation 216 (2010) 2380–2393 2385
3.1. Theorem 1
The formulation of the Theorem.
(a) A generalized differential operator D :¼ Q1Ds1 þ � � � þ QnDsn ; Q1; . . . ;Qn 2 Fs1 ;...;sn is used to calculate functions
pj :¼ Djs; pj 2 Fs1 ;...;sn ; j ¼ 0;1; . . ..
(b) A rule q0 :¼ 1; qjþ1 ¼Qjk¼0ðaþ bkÞ; j ¼ 0;1;2; . . . is used to construct a sequence ðqj; j 2 Z0Þ; a; b 2C are such con-
stants that qj – 0.
(A) Then, three following statements can have a sense (but can be individually true or not true):
(i) pj ¼ qj
Pmr¼1lrk
jr , where kk;lk 2 Fs1 ;...;sn ; kk – kl and lk – 0 when k– l and k; l ¼ 1;2; . . . ;m; m 2 N and is a
fixed constant; j ¼ 0;1;2; . . ..
(ii) Hr 1qjpj; j 2 Z0
� �
¼ n; n 2 N and is a fixed constant. Also, roots of Hankel characteristic equation
q1;q2; . . . ;qn 2 Fs1 ;...;sn are all different.
(iii) There exists a set of functions c1; c2; . . . ; c�n;r1;r2; . . . ;r�n 2 Fs0 ;...;sn�1(�n 2 N and is a fixed constant) which sat-
isfy following conditions:
ðaÞck – cl; rk – 0;
ðbÞr1 þ r2 þ � � � þ r�n ¼ s;
ðcÞDrk ¼ arkck;
ðdÞDck ¼ bc2k ;
ð39Þ
where k– l; k ¼ 1;2; . . . ; �n;a; b 2 C.
(B) Statements (i), (ii) and (iii) are equivalent.
Proof
1. The equivalency of statements (i) and (ii) follows from results proven in [14]. Also, m ¼ n and kk ¼ qk when
k ¼ 1;2; . . . ;m.
2. Lets assume that the statement (iii) holds true. Then, p0 ¼ D0s ¼ s ¼ q0s;
p1 ¼ DX
�n
r¼1
rr ¼X
�n
r¼1
Drr ¼ aX
�n
r¼1
rrcr ¼ q1
X
�n
r¼1
rrcr:
We make a proposition that Djs ¼ pj ¼Qj�1
l¼0ðaþ blÞP�nr¼1rrc
jr ; j 2 N and is a fixed constant. Then,
Djþ1s ¼ Dpj ¼Y
j�1
l¼0
ðaþ blÞX
�n
r¼1
ðDrrÞcjr þ ri Dcjr
� �
¼Y
j�1
l¼0
ðaþ blÞX
�n
r¼1
arrcjþ1r þ jrrbc
jþ1r
�
¼Y
j
l¼0
ðaþ blÞX
�n
r¼1
rrcjþ1r ¼ pjþ1:
ð40ÞThus, the statement (ii) (and the statement (i)) holds true.
3. Lets assume that statements (i) and (ii) hold true. Then,
Djþ1s ¼ qjDX
m
r¼1
lrkjr ¼ qj
X
m
r¼1
kjrðDlrÞ þ jlrkj�1r ðDkrÞ
� �
: ð41Þ
Thus, the following system of equations is produced:
X
m
r¼1
kjrðDlrÞ þ jlrkj�1r ðDkrÞ
� �
¼ 1
qj
qjþ1
X
m
r¼1
lrkjþ1r ; ð42Þ
which is satisfied if following identities hold: Dlr ¼ alrkr and Dkr ¼ blrk2r . Thus, the system of equations has at least one
solution. On the other hand, the extended Van-Der-Mond determinant of coefficients at unknowns Dlr and Dkr (when j
sweeps over any of natural numbers 0 6 j1 < j2 < � � � < j2mÞ
D ¼
kj11 k
j12 � � � kj1m j1l1k
j1�11 j1l2k
j1�12 � � � j1lmk
j1�1m
kj21 k
j22 � � � kj2m j2l1k
j2�11 j2l2k
j2�12 � � � j2lmk
j2�1m
� � � � � � � � � � � � � � � � � � � � � � � �kj2m1 k
j2m2 � � � kj2mm j2ml1k
j2m�11 j2ml2k
j2m�12 � � � j2mlmk
j2m�1m
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
¼ l1l2 � . . . � lm
kj11 k
j12 � � � kj1m j1k
j1�11 j1k
j1�12 � � � j1k
j1�1m
kj21 k
j22 � � � kj2m j2k
j2�11 j2k
j2�12 � � � j2k
j2�1m
� � � � � � � � � � � � � � � � � � � � � � � �kj2m1 k
j2m2 � � � kj2mm j2mk
j2m�11 j2mk
j2m�12 � � � j2mk
j2m�1m
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
ð43Þ
2386 Z. Navickas et al. / Applied Mathematics and Computation 216 (2010) 2380–2393
cannot be equivocally equal to zero. Thus, the solution of the system of equations is the only one. Finally, the validity of the
statement (iii) follows from the validity of statements (i) and (ii) at m ¼ �n; kk ¼ ck; lk ¼ rk; k ¼ 1;2; . . . ; �n; a ¼ a and
b ¼ b. h
It can be noted that Theorem 1 holds only when all roots of the Hankel characteristic equation are different.
