research paper on homotopy perturbation method

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• Published: 31 March 2016

Homotopy perturbation method: a versatile tool to evaluate linear and nonlinear fuzzy Volterra integral equations of the second kind

• S. Narayanamoorthy 1 &
• S. P. Sathiyapriya 1  

SpringerPlus volume  5 , Article number:  387 ( 2016 ) Cite this article

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In this article, we focus on linear and nonlinear fuzzy Volterra integral equations of the second kind and we propose a numerical scheme using homotopy perturbation method (HPM) to obtain fuzzy approximate solutions to them. To facilitate the benefits of this proposal, an algorithmic form of the HPM is also designed to handle the same. In order to illustrate the potentiality of the approach, two test problems are offered and the obtained numerical results are compared with the existing exact solutions and are depicted in terms of plots to reveal its precision and reliability.

Integral equations find special applicability within many scientific and mathematical disciplines. Indeed modeling physical problems using integral equations with the exact parameters is not always easy but also impossible in the real problems. For this purpose, one way is using uncertainty measures for handling such lack of information. One of the most recent approaches is using fuzzy concept (Zadeh 1965 ). So instead of using deterministic models of integral equations, we can go in for fuzzy integral equations. Hence there occurs a need to develop mathematical models and numerical procedures that would appropriately treat general fuzzy integral equations and solve them. In this paper, we apply homotopy perturbation method (HPM) to solve both linear and nonlinear fuzzy Volterra integral equations of the second kind (FVIE-2). Many research papers dealing with fuzzy integral equations exists in open literatures and some of them are reviewed and cited here for better understanding of the present analysis. We know that solving fuzzy integral equations requires appropriate definitions of fuzzy function and fuzzy integral of a fuzzy function. We refer to the reader (Dubois and Prade 1978 ) for basic arithmetic operations on fuzzy numbers and he also presented the elementary fuzzy calculus (Dubois and Prade 1982 ) based on extension principle. Alternative approaches were later suggested by others (Goetschel and Voxman 1986 ; Kaleva 1987 ). Various methods and applications of linear and nonlinear integral equations were also reported (Wazwaz 2011 ). The fuzzy integral equations (Friedman et al. 1999 ; Subrahmanyam and Sudarsanam 1996 ) is one of the most important fields of fuzzy set theory. The concepts of fuzzy integral equations and fuzzy integro-differential equations have motivated a large amount of research works (Mirzaee et al. 2012 ; Hussain and Ali 2013a , b ; Mosleh and Otadi 2013 ; Allahviranloo et al. 2014 ; Otadi and Mosleh 2014 ) in last decades because of its applications in scientific phenomena. Therefore investigating fuzzy integral equations by finding accurate and efficient methods for solving these equations has become an active research undertaking.

Homotopy perturbation method (He 1999 ) is a perturbation technique coupled with the homotopy technique was developed by He JH and was further improved by him (He 2000 , 2003 , 2004 ). In homotopy perturbation method, a complicated problem under study is continuously deformed into a simple problem which is easy to solve to obtain an approximate solution. HPM yields a very rapid convergence of the solution series in most cases, usually only few iterations lead to very accurate solutions. This method has been applied to many problems (Chun 2007 ; Filobello-Nino et al. 2014b ) and also in particular to integral equations (Abbasbandy 2007 ) and fuzzy integral equations (Matinfar and Saeidy 2010 ). In recent past, various analytical and numerical methods were used such as Adomian decomposition method, direct computation method, series solution method, successive approximation method and conversion to integral equations. However, these methods are not easy to use and require tedious calculations. But HPM was proved (Saberi-Nadjafi and Ghorbani 2009 ) to be an effective and reliable tool for handling most of the linear and nonlinear differential, ordinary and partial, as well as linear and nonlinear integral equations. The main reason consists in the fact that this gives flexibility in the choice of basis functions for the solution and does not involve linear inversion operators (as compared to the Adomian decomposition method), while still retaining a simplicity that makes the method easily understandable from the standpoint of general perturbation methods.

Fuzzy integral equations are important in studying and solving a large proportion of problems in various topics in applied mathematics and it also arise in many industrial fields, such as electromagnetic fields and thermal problems. It is well-known that a large class of initial and boundary value problems can be converted to Volterra integral equations and also the problem of heat conduction with a variable heat transfer coefficient is reduced to the solution of Volterra integral equations of the second kind. Several authors applied HPM to solve linear and nonlinear partial differential equations of fractional order (Momani and Odibat 2007a , b ), Volterra integral equations (Grover and Tomer 2011 ), singularly perturbed Volterra integral equations (Alnasr and Momani 2008 ) and n-th order fuzzy linear differential equations (Tapaswini and Chakraverty 2013 ). The reader is also referred to two recent papers where the authors (Filobello-Nino et al. 2014a ; Vazquez-Leal and Sarmiento-Reyes 2015 ) make use of HPM in their application oriented works. In the very recent studies, it has been found that homotopy perturbation method was also used to solve fractional fisher’s equation (Hamdi Cherif et al. 2016 ) and a system of nonlinear chemistry problems (Ramesh Rao 2016 ). Taking into account all these specifications, the proposed work is mainly dealt with fuzzy Volterra integral equations. Up to authors’ knowledge no research has been carried out so far by constructing a numerical algorithm using HPM for solving fuzzy Volterra integral equations of the second kind and our work is aimed at these.

