THE SUCCESSIVE DIFFERENTIATION METHOD FOR SOLVING BRATU
EQUATION AND BRATU-TYPE EQUATIONS
ABDUL-MAJID WAZWAZ
Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA
E-mail: [email protected]
Received January 10, 2016
In this work, we apply the successive differentiation method for solving the
nonlinear Bratu problem and a variety of Bratu-type equations. We use the successive
differentiation of any linear or nonlinear ordinary differential equation to determine the
values of the function’s derivatives at x = 0. The Taylor series of the solution can be
established by using the derived coefficients. The algorithm handles the problem in a
direct manner without any need to restrictive assumptions. We emphasize the power of
the method by applying it to Bratu equation and a variety of Bratu-type equations.
Key words: Successive differentiation method; Taylor series; Bratu equation.
1. INTRODUCTION
The ordinary differential equations (ODEs) [1] are used in mathematical modeling to describe a variety of real-world problems in science and engineering, such
as population growth models, the predator-prey equations, and chemical and biological models. The discovery of ODEs goes back to Leibniz, Newton, Bernoulli, and
others. Many traditional methods, analytic and numerical, were used in the literature
to solve ODEs and partial differential equations (PDEs), either linear or nonlinear
[1]-[20].
The main concern of many researchers is to find a unified method that works
for almost, but not all, ODEs and PDEs, linear or nonlinear. Researchers have been
attempting to discover new methods for solving differential and integral equations,
analytically and numerically. The Adomian decomposition method (ADM) [2]-[6],
[16, 17], developed by Adomian is a powerful method that can be used to handle ODEs, homogeneous or nonhomogeneous, linear or nonlinear ones. However,
the ADM requires the so-called Adomian polynomials to solve nonlinear equations,
ODEs or PDEs. To facilitate the computational work of the ADM, Wazwaz [2][4] developed the so-called modified decomposition method, where a slight change
was proposed in the first two components of the solution. The variational iteration
method (VIM) developed by He [7], is another useful systematic technique for solving differential and integral equations. The VIM requires the determination of the
Lagrange multipliers, which are developed optimally by using the variational theory.
RJP 61(Nos.
Rom.
Journ. Phys.,
5-6),
Vol.
774–783
61, Nos. 5-6,
(2016)
P. 774–783,
(c) 2016
Bucharest,
- v.1.3a*2016.7.20
2016
2
The successive differentiation method for solving Bratu-type equations
775
The Lagrange multipliers are found to be either constants or functions. The homotopy
perturbation method (HPM) is a series expansion method used to find the solution
of linear and nonlinear equations as well. The HPM was developed by merging the
standard homotopy and the perturbation techniques. Other methods were developed
and used in the literature. The aforementioned methods were adapted for obtaining
the series solution for the equation under discussion. The derived series solution was
studied for exact solutions or just for numerical purposes.
In a parallel manner, we aim to use a powerful technique for solving linear and
nonlinear, homogeneous and nonhomogeneous ODEs. The proposed scheme relies
mainly from using the ODE and finding the successive differentiation of that equation
without any need for transforming nonlinear terms or using Lagrange multipliers. In
this case, we are able to compute the values of the derivatives at x = 0, and this will
lead to the derivation of the Taylor series of the solution. The successive differentiation method (SDM) will be used to derive series solutions for the Bratu equation [21]
and three Bratu-type equations. It will also be used for solving a variety of Bratu-type
equations which were first introduced by Wazwaz [4].
The Bratu’s boundary value problem in 1-dimensional planar coordinates is
given as
00
u + λeu = 0, 0 < x < 1,
(1)
u(0) = u(1) = 0,
which was used to model a combustion problem in a numerical slab, the fuel ignition
of the thermal combustion theory, and appeared in the Chandrasekhar model of the
expansion of the universe. It stimulates a thermal reaction process in a rigid material
where the process depends on a balance between chemically generated heat and heat
transfer by conduction [1]-[7].
The exact solution to (1) reads [1]-[6]
(
)
cosh[(x − 21 ) 2θ ]
u(x) = −2 ln
,
(2)
cosh( 4θ )
where θ satisfies
√
θ
.
(3)
θ = 2λ cosh
4
The Bratu problem has zero, one or two solutions when λ > λc , λ = λc , and λ < λc
respectively, where the critical value λc satisfies the equation
θc
2
λc = 8 csch
.
(4)
4
It was evaluated in [1]-[6] that the critical value λc is given by
λc = 3.513830719.