Theorem 1 forms the theoretical background for the construction of general solutions of differential equations. Let a se-
quence ðDjs; j 2 Z0Þ satisfy equivalent statements (i), (ii) and (iii) of the Theorem 1. Then, a general solution
y ¼ yðx; s0; s1; . . . ; sn�1Þ of a differential equation can be expressed in the following form:
yðx; s0; s1; . . . ; sn�1Þ ¼X
þ1
j¼0
ðqj
X
m
r¼1
lrkjrÞðx� x0Þj
j!¼X
m
r¼1
lr
X
þ1
j¼0
qj
j!ðkrðx� x0ÞÞj ¼
X
m
r¼1
lrf ðkrðx� x0ÞÞ; ð44Þ
where f ðxÞ is a standard function defined in Definition 2.
3.2. Theorem 2
Roots of the Hankel characteristic equation coincide with roots of the standard characteristic equation
km � am�1km�1 � � � � � a1k� a0 ¼ 0 for linear differential equations with constant coefficients.
Proof. The initial value problem reads:
yðmÞx ¼ am�1y
ðm�1Þx þ � � � þ a1y
0x þ a0y; y
ðrÞx ðx; s0; . . . ; sm�1Þjx¼x0
¼ sr ;
r ¼ 0;1; . . . ;m� 1; a0; a1; . . . ; am�1 2 R:ð45Þ
We select qj ¼ 1; j ¼ 0;1;2; . . ..
Then, the generalized differential operator takes the following form (Eq. (13)):
D ¼ s1Ds0 þ s2Ds1 þ � � � þ sm�1Dsm�2þ ðam�1sm�1 þ � � � þ a1s1 þ a0s0ÞDsm�1
: ð46Þ
It can be noted that �a0s0 � a1Ds0 � � � � � am�1Dm�1s0 þ Dms0 ¼ 0 because
�a0s0 � a1s1 � � � � � am�1sm�1 þ am�1sm�1 þ � � � þ a1s1 þ a0s0 ¼ 0. Thus,
�a0Djs0 � a1D
jþ1s0 � � � � � am�1Djþm�1s0 þ Djþms0 ¼ 0; j ¼ 0;1;2; . . .. The last equalities yield:
dmþ1 ¼ det
D0s0 Ds0 � � � Dms0
Ds0 D2s0 � � � Dmþ1s0
� � � � � � � � � � � �Dms0 Dmþ1s0 � � � D2ms0
2
6
6
6
4
3
7
7
7
5
¼ 0: ð47Þ
But
dm ¼ det
D0s0 � � � Dm�1s0
� � � � � � � � �Dm�1s0 � � � D2m�2s0
2
6
4
3
7
5– 0: ð48Þ
Thus, the sequence of determinants is ðd1; d2; . . . ; dm;0; 0; . . .Þ. Therefore, HrðDjs; j ¼ 0;1;2; . . .Þ ¼ m. Moreover, b ¼ 0 because
qj ¼ 1. Next, one has to find roots of the Hankel characteristic equation q1;q2; . . . ;qm.
If all roots are different, then Djs0 ¼Pmr¼1lrðs0; s1; . . . ; sm�1Þqj
r . Thus, Dlr ¼ lrqr ;Dqr ¼ 0; r ¼ 1;2; . . . ;m� 1. Finally,
y ¼X
m
r¼1
lrðs0; s1; . . . ; sm�1Þ expðqrðx� x0ÞÞ: � ð49Þ
It can be noted that in general case not only roots but also their multiplicities of the Hankel characteristic equation coin-
cide with roots of the standard characteristic equation if only the equation is a linear differential equation with constant
coefficients.