The paper is organized as follows: the fuzzy Volterra integral equations are recalled with the required theorem and the basic concept of HPM is provided. Our main results are stated in the numerical technique, which presents the detailed description of the proposed method. Besides we introduce the algorithm for solving FVIE-2. Additionally test problems are included in the section ‘Illustrative examples’ and concluding results are made over them. Finally, a brief conclusion is provided.

Some of the basic notations and definitions that are not described in this paper are standard and usual. One can refer to the authors (Dubois and Prade 1978 ; Goetschel and Voxman 1983 , 1986 ; Puri and Ralescu 1983 , 1986 ; Kaleva 1987 ) for the notion of fuzzy numbers and its arithmetic operations, Hausdorff distance between fuzzy numbers, continuity and existence of definite integral of fuzzy functions.

• Fuzzy Volterra integral equations

The integral equations which are discussed in this section are the Volterra integral equations of the second kind. The Fredholm integral equation of the second kind (Hochstadt 1973 ) is given by

where λ  > 0, a and b are constants, k ( x ,  t ) is an arbitrary kernel function over the square a  ≤  x , t  ≤  b and f ( x ) is a function of a  ≤  x  ≤  b .

If the kernel satisfies k ( x ,  t ) = 0, x  >  t ,

we obtain the Volterra integral equation of the second kind and is given by

In the above equation a refers to constant and x is a variable. If f ( x ) is a crisp function, then the solutions of the following equations

are crisp as well, where \(\left( {\underline{f} \left( {x;\alpha } \right),\overline{f} \left( {x;\alpha } \right)} \right)\) is the parametric form of f ( x ). However, if f ( x ) is a fuzzy function these equations possess fuzzy solutions.

Let F : I  →  E 1 be a fuzzy function. The integral of F over I , denoted by \(\int_{I} {F\left( x \right)dx} {\text{ or }}\int_{a}^{b} {F\left( x \right)dx}\) , is defined levelwise by

for each 0 ≤  α  ≤ 1.

Now the parametric form of fuzzy Volterra integral equation of the second kind (FVIE-2) are as follows (Ghanbari 2010 )

Let ( \(\underline{f} \left( {x,\alpha } \right)\) , \(\overline{f} \left( {x;\alpha } \right)\) ) and ( \(\underline{F} \left( {x;\alpha } \right)\) , \(\overline{F} \left( {x;\alpha } \right)\) ), be the parametric forms of f ( x ) and F ( x ) respectively, 0 ≤  α  ≤ 1, a  ≤  x  ≤  b . Then the parametric forms of FVIE-2 are given by

for 0 ≤  α  ≤ 1, where

The following theorem provides sufficient conditions for the existence of a unique solution to Eq. ( 1 ), where f ( x ) is a fuzzy function.

(Wu and Ma 1990 ) Let \(k\left( {x, t} \right)\) be continuous for a   ≤   x, t   ≤   b, λ   >  0 , and f ( x ) is a fuzzy continuous function of x, a   ≤   x   ≤   b. If \(\lambda < \frac{1}{{M\left( {b - a} \right)}}\) , where \(M = \begin{array}{*{20}c} {\hbox{max} } \\ {\left( {a \le x, t \le b} \right)} \\ \end{array} \left| {k\left( {x,t} \right)} \right|\) , then the iterative procedure

converges to the unique solution of Eq.  ( 1 ) . Specifically,

where L   =   λM ( b   −   a ) . This refers that F k ( x ) converges uniformly in x to F ( x ) , i.e., given arbitrary ɛ   >   0 we can find N such that

The proof this theorem can be easily extended for fuzzy Volterra integral equations of the second kind, i.e., for Eq.  ( 2 ) where f ( x ) is a fuzzy function as well.

Basic concept of homotopy perturbation method and its application to Volterra integral equations

For convenience of the reader, we present the review of the HPM. The essential idea of this method is to introduce a homotopy parameter, say p , which takes the values from 0 to 1. When p  = 0, the system of equation usually reduces to a sufficiently simplified form, which normally admits a rather simple solution. As p gradually increases to 1, the system goes through a sequence of deformation, the solution of each of which is close to that at the previous stage of deformation. Eventually at p  = 1, the system takes the original form of the equation and final stage of deformation gives the desired solution.