RJP 61(Nos. 5-6), 774–783 (2016) (c) 2016 - v.1.3a*2016.7.20
(5)
776
Abdul-Majid Wazwaz
3
In addition to the standard Bratu problem, there are other Bratu-type problems
which were introduced and examined in the literature. In [5], a variety of Bratu-type
equations was introduced as
00
u − π 2 eu = 0, 0 < x < 1,
u(0) = u(1) = 0.
(6)
00
u + π 2 e−u = 0, 0 < x < 1
u(0) = u(1) = 0.
and
(7)
00
u − eu = 0, 0 < x < 1,
(8)
u(0) = u(1) = 0,
and will be referred to as type-I, type-II, and type-III, respectively. The last equation
(8) is of great interest in magnetohydrodynamics.
2. THE SUCCESSIVE DIFFERENTIATION METHOD
In what follows, we highlight the main steps of the SDM, where the given differential equation will be differentiated successively for many times, and the values of
the resulting functions at x = 0, or at other values such as x = 1, will be determined
for each obtained derivative. Having determined these values, we construct the Taylor series of the solution of the equation under discussion. As usual, the obtained
series converges to an exact solution, if this solution exists, otherwise, the series can
be used for numerical computations. The SDM can be used directly, for both linear
and nonlinear equations, and for homogeneous and nonhomogeneous cases, without
any need for any restrictive assumptions.
We consider a generalized ordinary differential equation of nth-order as
u(n) (x) = R(u(x), u0 (x), · · · , u(n−1) (x)) + g(x),
u(0) = α0 , u0 (0) = α1 , · · · , u(n−1) (0) = αn−1 ,
(9)
where R can be a linear or a nonlinear operator of u(x) and its derivatives with
constant coefficients or variable coefficients, and αi , 0 ≤ i ≤ (n − 1) are given initial
conditions. By differentiating both sides of (9) many times, say k times for example,
we obtain
(k)
u(n)
(x) = R(k) (u(x), u0 (x), · · · , u(n−1) (x)) + g (k) (x), k ≥ 1.
(10)
By substituting x = 0 at each step of differentiating we compute the values of the
0
00
000
functions u(n) (0), u(n) (0), u(n) (0), · · · . Having determined these values,
and by using the given initial conditions, the Taylor series of the solution u(x) follows
immediately.
RJP 61(Nos. 5-6), 774–783 (2016) (c) 2016 - v.1.3a*2016.7.20
4
The successive differentiation method for solving Bratu-type equations
777
For simplicity, we will introduce the dynamics of the successive differentiation
method for first order, and the second-order ODEs, and hence it can be generalized
to any linear or nonlinear differential equation of any order.
2.1. THE FIRST-ORDER ODEs
Consider the first-order ordinary differential equation
u0 (x) − f (x)u(x) = g(x), u(0) = α0 .
(11)
Following the analysis presented earlier, we set
u(x),
u0 (x) = f (x)u(x) + g(x),
u00 (x) = f (x)u0 (x) + f 0 (x)u(x) + g 0 (x),
u000 (x) = f (x)u00 (x) + 2f 0 (x)u0 (x) + f 00 (x)u(x) + g 00 (x),
u(iv) (x) = f (x)u000 (x) + 3f 0 (x)u00 (x) + 3f 00 (x)u0 (x) + f 000 (x)u(x) + g 000 (x),
..
.
(12)
Substituting x = 0 in each step of (12) gives the values of the functions as
u(0) = α0
u0 (0) = f (0)u(0) + g(0) = α1 ,
u00 (0) = f (0)u0 (0) + f 0 (0)u(0) + g 0 (0) = α2 ,
u000 (0) = f (0)u00 (0) + 2f 0 (0)u0 (0) + f 00 (0)u(0) + g 00 (0) = α3 ,
u(iv) (0) = f (0)u000 (0) + 3f 0 (0)u00 (0) + 3f 00 (0)u0 (0) + f 000 (0)u(0) + g 000 (0) = α4 ,
..
.
(13)
Having determined the values of the function at x = 0, we can easily set the Taylor
series of the solution u(x) as
α2
α3
α4
u(x) = α0 + α1 x + x2 + x3 + x4 + · · · .
(14)
2!
3!
4!
RJP 61(Nos. 5-6), 774–783 (2016) (c) 2016 - v.1.3a*2016.7.20
778
Abdul-Majid Wazwaz
5
2.2. THE SECOND-ORDER ODEs
Consider the second-order ordinary differential equation
u00 (x) − f (x)u0 (x) − h(x)u(x) = g(x), u(0) = α0 , u0 (0) = α1 .
