EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 1, 2018, 202-214
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Modification of Laplace Adomian decomposition
method for solving nonlinear Volterra integral and

integro-differential equations based on Newton Raphson
formula

Dimple Rani1,∗, Vinod Mishra1

1 Department of Mathematics, Sant Longowal Institute of Engineering and Technology,
Longowal-148106 (Punjab), India

Abstract. In this paper, we establish a modified Laplace transform Adomian decomposition
method for solving nonlinear Volterra integral and integro-differential equations. This technique is
different from the standard Laplace Adomian decomposition method because of the terms involved
in Adomian polynomials. Here, we have used Newton Raphson formula in place of the term ui in
Adomian polynomials. The proposed scheme is investigated with some illustrative examples and
has given reliable results.

2010 Mathematics Subject Classifications: 44A10, 45D05, 49M27, 65R10, 65R20.

Key Words and Phrases: Numerical Laplace Transform Method, Volterra Integral Equations,
Volterra Integro-differential Equations, Adomian Decomposition Method, Newton Raphson
Formula.

1. Introduction

For solving nonlinear functional equations, Adomian decomposition method was
introduced by George Adomian in 1980 [4, 19, 23]. Basically, the technique provides an
infinite series solution of the equation and the nonlinear term is decomposed into an
infinite series of Adomian polynomials [1, 2, 5, 6, 8, 10, 14–17, 20, 22–24, 26–29]. Several
linear and nonlinear ordinary, partial, deterministic and stochastic differential equations
are solved easily and adequately by Adomian decomposition method [4, 13, 14, 19, 23].
In this work, Laplace transform technique in combination with Adomian decomposition
method is presented and modified, which was first studied by Khuri in [14] to solve
nonlinear differential equations. In [13], the authors investigated the method for solving
coupled nonlinear partial differential equations. Laplace decomposition method was

∗Corresponding author.

Email addresses: dimplechawla23@yahoo.com (D. Rani),
vinodmishra.2011@rediffmail.com (V. Mishra)

http://www.ejpam.com 202 c© 2018 EJPAM All rights reserved.



D. Rani, V. Mishra / Eur. J. Pure Appl. Math, 11 (1) (2018), 202-214 203

employed to logistic differential equations to find the numerical solutions in [10].
Chanquing and Jianhua studied the Adomian decomposition method to solve the
nonlinear fractional differential equations in [27]. In [7], the technique was applied on
delay differential equations. A comparison was made between Adomian decomposition
and tau methods in [4] for finding the solution of Volterra integro-differential equations.
Magdy and Mohamed [20] practiced Laplace decomposition method and Pade
approximation to get the numerical solution of nonlinear system of partial differential
equations. Further, a modified Laplace decomposition method was adopted for
Lane-Emden type differential equations in [28]. Hence, there are numerous applications
where Laplace Adomian decomposition method is used by many researchers.

In the present paper, we focus to solve nonlinear Volterra integral and
integro-differential equations. Nonlinear Volterra integral equations arise in many
scientific fields such as the population dynamics, spread of epidemics and semi-conductor
devices [25]. Volterra integro-differential equations also emanated in many physical
applications such as biological species coexisting together with increasing and decreasing
rates of generating and in engineering applications such as heat transfer, diffusion
process in general [3, 4, 21, 24]. Recently, many researchers investigated the solution of
these problems. Extant methods are presented to solve these kinds of equations. In [18],
quasilinearization technique was employed to solve Volterra integral equations. Kamyad
et al. proposed the discretisation and interpolation method for Volterra integral
equations [12]. In [3], a comparison was made between Laplace decomposition method,
homotopy perturbation method and wavelet-Galerkin method for solving nonlinear
Volterra integro-differential equations. Laplace transform combined with Adomian
decomposition method is pertained already to solve nonlinear Volterra integral and
integro-differential equations [9, 19, 23]. Our work is inspired from these. In this paper,
we have followed the combined Laplace transform and Adomian decomposition method
but while decomposing the nonlinear term using Adomian polynomials, we have
substituted the term ui with Newton Raphson formula. As we know that Newton
Raphson formula is used for finding the better approximate solution of real valued
function. By adapting this change, we have achieved the approximate solutions which
are in good agreement with the exact one.

The paper is organized as follows: In Section 2, the modified Laplace Adomian
decomposition method is presented and discussed for Volterra integral equations. Section
3 summarizes the application of technique to nonlinear Volterra integro-differential
equations. In Section 2 and 3, some numerical results are also given to clarify the
method. The conclusions are drawn in last section.