To figure out how HPM works, consider the nonlinear integral equation

with boundary conditions \(B\left( {u, \frac{\partial u}{\partial n}} \right) = 0\) , \(r \epsilon \varGamma\) ,

where L is a linear operator, N is a nonlinear operator, B is a boundary operator, Γ is the boundary of the domain \(\varvec{\varOmega}\) and f ( r ) is a known analytic function. In order to use the HPM, a suitable construction of homotopy is of vital importance. If L ( u ) = 0 with some possible unknown parameter can best describe the original nonlinear system. Generally, a homotopy can be constructed as (He 1999 , 2000 )

where \(p \epsilon [0, 1]\) and \(r \epsilon {\varvec\varOmega }\) .

The nonlinear Volterra integral equation is given by

where γ ( x ) is an unknown function that will be determined, k ( x ,  t ) is the kernel of the integral equation, f ( x ) is an analytic function, R ( γ ) and N ( γ ) are linear and nonlinear function of γ respectively.

To illustrate the HPM, we reconstitute Eq. ( 14 ) as

with the solution u ( x ) =  γ ( x ) and we define the homotopy H ( U ,  p ) by

where F ( u ) is a functional operator with solution, say u 0 , which can be obtained easily. We can choose a homotopy

and continuously trace an implicitly defined curve from a starting point \(H\left( {u_{0} , 0} \right)\) to a solution function H ( γ , 1). The embedding parameter p monotonically increases from 0 to 1 as the trivial problem F ( u ) = 0 continuously deformed to the original problem L ( u ) = 0. The embedded parameter \(p \epsilon [0, 1]\) can be considered as an expanding parameter.

When p  → 1, Eq. ( 17 ) corresponds to Eq. ( 15 ) and Eq. ( 18 ) becomes the approximate solution of Eq. ( 14 ), i.e.,

The above series is convergent in most of the cases and also the rate of convergence depends on L ( u ) (He 2000 ).

In other words, it is very natural to assume that the solution of [Eqs. ( 16 ) and ( 17 )] can be expressed as

Equating the terms with identical powers of p , we get

Therefore when p  = 1, the approximate solution of above equation can be readily obtained as follows

The above series is convergent for most cases.

Proposed numerical technique for solving FVIE-2 using HPM

As HPM is considered as a combination of the classical perturbation technique and the homotopy (whose origin is in the topology), but not restricted to small parameters as occur with traditional perturbation methods, because HPM neither requires small parameter nor linearization, but only few iterations to obtain highly accurate solution. Hence we introduce the recursive scheme for solving FVIE-2 as follows

In view of homotopy, we can define the following convex homotopy,

where α is the fuzzy parameter (0 ≤  α  ≤ 1).

The solution of Eq. ( 23 ) is assumed in the following form

where \(\left( {\underline{V}_{i} , \bar{V}_{i} } \right) \forall i\) are unknown functions to be determined.

The initial approximation is taken as

Substituting Eqs. ( 24 ) and ( 25 ) in Eq. ( 23 ) reduces to

Equating the coefficients with like powers of p , we yield the following iterations

Finally, the solution of FVIE-2 is given by

In general, we obtain the following iteration formulae

These relations [Eq. ( 31 )] will enable us to determine the components \(\underline{u} \left( {x, \alpha } \right)\) and \(\bar{u}\left( {x, \alpha } \right)\) recursively for i  ≥ 0.

Algorithm of the approach

On the basis of our proposed results using HPM, which are discussed in the above section, we convert to a numerical HPM algorithm that is one of the main results of this paper. Hence, in an algorithmic form, HPM can be expressed and implemented as follows.

The Eqs. ( 5 ) and ( 6 ) with the iteration index i  ≥ 0, and a appropriate value for the tolerance (here \(Tol = 10^{ - 5}\) ).

For reckoning the initial data, let \(\underline{V}_{0} = \underline{V}\) and \(\bar{V}_{0}\)  =  \(\bar{V}\) where x  = 0

Use the recursive relations [Eq. ( 31 )] to compute the values of \(\underline{V}_{i + 1} \left( x \right)\) by using \(\underline{V}_{i} \left( x \right)\) and \(\bar{V}_{i + 1} \left( x \right)\) by using \(\bar{V}_{i} \left( x \right)\)

If \(\left\{ {\hbox{max} \left| {\underline{V}_{i + 1} \left( x \right) - \underline{V}_{i} \left( x \right)} \right| < Tol, max\left| { \bar{V}_{i + 1} \left( x \right) - \bar{V}_{i} \left( x \right)} \right| < Tol} \right\}\) , then go to step 4, else set i  =  i  + 1 and go to step 2

Print \(\underline{u} \left( x \right) = \sum\nolimits_{i = 0}^{\infty } {\underline{V}_{i} }\) and \(\bar{u}\left( x \right) = \sum\nolimits_{i = 0}^{\infty } {\bar{V}_{i} }\) as the approximates of the exact solutions

Illustrative examples

To give a clear overview and to demonstrate the efficiency of the homotopy perturbation method, we implement the proposed numerical technique by solving a linear FVIE-2 and a nonlinear FVIE-2 with known exact solutions.

