IHJPAS. 36(1)2023 325 This work is licensed under a Creative Commons Attribution 4.0 International License -Sumudu Transformation Homotopy Perturbation Technique on Fractional Gas Dynamical Equation Abstract Transformation and many other substitution methods have been used to solve non-linear differential fractional equations. In this present work, the homotopy perturbation method to solve the non-linear differential fractional equation with the help of He’s Polynomials is provided as the transformation plays an essential role in solving differential linear and non-linear equations. Here is the α-Sumudu technique to find the relevant results of the gas dynamics equation in fractional order. To calculate the non-linear fractional gas dynamical problem, a consumer method created on the new homotopy perturbation -Sumudu transformation method (HP𝑆TM) is suggested. In the Caputo type, the derivative is evaluated. -Sumudu homotopy perturbation technique and He’s polynomials are all incorporated in the HPSaTM. The availability of He’s polynomials could be used to conveniently manage the non-linearity. The suggested approach shows that the strategy is simple to implement and provides results that can be compared to the results gained from any other transformation technique. Keywords -Sumudu, Mittag-Leffler, homotopy perturbation method, He’s Polynomials. 1. Introduction Fractional modeling is a collection of classification algorithms concerned with fractions having random order integrals and derivatives [1]. In [2], fractional derivatives have explored damping laws. Fractional calculus has been used in seemingly unrelated scientific and technological doi.org/10.30526/36.1.3029 Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq Article history: Received 21 September 2022, Accepted 24 October 2022, Published in January 2023. Ali Moazzam University of Agriculture Faisalab, Pakistan alimoazzam7309723@gmail.com Adnan Shokat Comsats University, Islamabad, Pakistan, Adnanshokat4118@gmail.com Emad A. Kuffi Al-Qadisiyah University, College of Engineering, Iraq. emad.abbas@qu.edu.iq https://creativecommons.org/licenses/by/4.0/ mailto:alimoazzam7309723@gmail.com mailto:Adnanshokat4118@gmail.com mailto:emad.abbas@qu.edu.iq IHJPAS. 36(1)2023 326 domains over the last ten years. Increasingly, difficulties in the mechanics of fluid [3], acoustics, electromagnetics, diffusion, processing of signals, and various physical phenomena are formulated using fractional differential equations [4]. Approximation approaches to fractional calculus perturbation describe a set of equations solved using different methods [5]. However, there are some drawbacks in perturbation approaches observed by [6] while solving the space-time diffusion equation of the fractional form. In [7] substitution method for solving the reaction-diffusion equation has been described. A recent overview of various new exponential approaches for searching isolated values of fractional nonlinear differential equations has been published [8]. He was the first to propose the homotopy perturbation method (HPM) [9]. Many researchers have looked at using the HPM to solve linear [10] and nonlinear problems in various science-technical sectors [11]. Method of Adomian decomposition [12] and variational iteration [13] have also been applied to learn about numerous physical problems. In the latest paper, [14] Appreciated consideration of homotopy perturbation Sumudu transform method (HPSTM) has been given to analyze results of partial differential nonlinear and linear equations. In [15], the iterative variational method is used to find the results for a system of linear and nonlinear differential equations with the help of He’s Polynomials. Numerical solutions of the