3.3. The case when some roots of the Hankel characteristic equation are multiple
It is clear that Theorem 1 does not hold when some roots of the Hankel characteristic equation are multiple. In that case
the structure of the solution becomes more complex and cannot be expressed a form comprising only standard functions
(the independent variable also figures in the expression then). We will give a short discussion and an example though this
is an object of the future research.
Let Djs ¼ qj
Plr¼1
Pmr�1k¼0 lrk
jk
� �
qj�kr . In other words, multiplicities of roots of the Hankel characteristic equation
q1;q2; . . . ;ql are m1;m2; . . . ;ml accordingly. Then, Eq. (12) yields:
Z. Navickas et al. / Applied Mathematics and Computation 216 (2010) 2380–2393 2387
X
þ1
j¼0
qj
j
k
� �
qj�kr
ðx� x0Þjj!
�
�
�
�
�
c¼x0
¼ ðx� x0Þkk!
X
þ1
j¼k
qj
ðj� kÞ! ðqrðx� x0ÞÞj�k�
�
�
c¼x0
¼ ðx� x0Þkk!qk
r
X
þ1
j¼0
qj
j!ðqrðx� x0ÞÞj
!ðkÞ
x
�
�
�
�
�
�
c¼x0
¼ ðx� x0Þkk!qk
r
ðf ðqrðx� x0ÞÞÞðkÞx
�
�
�
c¼x0
: ð50Þ
Therefore,
y ¼ Pþ1
j¼0
qj
P
l
r¼1
P
mr�1
k¼0
lrk
j
k
� �
qj�kr
ðx�x0Þjj!
�
�
�
c¼x0
¼ Pl
r¼1
P
mr�1
k¼0
lrk
P
þ1
j¼0
qj
j
k
� �
qj�kr
ðx�x0Þjj!
�
�
�
c¼x0
¼P
l
r¼1
P
mr�1
k¼0
lrkðx�x0Þkk!qk
r
ðf ðqrðx� x0ÞÞÞðkÞx
�
�
�
c¼x0
:
ð51Þ
Example 5. y00 ¼ 4y0 � 4y; y ¼ yðx; s0; s1Þ; initial conditions yð0; s0; s1Þ ¼ s0; ðyðx; s0; s1ÞÞ0xjx¼0 ¼ s1.
The generalized differential operator for this differential equations is D ¼ s1Ds0 þ 4ðs1 � s0ÞDs1 .
Trivial transformations yield HrðDjs; j ¼ 0;1;2; . . .Þ ¼ 2. Roots of the Hankel characteristic equation are q1 ¼ q2 ¼ 2. Then,
l10 ¼ s0 and l11 ¼ s1 � s0. Thus, pj ¼ s02j þ ðs1 � 2s0Þ j
1
� �
2j�1; and finally,
yðx; s0; s1Þ ¼ s0 expð2xÞ þ ðs1 � 2s0Þx
2ðexpð2xÞÞ0x ¼ s0 expð2xÞ þ ðs1 � 2s0Þx expð2xÞ:
This trivial example illustrates the above-mentioned fact that a solution cannot be expressed in a form comprising standard
functions if roots of the Hankel characteristic equation are all not different.
4. The generalization of the Exp-function method
4.1. The algorithm for the construction of an exact solution of an ordinary differential equation
We will illustrate the algorithm for the initial problem of an nth order ordinary explicit differential equation:
dny
dxn ¼ Pn x; y;
dy
dx; . . . ;
dn�1
y
dxn�1
!
;
djy
dxjðx; s0; s1; . . . ; sn�1Þ
�
�
�
�
�
x¼x0
¼ sj; x0 2 R; Pnðc; s0; s1; . . . ; sn�1Þ 2 Fx;s1 ;...;sn : ð52Þ
(A) The change of the independent variable x ¼ uðz; aÞ; z ¼ wðx; aÞ; where a is an independent parameter (in general, the
variable change can comprise many independent parameters).