Linear fuzzy Volterra integral equations of the second kind

We consider the linear FVIE-2 given by

where λ  = 1, a  = 0, k ( x ,  t ) =  sinhx and

The exact solution in this case is given by

Here we have

To solve the given equation by HPM we construct a convex homotopy as follows

Now we make use of the iteration formulae [Eq. ( 31 )] and applying the numerical algorithm, the HPM solution series with few iterative terms are as follows

\(p^{0} :( \underline{V}_{0} \left( {x,\alpha } \right), \bar{V}_{0} \left( {x,\alpha } \right)\) ) is the initial fuzzy approximations,where

In the same way, the iterations \(p^{2} :\left( { \underline{V}_{2} \left( {x,\alpha } \right), \bar{V}_{2} \left( {x,\alpha } \right)} \right)\) and \(p^{3} :\left( {\underline{V}_{3} \left( {x,\alpha } \right), \bar{V}_{3} \left( {x,\alpha } \right)} \right)\) were also computed for calculating fuzzy approximate solutions which could not be stated here as the iterations contain very long expressions.

We approximate \(\underline{F} \left( {x,\alpha } \right)\) and \(\bar{F}\left( {x,\alpha } \right)\) by setting p  = 1 and obtain the following fuzzy solutions, which are given by

• Error analysis

Absolute errors are computed as

The components of iteration formulae [Eq. ( 31 )] were obtained by the Mathematica program (Mathematica package version 5.2) according to the numerical algorithm where the tolerance is given a suitable positive value. The numerical results of the obtained approximate solutions are compared with the exact solutions for different α - values and are presented in Table  1 . Moreover exact and approximate solutions are shown graphically in Fig.  1 .

Plots of exact and approximate solutions of Example 1

Nonlinear fuzzy Volterra integral equations of the second kind

We consider the nonlinear FVIE-2 given by

where λ  = 1, a  = 0, k ( x ,  t ) =  x 2 (1 − 2 t ) and

\(p^{0} :\left( {\underline{V}_{0} \left( {x,\alpha } \right), \bar{V}_{0} \left( {x,\alpha } \right)} \right)\) is the initial fuzzy approximations,

In the same way, the iterations \(p^{2} :\left( {\underline{V}_{2} \left( {x,\alpha } \right), \bar{V}_{2} \left( {x,\alpha } \right)} \right)\) and \(p^{3} :\left( {\underline{V}_{3} \left( {x,\alpha } \right), \bar{V}_{3} \left( {x,\alpha } \right)} \right)\) were also computed for calculating fuzzy approximate solutions which could not be stated here as the iterations contain very long expressions.

We approximate \(\underline{F} \left( {x,\alpha } \right)\) and \(\bar{F}\left( {x,\alpha } \right)\) by setting p  = 1 and the obtained fuzzy solutions are given by

By the same way of the previous example, we obtained the components of iteration formulae [Eq. ( 31 )] by using the Mathematica program (Mathematica package version 5.2) according to the numerical algorithm where the tolerance is given a suitable positive value. The numerical results of the obtained approximate solutions are compared with the exact solutions for different α - values and are presented in Table  2 . Moreover exact and approximate solutions are shown graphically in Fig.  2 .

Plots of exact and approximate solutions of Example 2

Concluding remarks on numerical results

By looking into the computed values presented above (Tables  1 , 2 ), one may conclude that the obtained approximate solutions by applying the proposed method (HPM) are in high agreement with the exact solutions. Furthermore, from the graphical representation (Figs.  1 , 2 ), it is apparent that the plot of exact and approximate solutions merely coincide due to the occurrence of very minimum amount of errors ( \(\underline{E} \left( {x,\alpha } \right)\) and \(\bar{E}\left( {x,\alpha } \right)\) ). The fuzzy approximate solutions are calculated at four iterations for the numerical examples considered. The results presented clearly shows that the iterative steps number 3–4 in the numerical algorithm accompany for the convergence of the fuzzy solution so that the computations are minimized and the accuracy of the approximate values is also achieved with exceptionally few iteration and overall errors are reduced.

Conclusions

In this paper, we developed a simple yet versatile numerical technique using HPM and successfully applied it by means of the proposed algorithm for solving the linear and nonlinear fuzzy Volterra integral equations of the second kind. Mathematica has been used for computations in this paper. The examples analyzed illustrates that the answers are trusty and reveals the effectiveness of the proposed method. The reliability of HPM due to its precise results and the reduction in computation gives HPM a wider applicability. Moreover, it avoids the tedious work needed by the traditional numerical methods. Advantage of the proposed method lies in the free selection of initial approximation in a straightforward manner and also it overcomes the drawbacks of handling larger equations. This method does not involve any discretization of variables and hence it is free from rounding off errors. One can apply this method to higher order equations also.

Further research can be focused on studying the existence and convergence of HPM for linear and nonlinear FVIE-2. Finally we conclude that this work provides the applicable computational technique will aid the practical applications already used in the design of various fuzzy dynamical systems and it is an issue of considerable importance at this current trend due to its versatile nature.