biological population model have been founded with the help of homotopy perturbation, and He’s Polynomial by [16].In [17], the numerical of the Newell-Whitehead-Segal equation with the help of the variational iteration method and He’s Polynomials. With the help of He’s Polynomial and homotopy perturbation, the iterative variation method has been used to solve the KS equation [18]. The solution of the Black school option evaluation model [19], the KDV equation of fifth order in magneto-acoustic waves of numerical type [20], has been previously done by the mean of He’s polynomials. [21] previously described the analytical and numeric solutions of vibration problems in nonlinear differential form. In [22-28], He’s polynomial, homotopy perturbation methods, transformation of Sumudu, and Laplace type are used in solving nonlinear differential models. In [29], HPSTM, which is a mixture of the Sumudu technique of homotopy perturbation and polynomials of He’s typing, is applied. We assume the nonlinear gas dynamical fractional equation. ( ) ( )2 1 1 0 2 D V V V V    + + − = (1.1) with the initial condition ( ) .0, −= eV (1.2) Where fractional derivative order is β. The probability distribution ( ),V   is demarcated as a probability density function, where  is the period and  is the spatially component. In the Caputo meaning, the derivation is grasped. The ordering of the fractional differential equation is described by a factor in the average response formulation, which may be changed to get different replies. The fractionated gas dynamical equation is reduced to the classic gas dynamical equation when 𝛽 = 1. The universal laws of preservation, such as mass conservation, momentum preservation, sustainable energy, and so on, are used to create the gas dynamical equations. The non-linear time- fractional of gas dynamics has been deliberate before by [30]. To address the non-linear time- fractional problem, we use HPSTM and Adomian decomposition method (ADM). The study aims to expand the HPSTM’s applicability to provide analytical and approximation answers to the time- fractional gas dynamical problem. The HPSTM has the benefit of being able to combine two effective approaches for finding precise and numerical approximation models for non-linear equations. It generates solutions in terms of convergent series with readily computed constituents without any need for linearization, perturbation, or restricted conditions. It is important to note IHJPAS. 36(1)2023 327 that, compared to conventional approaches, the HPSTM can reduce the quantity of computational effort while keeping a high accuracy output; the size reduction equates to an enhancement in the approach’s efficiency. 2. Materials and Methods Definition 1. [31] If ( )f t is defined on , + R then the -Sumudu transform is ( ) ( ) ( ) ( )  −   ==  0 1 .; 1 1 Rvdttfe v vtfSvF v t (1.3) Definition 2. [1] The integral operator of Riemann-Liouville fractional with order β>0 is given as ( ) ( ) ( ) ( ) ( )   −  = 0 ,0 1 dffJ (1.4) ( ) ( ).0 = ffJ (1.5) It can also be expressed in gamma relationship as ( ) ( ) . 1 1 +  ++ + =J (1.6) Definition 3. [32] In the Caputo interpretation, the fractional derivative of ( )tf is specified as ( ) ( ) ( ) ( ) ( )( )  −−−  −− == 0 1 , 1 df n fDJfD mmnm for 1,m m m  − is natural number and .0 For the relationship between the fractional integral of Riemann-Liouville and the fractional Caputo derivatives, we have: ( ) ( ) ( )( ) − =      +−= 1 0 . ! 0 m q q q q fffDJ (1.7) Definition 4. [31] Caputo fractional derivative’s -Sumudu transformation is described as:  ( ) ( ) ( )( )  − =  −     −= 1 0 . 