(15)
Proceeding as before, we set
u(x),
u0 (x)
u00 (x)
u000 (x)
(iv)
u (x)
=
=
=
=
+
..
.
,
f (x)u0 (x) + h(x)u(x) + g(x),
f (x)u00 (x) + (f 0 (x) + h(x))u0 (x) + h0 (x)u(x) + g 0 (x),
f (x)u000 (x) + (2f 0 (x) + h(x))u00 (x) + (f 00 (x) + 2h0 (x))u0 (x)
h00 (x)u(x) + g 00 (x),
(16)
Substituting x = 0 in each step of (16) gives the values of the function as
u(0)
u0 (0)
u00 (0)
u000 (0)
u(iv) (0)
=
=
=
=
=
+
..
.
α0
α1 ,
f (0)u0 (0) + h(0)u(0) + g(0) = α2 ,
f (0)u00 (0) + (f 0 (0) + h(0))u0 (0) + h0 (0)u(0) + g 0 (0) = α3 ,
f (0)u000 (0) + (2f 0 (0) + h(0))u00 (0) + (f 00 (0) + 2h0 (0))u0 (0)
h00 (0)u(0) + g 00 (0) = α4 ,
(17)
Having determined the values of the function at x = 0, we can easily set the Taylor
function of u(x) as
α2
α3
α4
u(x) = α0 + α1 x + x2 + x3 + x4 + · · · .
(18)
2!
3!
4!
In what follows, we will employ the SDM to the standard Bratu problem and
to a variety of Bratu-type problems, which were handled by other methods in the
literature.
3. THE BRATU EQUATION
The Bratu’s boundary value problem in 1-dimensional planar coordinates is
given as
00
u + λeu = 0, 0 < x < 1,
u(0) = u(1) = 0.
RJP 61(Nos. 5-6), 774–783 (2016) (c) 2016 - v.1.3a*2016.7.20
(19)
6
The successive differentiation method for solving Bratu-type equations
779
Using the SDM gives
u(x),
u(0) = 0,
0
u0 (0) = α,
u (x) =,
u00 (x) = −λeu(x) ,
u00 (0) = −λ,
u000 (x) = −λu0 (x)eu(x) ,
u000 (0) = −αλ,
u(iv) (x) = − λ(u0 (x))2 + λu00 (x) eu(x) ,
u(iv) (0) = −α2 λ + λ2 ,
u(v) (x) = − λ(u0 (x))3 + 3λu0 (x)u00 (x) + λu000 (x) eu(x) , u(v) (0) = −α3 λ + 4λ2 α,
..
.
.
(20)
In view of this, we obtain the series approximation
u(x) = αx −
λ 2 αλ 3 α2 λ − λ2 4 α3 λ − 4λ2 α 5
x −
x −
x −
x +··· .
2!
3!
4!
5!
(21)
To determine α, we use the other boundary conditions u(1) = 0 for the case λ =
3.0 < λc = 3.513830719, and by solving the resulting equation we find
α = 1.675116658, 3.515010276.
(22)
Substituting these two values of α, and for λ = 3, we obtain the two approximate
solutions
u(x) = 1.675116658x − 1.5x2 − 0.837558329x3 + 0.2774167694x4
+ 0.3850249013x5 + · · · , (23)
and
u(x) = 3.515010276x − 1.5x2 − 1.757505138x3 − 0.2262833017x4 (24)
− 0.03122183667x5 + · · · .
The obtained result is consistent with the fact that the Bratu equation has two solutions for λ < λc .
4. THE BRATU-TYPE EQUATION I
The Bratu-type I equation reads
00
u − π 2 eu = 0, 0 < x < 1,
u(0) = u(1) = 0.
RJP 61(Nos. 5-6), 774–783 (2016) (c) 2016 - v.1.3a*2016.7.20
(25)
780
Abdul-Majid Wazwaz
7
Proceeding as before, the SDM gives
u(x),
u(0) = 0,
0
u0 (0) = α,
u (x) =,
u00 (x) = π 2 eu(x) ,
u00 (0) = π 2 ,
u000 (x) = π 2 u0 (x)eu(x) ,
u000 (0) = π 2 α,
u(iv) (x) = π 2 (u0 (x))2 + u00 (x) eu(x) ,
u(iv) (0) = π 2 (π 2 + α2 ),
u(v) (x) = π 2 (u0 (x))3 + 3u0 (x)u00 (x) + u000 (x) eu(x) , u(v) (0) = π 2 (4απ 2 + α3 ),
..