2. Nonlinear Volterra integral equations of the second kind

Consider the following nonlinear Volterra integral equation with difference kernel i.e.
k(x, t) = k(x− t) defined as

u(x) = f(x) +

∫ x

0
k(x− t)F (u(t))dt, (1)



D. Rani, V. Mishra / Eur. J. Pure Appl. Math, 11 (1) (2018), 202-214 204

where f(x) is known real valued function and F (u(x)) is the nonlinear function of u(x).
Apply Laplace transform on both sides of (1). After that using the linear property and
convolution theorem of Laplace transform, we have

L[u(x)] = L[f(x)] + L[k(x− t)]L[F (u(x))]. (2)

The methodology consists of approximating the solution of (1) as an infinite series given
by

u(x) =

∞∑
n=0

un(x). (3)

However, the nonlinear term F (u(x)) is decomposed as

F (u(x)) =
∞∑
n=0

An(x), (4)

where A′ns are modified Adomian polynomials which are based on Newton Raphson
formula given by

An =
1

n!

dn

dλn

[
f

(
n∑
i=0

λi
(
ui −

F (ui)

F ′(ui)

))]
λ=0

, n ≥ 0. (5)

Substituting (3) and (4) into (2), we get

L

[ ∞∑
n=0

un(x)

]
= L[f(x)] + L[k(x− t)]L

[ ∞∑
n=0

An(x)

]
.

Using the linearity property of Laplace transform, we get

∞∑
n=0

L[un(x)] = L[f(x)] + L[k(x− t)]
∞∑
n=0

L[An(x)]. (6)

To determine the terms u0(x),u1(x),u2(x),u3(x). . . of infinite series, comparing both sides
of (6), we have the following iterative scheme

L[u0(x)] = L[f(x)], (7)

In general, the relation is given by

L[un+1(x)] = L[k(x− t)]L[An(x)]. (8)

Employing the inverse Laplace transform to (7) and (8), we get

u0(x) = L−1 [L[f(x)]] , (9)



D. Rani, V. Mishra / Eur. J. Pure Appl. Math, 11 (1) (2018), 202-214 205

un+1(x) = L−1 [L[k(x− t)]L[An(x)]] . (10)

Adapting the value of u0(x) into (5) gives the value of A0 and then using the general
iterative relation (10), we get the values of u1(x), u2(x), u3(x) and so on, which finally
gives the solution (3) to the given Volterra integral equation.

The effectiveness of modified technique for solving Volterra integral equations is
shown by following numerical examples. Here, we have also found the maximum absolute
error estimation to show the adequacy of technique given as:

ej = Max|uex − uapp|

where ej denotes the maximum absolute error at some xj in the given interval.
Example 1.Consider the following Volterra integral equation [11, 25]

u(x) = x+

∫ x

0
u2(t)dt, (11)

which has the exact solution as u(x) = tanx.
Solution.Taking Laplace transform on both sides of (11) and using the linearity property
of Laplace transform, we have

L[u(x)] = L[x] + L

[∫ x

0
u2(t)dt

]
,

that is

L[u(x)] =
1

s2
+

1

s
L[u2(x)],

Using above technique, we have

L

[ ∞∑
n=0

un(x)

]
=

1

s2
+

1

s
L

[ ∞∑
n=0

An(x)

]
, (12)

where the nonlinear term F (u(x)) = u2(x) is decomposed using the formula given by (5).
Certain terms of modified Adomian polynomials are as follows:

A0 =

(
1

2

)2

u20,

A1 =

(
1

2

)2

(u0u1),

A2 =

(
1

2

)2

(2u0u2 + u21),

A3 =

(
1

2

)2

(2u0u3 + 2u1u2).



D. Rani, V. Mishra / Eur. J. Pure Appl. Math, 11 (1) (2018), 202-214 206

Pairing both sides of (12), gives

L[u0(x)] =
1

s2
, (13)

In general

L[un+1(x)] =
1

s
L[An(x)]. (14)

Applying inverse Laplace transform on both sides of (13), gives

u0(x) = x, (15)

Using general relation, we have

u1(x) =
x3

12
,

Continuing in this manner, we get

u2(x) =
x5

120
,

u3(x) =
x7

20160
,

u4(x) =
31x9

362880
,

Subsequently, the approximate solution becomes

u(x) = x+
x3

12
+

x5

120
+

x7

20160
+

31x9

362880
. . . .