Abbreviations

homotopy perturbation method

fuzzy Volterra integral equation of the second kind

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Narayanamoorthy, S., Sathiyapriya, S.P. Homotopy perturbation method: a versatile tool to evaluate linear and nonlinear fuzzy Volterra integral equations of the second kind. SpringerPlus 5 , 387 (2016). https://doi.org/10.1186/s40064-016-2038-3

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Original research article, the aboodh transformation-based homotopy perturbation method: new hope for fractional calculus.

• 1 School of Mathematics and Statistics, Huanghuai University, Zhumadian, China
• 2 Department of Mathematics, Government College University, Faisalabad, Pakistan
• 3 School of Mathematics and Statistics, Nanyang Normal University, Nanyang, China

Fractional differential equations can model various complex problems in physics and engineering, but there is no universal method to solve fractional models precisely. This paper offers a new hope for this purpose by coupling the homotopy perturbation method with Aboodh transform. The new hybrid technique leads to a simple approach to finding an approximate solution, which converges fast to the exact one with less computing effort. An example of the fractional casting-mold system is given to elucidate the hope for fractional calculus, and this paper serves as a model for other fractional differential equations.

1 Introduction

Fractional calculus has triggered much interest in both physics and mathematics [ 1 , 2 ]. Traditional differential equations cannot accurately represent many physical problems, and the fractional partner can provide deeper insight into these complex physical phenomena with ease. In general, this newly developed field is for studying real-world applications in the fractal space, so most literature labeled it as the fractal–fractional calculus [ 3 – 5 ] or the local fractional calculus on the Cantor set [ 6 ]. A continuum medium, e.g., water or air, becomes a fractal space (porous medium) when we observe it on a molecule’s scale. Any phenomena arising in molecules’ perturbation have to be modeled by the fractal–fractional model [ 7 ]. As an example, we consider a nanoparticle’s motion in the air, which is stochastic and difficult to be modeled by the traditional differential equation; however, if the air is considered as a fractal space on a molecule’s scale, its motion is determinate and can be modeled by the fractal–fractional model. So, we need two scales for a porous medium; one is large enough so that the continuum assumption works, and the other is small enough so that the porosity can be measured, as pointed out by Ji-Huan He that “seeing with a single scale is always unbelieving” [ 8 ]. Another example is the motion of the Moon, which is naturally periodic; however, if we measure its motion at an extremely far distance, its motion becomes stochastic, and the Heisenberg-like uncertainty principle works for the Moon [ 9 ]. He and Qian showed that the fractal diffusion process in water depends on the fractal dimensions [ 10 ], and other scientists also discussed the fractional advection–reaction–diffusion process [ 11 ] and the fractal diffusion–reaction process [ 12 ]. A cocoon’s air/moisture permeability and its thermal property can best be revealed by the fractal–fractional model [ 13 , 14 ], and the fractal micro-electromechanical systems show even more amazing properties [ 15 – 18 ].

Fractional calculus is a good and reliable tool for scientists and engineers but a mixed blessing for practical applications because an intractable problem arises; that is, fractional models are extremely difficult to be solved. Researchers have been racing to test various analytical methods which were originally proposed to solve traditional differential equations. Though there are many famous analytical methods in the literature, for example, the homotopy perturbation method [ 19 – 23 ] and its various modifications [ 24 – 26 ], the decomposition method [ 27 ], the variational iteration method [ 28 – 30 ], the exp-function method [ 31 ], and the differential transform method [ 32 ], so far, there is not a universal approach to solving exactly fractional differential equations, and this paper offers a new hope for this purpose by coupling the homotopy perturbation method [ 33 , 34 ] and the Aboodh transform [ 35 ].

The homotopy perturbation method (HPM) was first proposed by Chinese mathematician Prof. Ji-Huan He in the later 1990s [ 33 ]; it is mathematically simple and physically insightful. The method is equally suitable for linear or non-linear, homogeneous or inhomogeneous, and initial and/or boundary value problems. The solution is expressed in an infinite series and typically converges to the exact solution. The HPM is now considered a matured tool for almost all kinds of problems, and many researchers have used this method for an accurate insight into the solution properties of a complex problem [ 36 – 38 ].

The Aboodh transform (AT) was proposed by Aboodh [ 35 ] and derived from the classical Fourier integral. This transform is now considered a simple technique for solving linear differential equations but is unable to solve non-linear ones. By coupling AT with the HPM, one has the capability to solve linear and non-linear problems, and a lot of literature works have been witnessed to utilize this coupling for solving various types of problems. Using AT–HPM, Manimegalai et al. [ 39 ] solved strongly non-linear oscillators with great success. Jani and Singh [ 40 ] found it had obvious advantages over the decomposition method, Yasmin [ 41 ] revealed the dynamic behavior of the fractional convection–reaction–diffusion process, and Jani and Singh [ 42 ] extended it to the soliton theory.