It can be noted that the selection of the invertible function uðz; aÞ is in many particular cases dependent on a subjec-
tive experience and knowledge of the area of application where the differential equation is used. The famous Exp-
function method is based on the variable change uðz; aÞ :¼ expðzÞ. Then the differential equation takes the form
defined by Eq. (34) and the image differential equation takes the form analogous to Eq. (35).
(B) Selection of the parameter b.
The sequence ðqj; j 2 Z0Þ is constructed as follows: q0 :¼ 1; qjþ1 ¼Qjk¼0ðaþ bkÞ; j ¼ 0;1;2; . . .. It can be noted that the
selection of the parameters a and b determines the standard function (Eq. (2)) which will figure in the expression of
the exact solution (in most cases a ¼ 1 is assumed).
(C) Construction of the generalized differential operator:
D :¼ Dc þ s1Ds0 þ s2Ds1 þ � � � þ sn�1Dsn�2þ Pnðc; s0; s1; . . . ; sn�1ÞDsn�1
: ð53Þ
(D) Calculation of the first elements of the sequence ðp̂j; j 2 Z0Þ (Eq. (1)):
p̂j :¼1
qj
Djs; j ¼ 0;1;2; . . . ;2m: ð54Þ
(E) Construction of first elements of the Hankel matrixes sequence:
H1 ¼ ½p̂0�; H2 ¼ p̂0 p̂1
p̂1 p̂2
�
; . . . ; Hmþ1 ¼
p̂0 p̂1 � � � p̂m
p̂1 p̂2 � � � p̂mþ1
� � � � � � � � � � � �p̂m p̂mþ1 � � � p̂2m
2
6
6
6
4
3
7
7
7
5
: ð55Þ
2388 Z. Navickas et al. / Applied Mathematics and Computation 216 (2010) 2380–2393
(F) Calculation of determinants of Hankel matrixes dk ¼ detHk up to the first zero:
ðd1;d2; . . . ; dm; 0Þ; d1;d2; . . . ; dm – 0; dmþ1 ¼ 0: ð56Þ
(G) Construction of the mth Hankel’s characteristic equation and determination of its roots q1;q2; . . . ;qm:
det
p̂0 p̂1 � � � p̂m
p̂1 p̂2 � � � p̂mþ1
� � � � � � � � � � � �p̂m�1 p̂m � � � p̂2m�1
1 q � � � qm
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
¼ 0: ð57Þ
(H) If all roots of the characteristic equation are different, the following linear algebraic system of equations is solved for
determination of l1;l2; . . . ;lm:
l1qj1 þ l2q
j2 þ � � � þ lmq
jm ¼ p̂j; j ¼ 0;1; . . . ;m� 1: ð58Þ
(I) Following equalities must be tested:
Dlr ¼ lrqr ;Dqr ¼ bq2r ; r ¼ 1;2; . . . ;m: ð59Þ
If these equalities hold, then statements (i), (ii) and (iii) of the Theorem 1 hold true too for the sequence
ðDjs; j ¼ 0;1;2; . . .Þ. Moreover, dmþ1 ¼ dmþ2 ¼ � � � ¼ 0 and the exact solution of the differential equation can be ex-
pressed in a form comprising the selected standard function.
(J) Construction of the exact solution of the image differential equation in a form of a series:
xðzÞ ¼X
þ1
j¼0
qj
j!
X
m
r¼1
lrqjr
!
ðz� z0Þj ¼X
m
r¼1
lr
X
þ1
j¼0
qj
j!qj
rðz� z0Þj: ð60Þ
(K) Expression of the exact solution in a form comprising the selected standard function f ðzÞ ¼Pþ1j¼0
qjj!zj:
xðzÞ ¼X
m
r¼1
lrf ðqrðz� z0ÞÞ: ð61Þ
(L) Construction of the exact solution of the original differential equation:
yðx; s1; . . . ; snÞ ¼X
m
r¼1
lrf ðqrðwðx; aÞ � wðx0; aÞÞÞ: ð62Þ
(M) Checking if the produced solution satisfies the original differential equation.
