Ratio Mathematica Volume 42, 2022

Abel fractional differential equations using
Variation of parameters method

N. Nithyadevi*
P. Prakash†

Abstract

The Variation of Parameters Method (VPM) is utilized throughout
the research to identify a numerical model for a nonlinear fractional
Abel differential equation (FADE). The approach given here is used
to solve the initial problem of fractional Abel differential equations.
There is no conversion, quantization, disturbance, structural change,
or precautionary concerns in the proposed method, although it is easy
with numerical solutions. The measured values are graphed and tab-
ulated to be compared with the numerical model.
Keywords: Abel fractional differential equations; Reimann-liouville
fractional integral; Reimann-liouville fractional derivative; variation
of parameters method.1

*N.Nithyadevi (Research Scholar, Department of Mathematics, Periyar University, Salem-
636011, INDIA); nithyadevi84bu@gmail.com.

†P. Prakash (Associate Professor, Department of Mathematics, Periyar University, Salem-
636011, INDIA); pprakashmaths@gmail.com.
1Received on February 25th, 2022. Accepted on June 20th, 2022. Published on June 30th, 2022.
doi: 10.23755/rm.v41i0.723. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper
is published under the CC-BY licence agreement.

135



N. Nithyadevi and P. Prakash

1 Introduction
Abel differential equations is well established, perturbed of implementations

in structure and function in dynamics, linear systems with Stochastic, modeling
technique, and linear algebra. Numerous research has been done mostly on meth-
ods of the Abel differential equations. we suggest several methods, including
the iterative method of the Abel differential equation achieved from the variation
iteration method. We consider the following nonlinear FADE:

Dβf(x) = P1f
3(x) + P2f

2(x) + P3f(x) + P4, 0 < β ≤ 1, 0 ≤ x ≤ R. (1)

with the initial condition

fk(0) = uk, k = 1, 2, 3, ....n − 1. (2)

where P1, P2, P3 and P4 are arbitrary number, Dβ is the fractional derivative for
order β and f(x) is unknown function of the crisp variable x, k is an integer. How-
ever, assume IVPs (0.2) and each x > 0 has a unique fuzzy solution. The VPM
calculation is a novel numeric plan created to examine and decipher the arrange-
ment of first and second request dubious IVPs. Computer simulation, biophysics,
synthetic science, applied mathematics, geophysics, material science, harmonies,
and other domains rely heavily on linear and non-linear fractional equations. This
is seeing fractional derivatives must store the record of the parameter under delib-
eration. The proposed approach targets constructing an answer of a VPM devel-
opment just as limiting remaining blunder capacities for processing the obscure
coefficients of VPM by applying a specific differential administrator without lin-
early or constraint on the structure. Again, we refer to see numerous qualities to
show and reconsider some radical strategies for managing the various problems
that occur in ordinary miracles.

2 Preliminaries
The definitions of significance and associated characteristics of the hypothesis

are examined in this section.

Definition 2.1. The fractional component of Riemann-Liouville, f the valued
function of the fuzzy number β is considered to include Jβαf(x) =

1
Γ(β)

∫ x
0

f(ξ)
(x−ξ)1−β dξ

, x > a where Γ(β) is the famous Gamma characteristic.

Definition 2.2. The Riemann-Liouville fractional order derivative β of the crisp
function f almost everywhere on I exists and can be represented by RLa D

βf(x) =
1

Γ(m−β)
dm

dxm

∫ x
0
f(ξ)(x − ξ)m−β−1dξ, where m − 1 ≤ β < m ∈ Z+.

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Abel fractional differential equations using Variation of parameters method

Definition 2.3. The modified Riemann-Liouville fractional order derivative β of
the crisp function f almost everywhere on I exists and can be represented by
RL
a D

βf(x) = 1
Γ(m−β)

dm

dxm

∫ x
0
(f(ξ) − f(0))(x − ξ)m−β−1dξ, where m − 1 ≤ β <

m, and m ≥ 1.

Remark 2.1. .