Though much work was achieved, in this study, we will show that AT–HPM is a universal tool for fractional calculus. As an example, we consider the time-fractional casting-mold system which is used in manufacturing various medical equipment, ranging from injections to the COVID-19 tool-kit [ 43 ]. The significant findings reveal that AT–HPM is an accurate and effective approach that reduces the computational work with fast convergence ratio.

2 Aboodh transform-based homotopy perturbation technique

This section is divided into two sections. In the first section, the methodology will be proposed, and the convergence of the suggested technique will be discussed in the second section.

2.1 Methodology

In this section, we give a brief introduction to the Aboodh transform [ 35 ] and homotopy perturbation method [ 33 , 34 ].

If f is a continuous piecewise function of time t , then the Aboodh transform of f t is F u that can be expressed as follows [ 35 ]:

where k 1 and k 2 are positive and can be finite or infinite. f t is considered a function of the exponential order, which assures the convergence of the integrand. e − u t is the kernel of the transform, and u is the transform variable. Table 1 includes the Aboodh transformation of some elementary functions helpful for this manuscript. This table can also be used for inverse Aboodh transform.

TABLE 1 . Aboodh transform of some elementary functions.

The Aboodh transform of the partial derivative of time can be obtained using the following formula:

where w is the independent variable. Now, suppose the general system of PDEs is expressed as

where L is the linear operator, N 1 , N 2 are the non-linear operators, x , y are the dependent variables, and g 1 , g 2 are the inhomogeneous functions. We assume the initial conditions as

where h 1 and h 2 are known functions of the independent variable w . The methodology composed of initially applying the Aboodh transform to both sides of the system of equations written in Eq. 3 and then employing the given initial conditions expressed in Eq. 4 , thus yielding

By employing the differential characteristic of Aboodh transform, we can express Eq. 3 as

and after using the initial conditions, we have

where K 1 w and K 2 w denote the terms arising from the initial condition. According to the standard homotopy perturbation method [ 33 , 34 ], the solution x and y can be expanded into an infinite series as

where p ∈ 0,1 is the embedding parameter. Also, the non-linear terms N 1 and N 2 can be written as

where H 1 n and H 2 n are He’s polynomials [ 44 ] and can be generated by the recursive formula

By substituting Eqs 7 , 8 in Eq. 6 , we get

Comparing the coefficients of like powers of p , we have

We can obtain the best approximation for the solution as

2.2 Convergence analysis

To show that the series solution of the system in Eq. 14 converges to the solution of Eq. 3 , we are to prove the sufficient condition of the convergence, and the following theorem will help us.

Theorem: We assume that X and Y are Banach spaces and M : X → Y is a non-linear contractive mapping such that

Then, according to Banach’s fixed point theorem, M has a unique fixed point μ , that is, M μ = μ . Supposing that the sequence in Eq. 14 can be written as

and considering that S 0 = s 0 ∈ B r s , w h e r e   B r s = s * ∈ X s * − s < r , we have

(i) S n ∈ B r s

(ii) lim n → ∞ S n = s

Proof: (i) By the principle of mathematical induction, for n = 1 , we have

Assuming S n − 1 − s ≤ λ n + 1 s 0 − s as an induction hypothesis, we get

By employing the definition of B r s , we have

(ii) As S n − s ≤ λ n s 0 − s   a n d   lim n → ∞ λ n = 0 ,

Hence, the given statement is proved.

3 Numerical examples

In this section, three examples are presented to illustrate the idea explained in Section 2. First, we will study the method for a homogeneous linear system of PDEs. Second, the analytical solution will be obtained for an inhomogeneous linear system of PDEs. Finally, the inhomogeneous non-linear system of PDEs will be examined.

3.1 The system of homogeneous linear PDEs

We consider the following linear system:

with initial conditions

By employing the Aboodh transform method, we have

Using the initial conditions given in Eq. 16 , we reach

The Aboodh transform-based homotopy perturbation method considers a series solution given by

By using the aforestated equation, the system of equations in Eq. 19 gets the form

By comparing like powers of p from the aforestated equation, we obtain

Hence, the series solution by using Eq. 14 can be expressed as

or in a closed form as

which is the exact solution of Eq. 15 .

3.2 The system of inhomogeneous linear PDEs

Suppose the following inhomogeneous linear system of PDEs:

Applying the Aboodh transform on each side of the equations in Eq. 24 and then putting on the given initial conditions, we obtain

By using the Aboodh transform-based homotopy perturbation method, the series solution is expressed by

The system of equations in Eq. 27 gets the following form after employing the aforestated equation:

By comparing the coefficient of like powers of p , we have

Therefore, the solution in the form of an infinite series by using Eq. 14 can be expressed as

or in its convergent form as

which is the exact solution of Eq. 24 .

3.3 The system of inhomogeneous non-linear PDEs

Suppose the following inhomogeneous non-linear system of PDEs:

Employing the Aboodh transform on each side of the equations in Eq. 36 and then applying the given initial conditions give

Taking the inverse Aboodh transform on each side, we obtain

According to the Aboodh transform-based homotopy perturbation method, the solution functions x w , t and y w , t are series solutions, and inserting these series into both sides of each equation of the system yields

where the non-linear terms x w y and x y w are denoted by He’s polynomials H 1 n x , y and H 2 n x , y , respectively. A few He’s polynomials are

which is the exact solution of Eq. 36 .