The exact solution and the one obtained by our technique corresponding to distinct values
of x are presented in Table 1 and demonstrated through figure 1. The absolute error laid
out in the table admit that the solutions are very much close to the exact solution and
the maximum absolute error is 0.0002.
Example 2. Solve the following Volterra integral equation [12]

u(x) = 2x− x4

12
+ 0.25

∫ x

0
(x− t)u2(t)dt, (16)

having exact solution u(x) = 2x.
Solution. Applying the modified decomposition method, we have

L[u(x)] = L

[
2x− x4

12

]
+ 0.25L[x]L[u2(x)],

The method assumes the series solution of function u(x)

L

[ ∞∑
n=0

un(x)

]
= L

[
2x− x4

12

]
+

1

4s2
L

[ ∞∑
n=0

An(x)

]
, (17)



D. Rani, V. Mishra / Eur. J. Pure Appl. Math, 11 (1) (2018), 202-214 207

Table 1: Numerical Results for Example 1.

x Exact Solution Approximate Solution Absolute Error

0 0 0 0.0000E+00
0.01 0.010000333 0.010000083 2.5001E-07
0.02 0.020002667 0.020000667 2.0004E-06
0.03 0.030009003 0.030002250 6.7530E-06
0.04 0.040021347 0.040005334 1.6013E-05
0.05 0.050041708 0.050010419 3.1289E-05
0.06 0.060072104 0.060018006 5.4097E-05
0.07 0.070114558 0.070028597 8.5961E-05
0.08 0.080171105 0.080042694 1.2841E-04
0.09 0.090243790 0.090060799 1.8299E-04
0.1 0.100334672 0.100083417 2.5126E-04

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0

0.02

0.04

0.06

0.08

0.1

0.12

x

u
(x

)

 

 

Exact Solution

Approximate Solution

Figure 1: Comparison of Exact Solution and Approximate Solution.

Comparing both sides of (17), gives the continual algorithm

L[u0(x)] = L

[
2x− x4

12

]
, (18)

In general

L[un+1(x)] =
1

4s2
L[An(x)]. (19)

Taking inverse Laplace transform on above iterative steps, implies

u0(x) = 2x− x4

12
, (20)

u1(x) =
x10

207360
− x7

2016
+
x4

48
,



D. Rani, V. Mishra / Eur. J. Pure Appl. Math, 11 (1) (2018), 202-214 208

u2(x) = − x16

4777574400
+

37x13

905748480
− 11x10

2903040
+

x7

8064
,

and so on.
Thus, the solution takes the form

u(x) = 2x − x4

16
− x7

2688
+

x10

967680
+

37x13

905748480
− x16

4777574400
+ . . . .

The numerical results shown in Table 2 and Figure 2 illustrate the performance of proposed

Table 2: Comparison of Approximate Solution with Exact solution for Example 2.

x Exact Solution Approximate Solution Absolute Error

0 0 0 0.0000E+00
0.05 0.1 0.099999609 3.9063E-07
0.1 0.2 0.19999375 6.2500E-06
0.15 0.3 0.299968359 3.1641E-05
0.2 0.4 0.399899995 1.0000E-04
0.25 0.5 0.499755837 2.4416E-04
0.3 0.6 0.599493669 5.0633E-04
0.35 0.7 0.69906187 9.3813E-04
0.4 0.8 0.798399391 1.6006E-03
0.45 0.9 0.89743572 2.5643E-03
0.5 1 0.996090845 3.9092E-03

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

u
(x

)

 

 

Exact Solution

Approximate Solution

Figure 2: Comparison of Presented Approximate Solution with Exact solution.

method and maximum absolute error is 0.003.



D. Rani, V. Mishra / Eur. J. Pure Appl. Math, 11 (1) (2018), 202-214 209

3. Nonlinear Volterra integro-differential equations of the second kind

The nonlinear Volterra integro-differential equation of the second kind with difference
kernel k(x, t) = k(x− t) is defined as

u(i)(x) = f(x) +

∫ x

0
k(x− t)F (u(t))dt, (21)

where u(i)(x) denotes the ith derivative of u(x) w.r.to x, f(x) is known as source term
and F (u(x)) is the nonlinear function of u(x).
The derivative property of Laplace transform is defined by

L[u(i)(x)] = siL[u(x)]− si−1u(0)− si−2u′(0)− . . .− u(i−1)(0). (22)

Taking Laplace transform on both sides of (21) and using the properties of Laplace
transform, we get

siL[u(x)]− si−1u(0)− si−2u′(0)− . . .− u(i−1)(0) = L[f(x)] +L[k(x− t)]L[F (u(x))]. (23)

which implies

L[u(x)] =
1

s
u(0) +

1

s2
u′(0) + . . .+

1

si
u(i−1)(0) +

1

si
L[f(x)] +

1

si
L[k(x− t)]L[F (u(x))].