5. Examples
5.1. The exact solution is expressed in a ratio of sums of exponential functions
The initial problem reads:
y0x ¼ y2 þ y� 6; y ¼ yðx; sÞ; yð0; sÞ ¼ s: ð63Þ
A variable change x ¼ 1a ln z; z ¼ expðaxÞ yields:
x0z ¼
1
azðx2 þx� 6Þ; x ¼ xðz; sÞ;xð1; sÞ ¼ s: ð64Þ
We select b ¼ 1. Thus, qj ¼ j!; j 2 Z0. The generalized differential operator reads:
D ¼ Dc þ1
acðs2 þ s� 6ÞDs ¼ Dc þ
ðsþ 3Þðs� 2Þac
Ds: ð65Þ
Computational symbolic transformations yield ðpj :¼ DjsÞ:
p0 ¼ s; p1 ¼ ðsþ 3Þðs� 2Þac
; p2 ¼ ðsþ 3Þ2ðs� 2Þ2ð2s� 1Þa2c2
� ðsþ 3Þðs� 2Þac2
;
p3 ¼ ðsþ 3Þðs� 2Þð6s2 þ 6s� 11� 6sa� 3aþ 2a2Þa3c3
; . . . :
ð66Þ
Z. Navickas et al. / Applied Mathematics and Computation 216 (2010) 2380–2393 2389
Next, determinants of Hankel matrixes dn ¼ detðHnÞ; n ¼ 1;2; . . . are calculated:
d1 ¼ s; d2 ¼ ðsþ 3Þðs� 2Þð�sþ sc � 12Þ2a2c2
;
d3 ¼ ðsþ 3Þ2ðs� 2Þ2ða� 5Þðaþ 5Þð2a2s� 36aþ 3saþ 25sÞ144a6c6
:
ð67Þ
It can be noted, that equality a ¼ 5 produces d3 ¼ 0 at any s. Thus, Hr 1j!Djs�
�
�
a¼5; j 2 Z0
� �
¼ 2. Therefore, the variable change
x ¼ 15ln z; z ¼ expð5xÞ appears to be appropriate.
Hankel characteristic equation reads:
s Ds 12D2s
Ds 12D2s 1
6D3s
1 q q2
�
�
�
�
�
�
�
�
�
�
�
�
�
�
¼ 0: ð68Þ
Elementary transformations yield q2 � s�25cq ¼ 0. Roots are q1 ¼ 0; q2 ¼ s�2
5c. Thus, l1 ¼ �3; l2 ¼ sþ 3. We check the validity
of the Theorem 1:
Dl1 ¼ l1q1 ¼ 0; Dl2 ¼ Dðsþ 3Þ ¼ ðsþ 3Þðs� 2Þ5c
¼ l2q2;
Dq1 ¼ 0 ¼ q21; Dq2 ¼ D
s� 2
5c¼ s� 2
5c
� �2
¼ q22:
ð69Þ
Thus, the sequence ðqj; j 2 Z0Þ and the variable change were selected appropriately. Now,
y ¼Pþ1j¼0 j! �3 � 0j þ ðsþ 3Þ s�2
5c
� j� ��
�
�
c¼1
ðz�1Þjj!
¼ �3þ ðsþ 3ÞPþ1j¼0
s�2ð Þðz�1Þ5
� �j
, when jz� 1j < 5js�2j. Now we expand the pro-
duced series into the whole z-axis, except the point z ¼ sþ3s�2
. Then,
y ¼ �3þ 5sþ 15
5� szþ zþ 2z� 2¼ 2ð3þ sÞ � 3ð2� sÞz
3þ sþ ð2� sÞz : ð70Þ
Finally, the exact solution of the differential equation reads:
yðx; sÞ ¼ 2ð3þ sÞ � 3ð2� sÞ expð5xÞ3þ sþ ð2� sÞ expð5xÞ ¼ 2ð3þ sÞ expð�3xÞ � 3ð2� sÞ expð2xÞ
ð3þ sÞ expð�3xÞ þ ð2� sÞ expð2xÞ : ð71Þ
5.2. The exact solution is expressed in a form comprising a square root
The initial problem reads:
y0x ¼ y3; y ¼ yðx; sÞ; yðc; sÞ ¼ s: ð72Þ
Now we do not make a variable change, but select b :¼ 2 and q0 ¼ 1; qj ¼ ð2j� 1Þ!!; j ¼ 1;2; . . . instead.
The generalized differential operator reads D ¼ s3Ds. Then, p0 ¼ s; pj ¼ 1qjDjs ¼ s2jþ1; j ¼ 1;2; . . ..
It is easy to see that the sequence of determinants of Hankel matrices is ðs; 0;0; . . .Þ; thus Hrðpj; j 2 Z0Þ ¼ 1. The Hankel
characteristic equation readss s3
1 q
�
�
�
�
�
�
�
�
¼ 0; or q ¼ s2. Then, l ¼ s.