4 Time-fractional casting-mold system

Now, we turn back to a time-fractional casting-mold system which models the temperature distribution in the casting and molding processes. For this, two heat conduction equations are used with initial and Dirichlet boundary conditions [ 45 ]. The mathematical model is depicted as follows:

where a , b are parameters, Z , N are functions of time t and space x that represent the temperature on casting and molding plates, respectively, and β is the fractal dimension. For more details on the modeling aspect of the aforementioned model, readers can see [ 45 ].

It is necessary to point out that Eq. 48 was originally studied in [ 45 ], where the series solution was presented and no closed-form solution was formulated. Our aim here is to overcome the main shortcomings in [ 45 ] and to offer a totally new hope for numerical approximation. To this end, applying the Aboodh transform in the aforementioned system, we have

Now, by inverse Aboodh transformation, we obtain

which can further be written as

According to the standard HPM [ 33 , 34 ], the solution Z and N can be expanded into a finite series as

By substituting Eq. 52 in Eq. 51 , the solution can be written as

Equating coefficients of powers of p , we yield the following:

The approximate solution can be obtained as

4.1 Example

We consider the system expressed in Eq. 48 for the case a = 1 , b = 1 , Z 0 , x = e 2 x , N 0 , x = e x . By utilizing Eqs 54 – 56 , we have

By employing Eq. 57 , the solution can be written as

The expressions are similar to those obtained by the fractional complex transform [ 46 – 49 ]. In the closed form, we obtain

where E β t β is the Mittag-Leffler function [ 50 ]. One can check that Eq. 59 is an exact solution of Eq. 48 for the said parameters.

4.2 Results and discussion

This section is devoted to test the applicability and validity of the suggested technique for the time-fractional casting-mold system over the series-based solution of the same model.

Figures 1 , 2 present the errors of the series solutions obtained by the HPM [ 45 ] for the fractal dimension β = 1. It is observed that for all the parameters and for both casting and molding processes, the errors grow exponentially for the case of a series solution [ 45 ] and can be reduced by adding more terms in the solution. On the other hand, the suggested solution has the exact solution, and there is no chance of error even for a larger range of t. Therefore, based on these findings, we can say that the proposed technique is more effective than the previous method [ 45 ].

FIGURE 1 . Error estimations for the casting process at β = 1 and x = 0.5, 1, 2.

FIGURE 2 . Error estimations for the molding process at β = 1 and x = 0.5, 1, 2.

5 Conclusion

The Aboodh transform-based homotopy perturbation method is successfully employed to solve traditional differential equations and fractional differential equations successfully. This approach has been shown to have the potential to solve both linear and non-linear problems. For a linear system, the exact solution is predicted, while for a non-linear system, with the help of He’s polynomials, a series solution is obtained, which converges fast to the exact one. So, the method pushes the progress of non-linear science and will make a “big change” to increase the number of practical applications, and this paper serves as a model for other applications.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Materials; further inquiries can be directed to the corresponding author.

Author contributions

Conceptualization: HT and NA; methodology: NA and YY; validation: NA and YY; writing—original draft preparation: HT and YY; writing—review and editing: HT, NA, and YY; supervision: HT and YY; and funding acquisition: YY. All authors read and agreed to the published version of the manuscript.

The study was supported by the Natural Science Foundation of Henan Province (No. 222300420507); National Natural Science Foundation of China (No. 12171193), Key Scientific Research Project of High Education Institutions of Henan Province (No. 23A110019), Science and Technology Research Projects of Henan Province (No. 182102110292), Basic and Frontier Technology Research Project of Henan Province (Nos. 12300410398 and 132300410084), and Zhumadian Key Laboratory of Statistical Computing and Data Modeling [No. (2022)12].

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Nomenclature

Keywords: homotopy perturbation method, Aboodh transform, He’s polynomials, fractional differential equation, two-scale fractal theory

Citation: Tao H, Anjum N and Yang Y-J (2023) The Aboodh transformation-based homotopy perturbation method: new hope for fractional calculus. Front. Phys. 11:1168795. doi: 10.3389/fphy.2023.1168795

Received: 18 February 2023; Accepted: 03 April 2023; Published: 27 April 2023.

Reviewed by:

Copyright © 2023 Tao, Anjum and Yang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Naveed Anjum, [email protected]

This article is part of the Research Topic

Analytical Methods for Nonlinear Oscillators and Solitary Waves

Homotopy Perturbation Method

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Approximate solution of fractional order random ordinary differential equations using homotopy perturbation method

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Sahar A. Mohammed , Fadhel S. Fadhel , Kasim A. Hussain; Approximate solution of fractional order random ordinary differential equations using homotopy perturbation method. AIP Conf. Proc. 7 May 2024; 3097 (1): 080005. https://doi.org/10.1063/5.0209932

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In this paper, the homotopy perturbation method will be applied to find the approximate solution of fractional order random ordinary differential equations, in which the fractional order derivatives and integrals are defined using Caputo and Riemann-Liouville definitions of fractional derivatives and integrals, respectively. Also, the convergence of the approximated solution is stated and proved in this work. The work is verified for three different examples, which are simulated for different generations of Brownian motion.

