1. Introduction
In recent years, fractional calculus has made a great contribution to the fields of science and engineering due to its many applications in the fields of damping visco elasticity, biology, electronics, genetic algorithms, signal processing, robotic technology, traffic systems, telecommunication, chemistry, physics, and economics and finance. This has all been possible due to such mathematicians as Riemann, Liouville, Leibniz, Euler, Bernoulli, Wallis, and L’ Hospital, who played an important role in the development of fractional calculus. In this regard, our research focuses on the following fractional order equations of Helmholtz, which are important in fractional calculus.
and initial condition
The Helmholtz equation is used in the study of physical problems consisting of partial differential equations in space-time, such as scattering problems in electromagnetism and acoustics in many areas, i.e., in aeronautics, marine technology geophysics, and optical problems. For further applications and studies about the concern problem, see the research study [
1,
2,
3] and references therein.
In fact, in fractional calculus, many researchers have focused on the various schemes and aspects of partial differential equations and fractional order partial differential equations (FPDEs) as well, see [
4,
5,
6,
7,
8,
9,
10,
11]. In this regard, various types of techniques have been developed for numerical solutions of non-linear and linear differential equations of integer order. However, there are very few schemes that have been extended to find the solution of linear and nonlinear differential equations of fractional order; for reading, see [
12,
13,
14,
15,
16,
17]. In this article, we desire to contribute to and extend the recent technique asymptotic homotopy perturbation method (AHPM) for the solution of real-world problems. In 2019, AHPM [
18] was used for the first time for the solution of the Zakharov–Kuznetsov equation. The main aim of our work is to introduce the proposed technique, which is easy to apply and more efficient than existing procedures. In this concern, we introduce and apply the asymptotic homotopy perturbation method (AHPM) to obtain the approximate solution of fractional order Helmholtz Equations (
1) and (
2). In addition, we compare the AHPM solution to the exact solution as well as to the HPETM solution.
3. Construction of the Method (AHPM)
Here, in this section, we will discuss that the best way to establish AHPM procedure to solve fractional order problem in the following form
where
is a differential operator that may consist of ordinary, partial, or time-fractional differential or space fractional derivative.
can be expressed for fractional model as follows:
and condition
where
denotes the Caputo derivative operator;
N may be linear or non linear operator;
B denotes a boundary operator;
is unknown exact solution of above equation;
denotes known function; and
denote special and temporal variables, respectively.
Let us construct a homotopy
satisfies
where
is said to be an embedding parameter. In this phase, the proposed deformation Equation (
10) is an alternate form of the deformation equations as:
and
in HPM, HAM, and OHAM proposed by Liao in [
20], He in [
21], and Marinca in [
22], respectively.
Basically, according to homotopy definition, when and we have
Obviously, when the embedding parameter
p varies from 0 to 1, the defined homotopy ensures the convergence of
to the exact solution
. Consider
in the form
and assuming
as follows
where
are arbitrary auxiliary functions, will be discussed later. Thus, if
and
in Equation (
10), we have
respectively.
It is obvious that the construction of introduced auxiliary function in Equation (
15) is different from the auxiliary functions that are proposed in articles [
20,
21,
22]. Hence, the procedure proposed in our paper is different from the procedures proposed by Liao, He, and Marinca in aforesaid papers [
20,
21,
22] as well as optimal homotopy perturbation method (OHPM) in [
23].
Furthermore, when we substitute Equations (14) and (15) in Equation (
10) and equate like power of
p, the obtained series of simpler linear problems are
and
kth order iteration is
We obtain the series solutions by using the integral operator on both sides of the above simple fractional differential equation. The convergence of the series solution Equation (
14) to the exact solution depends upon the auxiliary parameters (functions)
. The choice of
is purely on the basis of terms that appear in non-linear section of the Equation (
6). Equation (
14) converges to the exact solution of Equation (
6) at
:
Particularly, we can truncate Equation
into finite m-terms to obtain the solution of nonlinear problem. The auxiliary convergence control constants
can be found by solving the system
It can be verified to observe that HPM is only a case of Equation (
10) when
and
The HAM is also a case of Equation (
10) when
and
The OHAM is also another case when
in Equation (
15), we obtain exactly the series problems that are obtained by OHAM after expanding and equating the like power of
p in deformation equation. Furthermore, concerning the optimal homotopy asymptotic method (OHAM) mentioned in this manuscript and presented in [
22], the version of OHAM proposed in 2008 was improved over time, and the most recent improvement, which also contains auxiliary functions, is presented in papers [
24,
25]. We also have improved the version of OHAM by introducing a very new auxiliary function in Equation (
15). This paper uses a new and more general form of auxiliary function:
that depends on arbitrary parameters
and is useful for adjusting and controlling the convergence of nonlinear part as well as linear part of the problem with simple way.