Now we need to check the validity of Theorem 1: Dl ¼ s3Dss ¼ s3 ¼ lq; Dq ¼ s3Dss2 ¼ 2s4 ¼ 2q2. Thus, the selection of
the sequence q0 ¼ 1; qj ¼ ð2j� 1Þ!!; j ¼ 1;2; . . . is appropriate for this differential equation.
Now, the structure of the standard function yields:
yðxÞ ¼ sþX
þ1
j¼1
ð2j� 1Þ!!j!
s � s2jðx� cÞj ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 2s2ðx� cÞp ; jx� cj < 1
2s2: ð73Þ
Finally, the exact solution of the differential equation reads:
yðx; cÞ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 2s2ðx� cÞp ; x <
1
2s2þ c if it is required that yðx; cÞ 2 R: ð74Þ
5.3. Partial solutions of the Liouville’s equation
A natural question is if our developed technique could be applied for the construction of an exact analytical solution of an
equation which is already solved by the classical Exp-function method. We select the Liouville’s equation which is analyzed
in [17]:
2390 Z. Navickas et al. / Applied Mathematics and Computation 216 (2010) 2380–2393
y � y00xx � y0x� 2 þ 2y3 ¼ 0; ð75Þ
where the partial solution y ¼ yðxÞ satisfies initial conditions yð0Þ ¼ s and y0xðxÞ�
�
x¼0¼ t. We will seek only such solutions
which can be expressed in a ratio of finite sums of exponential functions. We perform the variable change defined by Eq.
(36) at a :¼ 1 because it helps to simplify technical computations. The image differential equation of Eq. (75) then takes
the following form:
x z2x00zz þ zx0
z
�
� z2 x0z
� 2 þ 2x3 ¼ 0; ð76Þ
where x ¼ xðzÞ; xð1Þ ¼ s and x0zðzÞ
�
�
z¼1¼ t. The generalized differential operation of the image differential equation reads:
D ¼ Dc þ tDs þc2t2 � cst � 2s3
c2s� Dt: ð77Þ
Then,xðzÞ ¼Pþ1
j¼0ðz�1Þj
j!� Djs
�
�
�
c¼1and yðxÞ ¼ xðexpðxÞÞ. We select qj :¼ j! (Eq. (3)) and construct the sequence ðp̂j; j ¼ 0;1;2; . . .Þ
where p̂j ¼ 1j!Djs.
The sequence of determinants of Hankel matrixes is:
detH1 ¼ jp̂0j ¼ s;
detH2 ¼ p̂0 p̂1
p̂1 p̂2
�
�
�
�
�
�
�
�
¼ � c2t2 þ sct þ 2s3
2c2;
detH3 ¼ � c2t2 � s2 þ 4s3
144s3c6ðc4t4 þ 3sc3t3 þ 4c2t2s3 þ 2t2s2c2 þ 6ts4c � 12s6Þ;
detH4 ¼ ðc2t2 � s2 þ 4s3Þ21036800s8c12
�88t2s7c2 � 72c3t3s6 þ 42c4t4s5 þ 48c5t5s4 þ 10c6t6s3 þ 1112t2s9c2þþ212s6c4t4 þ 360s11 þ 236c2t2s8 þ 360ts9c þ 1440s12 þ c8t8 þ 6sc7t7��18s3c5t5 þ 7s2c6t6 � 44s4c4t4 � 24s5c3t3 þ 1440cts10 þ 456t3s7c3
0
B
@
1
C
A:
Thus, detH3jc¼1 ¼ detH4jc¼1 ¼ 0 when
t2 � s2 þ 4s3 ¼ 0; ð78Þ
or t ¼ �sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp
; s 6 1=4. The graph of the curve (78) is illustrated in Fig. 1.
Thus, the solution of the differential Eq. (75) can be expressed in a ratio of finite sums of exponential functions when vari-
ables of initial conditions s and t satisfy equality (78). Now the algorithm described in Section 4.1 can be used to construct
the function xðzÞ. It is sufficient to check if this function satisfies the original differential equation to conclude that it is a
partial solution.
First of all, we construct the Hankel characteristic Eq. (57):
det
s Ds 12D2s
Ds 12D2s 1
6D3s
1 q q2
2
6
4
3
7
5
�
�
�
�
�
�
� c ¼ 1
t ¼ �sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp
¼ 0; ð79Þ
Fig. 1. The graph of the relationship between initial conditions s and t.