Adopting the same process as described in Section 2, we obtain

L

[ ∞∑
n=0

un(x)

]
=

1

s
u(0)+

1

s2
u′(0)+. . .+

1

si
u(i−1)(0)+

1

si
L[f(x)]+

1

si
L[k(x−t)]L

[ ∞∑
n=0

An(x)

]
.

(24)
The linearity property of Laplace transform gives

∞∑
n=0

L[un(x)] =
1

s
u(0)+

1

s2
u′(0)+ . . .+

1

si
u(i−1)(0)+

1

si
L[f(x)]+

1

si
L[k(x− t)]

∞∑
n=0

L[An(x)].

(25)
Matching both sides, we have the following recurrence relation

L[u0(x)] =
1

s
u(0) +

1

s2
u′(0) + . . .+

1

si
u(i−1)(0) +

1

si
L[f(x)], (26)

In general, the relation is given by

L[un+1] =
1

si
L[k(x− t)]L[An(x)]. (27)

Applying the inverse Laplace transform to (26), we get the value of u0(x), that will
define the value of A0. Using the value of A0(x), u1(x) is obtained. Continuing in this
manner we will find un(x) from the general relation given by (27). After finding the
components of infinite series, the series solution (3) follows. The proposed method will



D. Rani, V. Mishra / Eur. J. Pure Appl. Math, 11 (1) (2018), 202-214 210

be illustrated by using the following example.

Example 3. Consider the nonlinear Volterra integro-differential equation [3, 19, 21]

u′(x) = −1 +

∫ x

0
u2(t)dt, u(0) = 0. (28)

Solution.Taking Laplace transform on both sides of (28) and employing the initial
condition, we get

L[u′(x)] = L

[
−1 +

∫ x

0
u2(t)dt

]
,

L[u(x)] = − 1

s2
+

1

s2
L[u2(x)],

Substituting the series form of u(x) gives

L

[ ∞∑
n=0

un(x)

]
= − 1

s2
+

1

s2
L

[ ∞∑
n=0

An(x)

]
, (29)

Matching both sides of (29) gives the iterative algorithm

L[u0(x)] = − 1

s2
, (30)

L[un+1(x)] =
1

s2
L[An(x)]. (31)

Taking inverse Laplace transform on both sides of (30) and using the recursive relation
(31), we get

u0(x) = −x,

u1(x) =
x4

48
,

u2(x) = − x7

4032
,

u3(x) =
x10

387072
,

u4(x) = − x13

40255488
,

The series solution is therefore given by

u(x) = −x+
x4

48
− x7

4032
+

x10

387072
− x13

40255488
. . . .

Table 3 and Figure 3 show that the approximate numerical solution compared with exact
[21] is very superior having maximum absolute error 0.003.



REFERENCES 211

Table 3: Computed Exact and Approximation Solution for Example 3.

x Exact Solution Approximate Solution Absolute Error

0 0 0 0.0000E+00
0.0625 -0.0625 -0.062499682 3.1789E-07
0.125 -0.12498 -0.124994914 1.4914E-05
0.1875 -0.1874 -0.187474253 7.4253E-05
0.25 -0.24967 -0.249918635 2.4863E-04

0.3125 -0.31171 -0.31230139 5.9139E-04
0.375 -0.37336 -0.374588271 1.2283E-03
0.4375 -0.43446 -0.436737503 2.2775E-03

0.5 -0.49482 -0.498699852 3.8799E-03

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

x

u
(x

)

 

 

Exact Solution

Approximate Solution

Figure 3: Comparison of Presented Approximate Solution with Exact solution.

4. Concluding Remarks

Newton Raphson formula using as a term in Adomian polynomials exhibits the
tenability of combining Laplace transform technique and Adomian decomposition
method to solve the nonlinear Volterra integral and integro-differential equations. This is
the first time that the Adomian polynomials are modified using Newton Raphson
formula. The solution given in tables and demonstrated through figures reveals that the
approximate solution using the modified technique is very close to exact solution. Thus,
the proposed technique is easy to implement and manifest the accuracy of solution.

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