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American Journal of Mathematics and Statistics

p-ISSN: 2162-948X    e-ISSN: 2162-8475

2019;  9(3): 136-141

doi:10.5923/j.ajms.20190903.04

Homotopy Perturbation Method for Solving Highly Nonlinear Reaction-Diffusion-Convection Problem

M. Tahmina Akter 1 , M. A. Mansur Chowdhury 2

1 Department of Mathematics, Chittagong University of Engineering & Technology, Chittagong, Bangladesh

2 Jamal Nazrul Islam Research Center for Mathematical and Physical Sciences (JNIRCMPS), University of Chittagong, Chittagong, Bangladesh

Copyright © 2019 The Author(s). Published by Scientific & Academic Publishing.

An elegant and powerful technique is Homotopy Perturbation Method (HPM) to solve linearand nonlinear partial differential equations. Using the initial conditions this method provides an analytical or exact solutions. In this article, we shall be applied this method to get most accurate solution of a highly non-linear partial differential equation which is Reaction-Diffusion-Convection Problem. This article confirms the power, simplicity and efficiency of the method compared with the exact solution. This article also confirmed that this method is suitable method for solving any types of partial differential equations. A graphical representation of the result has been shown which provides the most accurate physical situation and accuracy of the solution. The HPM allows to find the solution of the nonlinear partial differential equations which will be calculated in the form of a series with easily computable components. From the calculation and its graphical representation it is clear that how the solution of the equation and its behavior depends on the initial conditions.

Keywords: Homotopy perturbation method, Approximate solution, Exact solution, Nonlinear Reaction-Diffusion-Convection problem

Cite this paper: M. Tahmina Akter, M. A. Mansur Chowdhury, Homotopy Perturbation Method for Solving Highly Nonlinear Reaction-Diffusion-Convection Problem, American Journal of Mathematics and Statistics , Vol. 9 No. 3, 2019, pp. 136-141. doi: 10.5923/j.ajms.20190903.04.

Article Outline

1. introduction, 2. homotopy perturbation method, 3. application, 4. graphical representation of above equation, 5. result and discussion of the solution of equation (10), 6. conclusions.

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A new modification of the homotopy perturbation method for solving a class of evolution equations

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Modal truncation method for continuum structures based on matrix norm: modal perturbation method

• Original Paper
• Published: 11 May 2024

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• Houjun Kang   ORCID: orcid.org/0000-0002-2452-9844 1 , 2 ,
• Quan Yuan 1 ,
• Xiaoyang Su 1 , 2 ,
• Tieding Guo 1 , 2 &
• Yunyue Cong 1 , 2  

Modal analysis is a widely applied method to study the vibration phenomenon of continuum structures, but there is no clear method to solve the modal truncation problem at present. To determine the contribution of different modes to the whole system, a new mode truncation method based on perturbation theory is proposed in this paper. The modes are subjected to perturbation parameters during discretization, and using norm error analysis on the stiffness matrix in different degrees of freedom (DOFs) systems confirms the model number of the continuum structure system. The results show that the DOF identified by the modal perturbation method is related to the perturbation parameter, and the smaller the perturbation parameter is, the fewer modes need to be considered. When the perturbation parameter is large enough, the response of the system can only be accurately explained by truncation to higher-order modes. Finally, the perturbation parameter is fixed to 1, and the traditional Galerkin method is connected to the modal perturbation, making traditional discretization a unique case for the modal perturbation method. This method can significantly reduce the modal truncation error, which is of great significance to the dynamic analysis of engineering applications.

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The authors wish to acknowledge the support of the National Natural Science Foundation of China (11972151, 12372006, 12302007, and 12202109), and also the Specific Research Project of Guangxi for Research Bases and Talents (AD23026051).

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School of Civil and Architectural Engineering, Guangxi University, Nanning, 530004, China

Houjun Kang, Quan Yuan, Xiaoyang Su, Tieding Guo & Yunyue Cong

Scientific Research Center of Engineering Mechanics, Guangxi University, Nanning, 530004, China

Houjun Kang, Xiaoyang Su, Tieding Guo & Yunyue Cong

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Kang, H., Yuan, Q., Su, X. et al. Modal truncation method for continuum structures based on matrix norm: modal perturbation method. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09628-2

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Received : 09 December 2023

Accepted : 12 April 2024

Published : 11 May 2024

DOI : https://doi.org/10.1007/s11071-024-09628-2

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• Continuum structures
• Modal truncation
• Perturbation
• Matrix norm

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