.
.
(26)
In view of this, we obtain the series approximation
u(x) = αx +
π 2 2 π 2 α 3 π 2 (π 2 + α2 ) 4 π 2 (4απ 2 + α3 ) 5
x +
x +
x +
x +··· .
2!
3!
4!
5!
(27)
Notice that
u(x) = − ln[1 − sin(αx)],
(28)
gives the Taylor series
u(x) = αx +
α2 2 α3 3 α4 2 α5 5 α6 6
x + x + x + x + x +··· .
2
6
12
24
45
(29)
However, note that
u(x) = − ln[1 − sin(πx)],
(30)
gives the following Taylor series
u(x) = πx +
π2 2 π3 3 π4 2 π5 5 π6 6
x + x + x + x + x +···
2
6
12
24
45
(31)
It is obvious that (30) satisfies the boundary condition u(1) = 0, and by comparing
the two Taylor series we find that α = π. This clearly means that (30) is the exact
solution of the Bratu-type problem (25).
5. THE BRATU-TYPE EQUATION II
The Bratu-type II equation reads
00
u + π 2 e−u = 0, 0 < x < 1,
u(0) = u(1) = 0.
(32)
In a manner parallel to the analysis presented before, the successive differentiRJP 61(Nos. 5-6), 774–783 (2016) (c) 2016 - v.1.3a*2016.7.20
8
The successive differentiation method for solving Bratu-type equations
781
ation method gives
u(x),
u(0) = 0,
0
u0 (0) = α,
u (x) =,
u00 (x) = −π 2 e−u(x) ,
u00 (0) = −π 2 ,
u000 (x) = π 2 u0 (x)e−u(x) ,
u000 (0) = απ 2 ,
u(iv) (x) = −π 2 (u0 (x))2 − u00 (x) e−u(x) ,
u(v) (x) = π 2
..
.
u(iv) (0) = −π 2 (π 2 + α2 ),
(u0 (x))3 − 3u0 (x)u00 (x) + u000 (x) eu(x) , u(v) (0) = π 2 (4απ 2 + α3 ),
.
(33)
In view of this, we obtain the series approximation
u(x) = αx −
π 2 2 π 2 α 3 π 2 (α2 + π 2 ) 4 π 2 (4απ 2 + α3 ) 5
x +
x −
x +
x +··· .
2!
3!
4!
5!
(34)
To determine α, we use the boundary condition u(1) = 0 to find that α = π. This in
turn gives the exact solution as
u(x) = ln[1 + sin(πx)],
(35)
6. THE BRATU-TYPE EQUATION III
The Bratu-type equation III reads
00
u − eu = 0, 0 < x < 1,
u(0) = u(1) = 0,
(36)
which is of great interest in magnetohydrodynamics [6]. Proceeding as before, we
find
u(x),
u(0) = 0,
0
u (x) =,
u0 (0) = α,
u00 (x) = eu(x) ,
u00 (0) = 1,
u000 (x) = u0 (x)eu(x) ,
u000 (0) = α,
u(iv) (x) = (u0 (x))2 + u00 (x) eu(x) ,
u(iv) (0) = 1 + α2 ,
u(v) (x) = (u0 (x))3 + 3u0 (x)u00 (x) + u000 (x) eu(x) , u(v) (0) = 4α + α3 ,
..
.
.
RJP 61(Nos. 5-6), 774–783 (2016) (c) 2016 - v.1.3a*2016.7.20
(37)
782
Abdul-Majid Wazwaz
9
The series approximation is therefore given by
1 2 α 3 1 + α2 4 4α + α3 5
x + x +
x +
x +··· .
2!
3!
4!
5!
Using the boundary condition u(1) = 0 gives
u(x) = αx +
a = u0 (0) = −0.463639988227675.
(38)
(39)
This in turn gives the series approximation [6]
u(x) = −0.463639988227675x + 0.5x2 − 0.07727333137x3
+ 0.05062341828x4 − 0.01628520791x5 + 0.008903876534x6
− 0.003646125737x7 + 0.001868477052x8 − 0.000851751709x9
+ 0.0004283517518x10 − 0.0002057749479x11
12
+ 0.0001033495701x − 0.00005105422613x
(40)
13
+ 0.00002576095109x14 + · · · .
However, an exact solution [6] is given by
where
u(x) = − ln 2 + ln[λ(x)],
(41)
c(2x − 1) 2
,
λ(x) = c sec
4
(42)
and c is the root of
h
The root c lies between 0 and
π
2,
c sec
c i2
= 2.