Z. Navickas et al. / Applied Mathematics and Computation 216 (2010) 2380–2393 2391
which yields q1 ¼ q2 ¼ 12
�1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp� �
. We assume q ¼ q1 ¼ q2 ¼ 12
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp
� 1� �
and construct equalities:
Djs
j!
�
�
�
�
�c ¼ 1
t ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp
¼ l1qj þ l2
j
1
� �
qj�1; j ¼ 0;1;2; . . . : ð80Þ
Eq. (80) yields a system of linear algebraic equations for determination of coefficients l1 and l2:
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp
� 1� �
� �j
l1 þ1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp
� 1� �
� �j�1
� j � l2 ¼ Djs
j!
�
�
�
�
�c ¼ 1
t ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp
; j ¼ 0;1: ð81Þ
It can be noted that the assumption q ¼ q1 ¼ q2 ¼ 12
�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp
� 1� �
would yield the same function x ¼ x zð Þ.
The system of Eq. (81) yields l1 ¼ s and l2 ¼ s2
1þffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp� �
. Finally,
xðzÞj c ¼ 1
t ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp
¼X
þ1
j¼0
ðz� 1Þj s1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp
� 1� �
� �j
þ s
21þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp� �
j1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp
� 1� �
� �j�1 !
¼2s 1� 2sþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp� �
z
2szþ 1� 2sþffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp� �2
¼1�2sþ
ffiffiffiffiffiffiffiffi
1�4sp
2s� z
zþ 1�2sþffiffiffiffiffiffiffiffi
1�4sp
2s
� �2ð82Þ
Further computations can be simplified introducing an auxiliary parameter b :¼ 1�2sþffiffiffiffiffiffiffiffi
1�4sp
2s; b 2 R. Now it is easy to check that
the function �xðzÞ :¼ xðzÞj c ¼ 1t ¼ s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp ¼ b�z
ðbþzÞ2 satisfies differential Eq. (76). Therefore,
yðxÞ ¼ �xðexpðxÞÞ ¼ b expðxÞðexpðxÞ þ bÞ2
ð83Þ
is a partial solutions of the differential Eq. (75) satisfying initial conditions yð0Þ ¼ b
ðbþ1Þ2 and y0xðxÞjx¼0 ¼ bðb�1Þðbþ1Þ3 for all b (except
b ¼ �1).
Thus, Hr 1j!Djs�
�
� c ¼ 1t ¼ �s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 4sp ; j ¼ 0;1;2; . . .
0
B
@
1
C
A¼ 2 and equalities (80) hold true.
It can be noted, that Eq. (83) coincides with the solution presented in [17]. But we would like to stress that we did not
guess the structure of the solution; it has been automatically identified by the direct application of the algorithm described
in Section 4.1. This is the main difference between classical Exp-function type methods and our proposed technique.
But even more important fact is that our approach enabled to show that the solution of the differential Eq. (75) takes the
form (83) only when initial conditions satisfy Eq. (78). This is an important fact, and we argue that the conclusion done in
[17] is in general incorrect. The partial solution cannot be expressed in the form represented by Eq. (83) for any initial con-
ditions. Our technique enabled the identification of the constraint linking two initial conditions. Classical Exp-function type
methods are incapable of finding such constrains and may produce wrong results in general, what is clearly illustrated by
this example.
6. Conclusions
We have constructed an analytical criterion determining if a solution of a nonlinear ordinary differential equation can be
expressed in a form comprising standard functions. This criterion is much more general compared to the criterion presented
in [12]. First of all, not only exponential functions are considered in the exact solution of a nonlinear differential equation. In
fact, the new criterion works with any standard function. But the most important result is that this new criterion can be used
to identify the structure of the solution which can be muchmore complex than a sum of standard functions or a ratio of sums
of standard functions. That opens new possibilities for finding exact solutions of nonlinear differential equations.
New computational algorithm for automatic identification of the structure of the solution is constructed. Several exam-
ples are used to illustrate the proposed concept.
Recent developments of the Exp-function method were summarized in [18,19]; the Exp-function method has been used
to solve differential-difference equations and stochastic equations [20–22]. Applicability of our technique for these problems
is a definite object of future research.
2392 Z. Navickas et al. / Applied Mathematics and Computation 216 (2010) 2380–2393
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