0 m k k k v f v vG fDS (1.8) Solution by Homotopy Perturbation of Caputo Fractional Differential Equations General Idea of HPSαTM Let us assume that the general non-homogeneous and non-linear partial fractional differential equation to demonstrate the general idea of this technique. We have ( ) ( ) ( ) ( ),,,,, =++ gNVRVVD (2.1) ( ) ( ),0, = fV (2.2) where R indicates the linearly difference operator, N denotes the generic non-linear differential controller and ( )   ,VD is the Caputo derivative of functions. By performing the -Sumudu transform on (2.1), we get  ( ) ( )( ) ( )( ) ( ) .,,,, =++    gSNVSRVSVDS (2.3) Applying the rules of the -Sumudu technique and after simplification, we get IHJPAS. 36(1)2023 328 ( )  ( ) ( )  ( ) ( ) .,,,, +−+=        NVRVSvgSvfVS (2.4) Using the inverse of -Sumudu on (2.4), we find ( ) ( )  ( ) ( ) ,,,,, 1 +−=    −  NVRVSSgV (2.5) where ( ),g is the source term. Now we apply the HPM: ( ) ( )  = = 0 ,, n n n VpV (2.6) and the non-linear part can be decomposed as: ( ) ( )  = = 0 , n n n VHpNV . (2.7) For little He’s described polynomials ( ),nH that is given below as ( ),...,,, 10 nn VVVH ....,2,1,0; ! 1 00 =                   =  =  nVpN pn pi i i n n (2.8) By putting the values of (2.6)-(2.7) into (2.5), we find ( ) ( ) ( ) ( ) .,,, 0 0 0 1     =  =  =    −                          +−= n n n n n n n n n VHpVpRSSpgVp (2.9) It is also the He’s polynomials-based connection of the -Sumudu technique and the HPM. The preceding estimates are produced by analyzing the factors of like powers of p, ( ) ( ) ( )  ( ) ( )  ( )  ( ) ( )  ( )  ( ) ( )  . ,,,: ,,,: ,,,: ,,: 22 1 3 3 11 1 2 2 00 1 1 1 0 0            +−= +−= +−= =    −     −     −  VRVSSVp VRVSSVp VRVSSVp gVp (2.10) The remaining elements of the ( ),nV may be acquired in almost the same way, and the appropriate route can therefore be found completely. Ultimately, we use a curtailed series to estimate the analytic answer ( ):, V ( ) ( )  = → = 0 .,lim, n n N VV (2.11) The series mentioned in (2.11) generally converges fastly. 3. Results Solution of Gas Dynamic Problem Let us assume that the proceeding non-linear gas dynamic time-fractional equation: IHJPAS. 36(1)2023 329 ( ) ( )2 1 1 0 2 D V V V V    + + − = (3.1) with given initial condition ( ) .0, −= eV (3.2) By performing the -Sumudu technique, and with the help of (3.2), equation (3.1) becomes ( )  ( ) ( ) .1 2 1 , 2     −+−=    −  VVVSveVS (3.3) By performing -Sumudu inverse transform, ( ) ( ) ( ) .1 2 1 , 21             −+−=    −  − VVVSvSeV (3.4) By HPM technique, ( )  =  0 , n n n Vp ( ) ( ) ( ) ., 2 1 0 00 1                         +−         −=    =  =   =    −  − n n n n n n n n n VHpVpVHpSvSe (3.5) In the above equation, ( )VH n and ( )VH n  are the pre-described He’s polynomials denoting the non-linear part. So, polynomials are illustrated by ( ) ( )  =  = 0 2 . 2 1 n n n VVHp (3.6) The first limited constituents of He’s polynomials are given by ( ) ( ) ( ) ( ) ( ) ( )  +=== 20 2 12101 2 00 ,, VVVVHVVVHVVH (3.7) and for ( ),VH n  we use ( )  =  = 0 2 , n n n VVHp ( ) ( ) ( ) .2,2, 0 2 12101 2 00 VVVVHVVVHVVH +===  (3.8) Equating the factors of powers of p, we get ( ) ,,: 0 0  = eVp ( ) ( ) ( ) ( ) , 12 1 ,: 000 1 1 1 +  =             +−=  −    −  eVHVVHSvSVp ( ) ( ) ( ) ( ) , 122 1 ,: 2 111 1 2 2 +  =             +−=  −    −  eVHVVHSvSVp ( ) ( ) ( ) ( ) . 132 1 ,: 3 222 1 3 3 +  =             +−=  −    −  eVHVVHSvSVp (3.9) Consequently, the solution series is ( ) ( ) ( ) ( ) . 13121 1, 32       +  + +  + +  +=  − eV (3.10) Setting 1= in equation (3.10), we repeat the solution of the equation as follows: IHJPAS. 36(1)2023 330 ( ) . !3!2 1, 32       +  +  ++= − eV (3.11) This result is comparable to the precise result in the form: ( ) ., − = eV (3.12) Equation (3.12) provides the exact solution of gas dynamic fractional equation. 5. 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