4
namely c = 1.336055695, to ten figures.
(43)
7. CONCLUSIONS
We applied the successive differentiation method for reliable treatment of the
standard Bratu problem and for three kinds of Bratu-type equations. The method
depends mainly on differentiating the given equation, and evaluating the obtained
derivatives at x = 0. We then obtain the Taylor series for the solution of the equation. The successive differentiation method handles both linear and nonlinear ordinary differential equations, either homogeneous or nonhomogeneous ones, in a direct
manner without any need to restrictive conditions. The method works effectively to
the Volterra integral equations as will be discussed in a forthcoming work.
REFERENCES
1. W. E. Boyce and R. C. DiPrima, Elementary Differential Equations (10th edn., Wiley, New York,
2013).
RJP 61(Nos. 5-6), 774–783 (2016) (c) 2016 - v.1.3a*2016.7.20
10
The successive differentiation method for solving Bratu-type equations
783
2. A. M. Wazwaz, A reliable modification of the Adomian decomposition method, Appl. Math. Comput. 102 , 77–86 (1999).
3. A. M. Wazwaz, The modified decomposition method and the Pade’ approximants for solving
Thomas-Fermi equation, Appl. Math. Comput. 105, 11–19 (1999).
4. A. M. Wazwaz, The modified decomposition method applied to unsteady flow of gas through a
porous medium, Appl. Math. Comput. 118, 123–132 (2001).
5. A. M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu–type equations, Appl. Math. Comput. 166, 652–663 (2005).
6. A. M. Wazwaz, On the decomposition method for the approximate solution of nonlinear ordinary
differential equations, Int. J. Math. Educ. Sci. Technol. 32, 525–539 (2001).
7. J. H. He, Variational iteration method - Some recent results and new interpretations, J. Comput.
Appl. Math. 207, 3–17 (2007).
8. D. J. Frantzeskakis, H. Leblond, and D. Mihalache, Nonlinear optics of intense few-cycle pulses:
An overview of recent theoretical and experimental developments, Rom. J. Phys. 59, 767–784
(2014).
9. D. Mihalache, Localized structures in nonlinear optical media: A selection of recent studies, Rom.
Rep. Phys. 67, 1383–1400 (2015).
10. D. Mihalache, Localized optical structures: An overview of recent theoretical and experimental
developments, Proc. Romanian Acad. A 16, 62–69 (2015).
11. H. Leblond and D. Mihalache, Models of few optical cycle solitons beyond the slowly varying
envelope approximation, Phys. Rep. 523, 61–126 (2013).
12. G. Adomian and R. Rach, Inversion of nonlinear stochastic operators, J. Math. Anal. Appl. 91,
39–46 (1983).
13. A. M. Wazwaz, Multiple soliton solutions for two integrable couplings of the modified Korteweg-de
Vries equation, Proc. Romanian Acad. A 14, 219–225 (2013).
14. A. M. Wazwaz, Multiple kink solutions for the (2+1)-dimensional integrable Gardner equation,
Proc. Romanian Acad. A 15, 241–246 (2014).
15. A. M. Wazwaz, New (3+1)-dimensional nonlinear evolution equations with Burgers and SharmaTasso-Oliver equations constituting the main parts, Proc. Romanian Acad. A 16, 32–40 (2015).
16. R. Rach, A new definition of the Adomian polynomials, Kybernetes 37, 910–955 (2008).
17. R. Rach, A. M. Wazwaz, and J.-S. Duan, The Volterra integral form of the Lane-Emden equation:
new derivations and solution by the Adomian decomposition method, J. Appl. Math. Comput. 47,
365-379 (2015).
18. A. M. Wazwaz, Solving Schlömilch’s integral equation by the regularization-Adomian method,
Rom. J. Phys. 60, 56–71 (2015).
19. H. Fatoorehchi, R. Rach, and H. Abolghasemi, A novel family of iterative schemes for computation
of matrix inverses by the Adomian decomposition method, Rom. J. Phys. 60, 1315–1327 (2015).
20. A. Jafarian, P. Ghaderi, A. K. Golmankhaneh, and D. Baleanu, Analytical treatment of system of
Abel integral equations by homotopy analysis method, Rom. Rep. Phys. 66, 603–611 (2014).
21. G. Bratu, Sur les équations intégrale non linéaires, Bull. Soc. Math. France 42, 113–142 (1914).
RJP 61(Nos. 5-6), 774–783 (2016) (c) 2016 - v.1.3a*2016.7.20