(i) Riemann-Liouville derivative does not satisfy Dαα(1) = 0 (D
α
α(1) = 0 for

the Caputo derivative), if α is not a natural number

(ii) All fractionals do not satisfy the known formula of the derivative of product
of the two functions:

Dαα(fg) = f(D
α
α(g)) + g(D

α
α(f))

(ii) All fractionals do not satisfy the known formula of the derivative of quotient
of the two functions:

Dαα(
f
g
) =

g(Dαα(f))−f(Dαα(g))
g2

3 Variation of parameters method
We consider the extensive expression to derive the main definition of the VPM

L(u) + N(u) + R(u) = f(x), a ≤ x ≤ b (3)

where L, N operators are linear and non-linear. R is a linear differential operator
but L has the highest order than R, f(x) is a source term in the given domain [a, b].
We have the following equation solution by using the VPM

u(x) =
∑k−1

l=0
pl+1x

l

l!
+
∫ x
0
f(x, α)(−N(u)(α) − R(u)(α) + f(α))dα, (4)

where k represent the order of given differential equation and Cl where l =
1, 2, 3,are unknown. So

u(x) =
∑k−1

l=0
pl+1x

l

l!
(5)

For homogeneous solution which is used by

L(u) = 0. (6)

137



N. Nithyadevi and P. Prakash

Another component obtained from definition (0.2) from VPM is∫ x
0
f(x, α)(−N(u)(α) − R(u)(α) + f(α))dα. (7)

Where, f(x, α) is a Lagrange multiplier which that eliminates the incremental use
of integers in the inverse problem and is dependent on the order of equations. The
explore description is based on calculating the function of the variable f(x, α)
from either a set of numbers.

f(x, α) =
∑k−1

l=0
(−1)l−1αl−1xk−1

(l−1)!(k−1)! =
(x−α)k−1
(k−1)! , (8)

The sequence of the specific differential equations differs with k. We have always
had the following criteria to explore:

k = 1, f(x, α) = 1,
k = 2, f(x, α) = (x − α)
k = 3, f(x, α) = x

2

2!
+ α

2

2!
− αx

(9)

As a result, we utilize its investigation to improve, the system to solve equations

un+1 = u0 +
∫ x
0
f(x, α)(−N(u)(α) − R(u)(α) + f(α))dα. (10)

Using initial conditions, we can obtain the initial guess u0(x). We improve our
estimate by using a specific value for the input parameter in each iteration. We
are using Reimann-Liouville to solve the fractional Abel differential condition.
When we combine VPM with a fractional integral for the arrangement process,
the iterative plan for fractional equations is

un+1 = u0 +
1

Γ(β)

∫ x
0
f(x, α)x−α(−N(u)(α) − R(u)(α) + f(α))dα. (11)

4 Numerical Examples
Example 4.1. We Consider the following fractional Abel problem,

Dβf(x) − 3f3(x) + f(x) = 1, 0 < β ≤ 1, x > 0, (12)

with initial condition f(0) = 1
3

can be found as follows:

f(x) = 1√
6e2x+3

. (13)

For the above problem, we create the iterative scheme shown below

fn+1 = f0 +
1

Γ(β)

∫ x
0
(x − α)β−1(3f3(x) − f(x))dα (14)

138



Abel fractional differential equations using Variation of parameters method

x Exact value VPM Absolute error
0.0 [0.3333] [0.2916] [0.0417]
0.1 [0.3111] [0.3062] [0.0049]
0.2 [0.2892] [0.2730] [0.0162]
0.3 [0.2679] [0.2531] [0.0148]
0.4 [0.2472] [0.2374] [0.0098]
0.5 [0.2275] [0.2164] [0.0111]
0.6 [0.2088] [0.1943] [0.0145]
0.7 [0.1912] [0.1804] [0.0108]
0.8 [0.1748] [0.1692] [0.0056]
0.9 [0.1595] [0.1439] [0.0156]
1.0 [0.1453] [0.1379] [0.0074]

Table 1: Value of f(x), β = 1

x β = 0.7 β = 0.8 β = 0.9
0.0 0.3201 0.3197 0.2932
0.2 0.2817 0.2809 0.2782
0.4 0.2402 0.2397 0.2381
0.6 0.2007 0.1991 0.1962
0.8 0.1731 0.1706 0.1699
1.0 0.1497 0.1399 0.1388

Table 2: Different values of β

Taking initial condition f(0) = 1
3
, the following results for β = 1 are pro-

duced:

f1(x) =
1
3
− 2

9
x

f2(x) =
7
24

− 4
9
x + 4

81
x3 − 2

243
x4

f3(x) =
7
24

− 3049
4608

x + 5
96
x2 + 2

9
x3 − 133

1728
x4 − 881

3880
x5 + 13

729
x6 − 2

1701
x7

− 13
8748

x8 + 5
13122

x9 + 8
885735

x10 − 32
1948617

x11 + 4
1594323

x12 − 8
62178597

x13.

Table 1 show a approximate solution and exact solution for β = 1. Table 2
shows different values of β. Fig 1 represents the exact and approximate solution.

139



N. Nithyadevi and P. Prakash

Figure 1: Value of f(x)

Example 4.2. We Consider the following fractional Abel problem,

Dβf(x) + f3(x) − f(x) = 1, 0 < β ≤ 1, x > 0, (15)

with initial condition f(0) = 1
3

can be found as follows:

f(x) = e
t

√
e2t+8

. (16)

For the above problem, we create the iterative scheme shown below

fn+1 = f0 +
1

Γ(β)

∫ x
0
(x − α)β−1(f(x) − f3(x))dα (17)

Taking initial condition f(0) = 1
3
, the following results for β = 1 are produced:

f1(x) =
1
3
+ 8

27
x

f2(x) =
31
96

+ 16
27
x + 8

81
x2 − 64

2187
x3 − 128

19683
x4

f3(x) =
31
96
+ 780193

884736
x+ 1045

3456
x2− 11201

93312
x3− 38591

419904
x4− 2657

157464
x5+ 784

177147
x6+ 3776

1594323
x7

− 2240
14348907

x8 − 100864
1162261467

x9 − 851968
52301766015

x10 − 131072
345191655699

x11 + 262144
847288609443

x12 +
2097152

99132767304831
x13.

140



Abel fractional differential equations using Variation of parameters method

x Exact value VPM Absolute error
0.0 [0.3333] [0.3229] [0.0104]
0.1 [0.3639] [0.3508] [0.0131]
0.2 [0.3964] [0.3872] [0.0092]
0.3 [0.4307] [0.4139] [0.0168]
0.4 [0.4665] [0.4477] [0.0188]
0.5 [0.5035] [0.4972] [0.0063]
0.6 [0.5415] [0.5128] [0.0287]
0.7 [0.5799] [0.5524] [0.0275]
0.8 [0.6183] [0.6106] [0.0077]
0.9 [0.6561] [0.6392] [0.0169]
1.0 [0.6929] [0.6548] [0.0381]

Table 3: Value of f(x), β = 1

x β = 0.7 β = 0.8 β = 0.9
0.0 0.3301 0.3299 0.3271
0.2 0.3956 0.3907 0.3882
0.4 0.4641 0.4596 0.4481
0.6 0.5364 0.5209 0.5197
0.8 0.6180 0.6159 0.6132
1.0 0.6877 0.6792 0.6674

Table 4: Different values of β

Table 3 shows a approximate solution and exact solution for β = 1. Table 4
shows different values of β. Fig 2 represents the exact and approximate solution.

141



N. Nithyadevi and P. Prakash

Figure 2: Value of f(x)

5 Conclusions
In this study, we examined the formulation to fractional Abel equation with

the Variation of Parameters method. We demonstrate that VPM is a functional,
efficient approach for the achievement of empirical and numerical testing for a
wide variety of nonlinear fractional equations. We discovered that obtained esti-
mate effects for different beta values interact simultaneously before the first-order
derivative is exceeded.

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