Homotopy Perturbation Sumudu Transform Method Research Articles

Numerical study of nonlinear time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising in propagation of waves

In this manuscript, the numerical approximation of the solution for time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) nonlinear partial differential equation is described involving the Caputo fractional derivative with the help of two novel techniques namely, Sumudu Residual Power Series (SRPS) method and Homotopy Perturbation Sumudu Transform (HPST) method. CDGSK equation describe the propagation of capillary waves and shallow water waves. The results obtained through numerical approximation shows that both SRPS and HPST under consideration are reliable and efficient. The present framework precisely reflects the behavior of the obtained result for various fractional orders. Additionally, we conduct a comparison between the exact solution and an approximate series solution to affirm the efficacy of the proposed techniques. This analysis highlights the efficiency and precision of both techniques in solving nonlinear fractional differential equations. All the numerical simulations are demonstrated graphically using Maple and Matlab.

Impact of new similarity transformations on heat transfer analysis of Casson tri-hybrid nanofluid in blood with thermal radiation through a stretching sheet: a homotopy perturbation Sumudu transformation approach

ABSTRACT The investigation of nonlinearly stretching surface has been conducted using the Casson model, focusing on a unique mixture known as a ternary nanofluid. This specific nanofluid comprises a combination of blood with suspended aluminum alloys, namely AA 7072 and AA 7075 , as well as A l 2 O 3 oxide. The nanoparticles dispersed in the underlying fluid are envisioned to possess a sphere-like structure, thereby ensuring optimal contact between the nanoparticle’s extensive surface area and the base fluid. Consequently, this design enables enhanced heat absorption from the surface. Additionally, the unique combination of the nanoparticles facilitates swift mobility within the nanofluid, expediting their movement. We derived a novel set of similarity transformations, in which the similarity variable is dimensionless and appears in the formulation as a function of all independent variables. The Runge-Kutta Fehlberg 4- 5 th methodology was used to solve the modified equations numerically. In previous research papers, the Homotopy perturbation Sumudu transform method (HPSTM) was employed to derive the analytical solution solely for the momentum equation. However, in this current study, the Homotopy perturbation Sumudu transform method (HPSTM), is applied to solve the coupled momentum and heat equations of trihybrid nanofluid, resulting in an innovative analytical solution that has not been previously explored. The comparison with the numerical results demonstrates the precision of the present work. The repercussions of unique components on thermal and velocity plots are explained in graphical records for each case of nanofluid, hybrid nanofluid and trihybrid nanofluid separately. This study encourages the application of Homotopy perturbation Sumudu transform technique in more fluid flow problems.

SOLVING FUZZY FRACTIONAL HEAT EQUATION USING HOMOTOPY PERTURBATION SUMUDU TRANSFORM METHOD

This paper extents the Homotopy Perturbation Sumudu Transform Method (HPSTM) to solve fuzzy fractional heat equations. To illustrate the reliability of the method, some examples are presented. The results reveal that HPSTM is a highly effective scheme for obtaining approximate analytical solutions of fuzzy fractional heat equations. Figures and numerical examples demonstrate the expertise of the suggested approach. This method is applied for both linear and nonlinear ordinary and partial FFDEs. The proposed approach is rapid, exact, and simple to apply and produce excellent outcomes.

Semianalytical investigation on heat transfer in porous fins with temperature‐dependent thermal conductivity via the homotopy perturbation Sumudu transform approach

AbstractA unique investigation has been undertaken to analyze the heat transmission by convective and radiative mechanisms in a fully saturated penetrable fin of a longitudinal structure positioned on a leaning surface. This study introduces the fusion of the realms of Homotopy perturbation and Sumudu transform techniques to address a previously unexplored problem involving a moving fin with temperature‐dependent thermal conductivity. In prior research papers, the Homotopy Perturbation Sumudu Transform Method (HPSTM) was utilized to obtain analytical solutions for fins featuring temperature‐dependent thermal conductivity. However, in our current study, we employ the HPSTM to tackle a novel problem involving a moving porous fin. This fin exhibits temperature‐dependent thermal conductivity and is subjected to convection and radiation effects. Through a comparison with numerical results, the present study has validated the dependability of its findings. The dimensionless temperature profile has been investigated by studying its relationship with several parameters. Here we observed that when the Peclet number is augmented by 400%, there is a corresponding 1.11% increase in thermal outline at the fin's extremity. Enhancing the value of radiation parameter by 400% declines the temperature of the fin tip by 14.079%. This study encourages the application of the Homotopy perturbation Sumudu transform technique in more complex fin problems.

Approximate Solution for Solving Fractional order Lotka-Volterra Equation

A lot of work regarding prey-predator type Lotka-Volteara model has already been done by many researchers during the last century. However, the work devoted to the concept of model related to fractional order derivatives is still inadequate. We employ Homotopy perturbation Sumudu transform method (HPSTM) to obtain the approximate solutions of these equations.
 Homotopy perturbation Sumudu transform method gives a new approach to the solution of these kind of problems. And we plot figures to illustrate this technique and it is seen that the (HPSTM) is very useful and effective method to get the approximate solutions.
 The most significant part of this study is the decay of both prey-predator populations with increase of the fractional time derivative. Another important part is increase of time taken for the meeting of prey-predator with increase of fractional time derivative. The article aims to motivate both engineers and biologists who are working on the pre-predator model to carry on further researches in this area.

Numerical investigation of fractional model of phytoplankton–toxic phytoplankton–zooplankton system with convergence analysis

In this paper, a fractional order model of the phytoplankton–toxic phytoplankton–zooplankton system with Caputo fractional derivative is investigated via three computational methods, namely, residual power series method (RPSM), homotopy perturbation Sumudu transform method (HPSTM) and the homotopy analysis Sumudu transform method (HASTM). This model is constituted by three components: phytoplankton, toxic phytoplankton and zooplankton. Phytoplankton species are self-feeding members of the plankton community and play a very significant role in ecosystems. A wide range of sea creatures get food through phytoplankton. This paper focuses on the implementation of the three above-mentioned computational methods for a nonlinear time-fractional phytoplankton–toxic phytoplankton–zooplankton (PTPZ) model with a perception to study the dynamics of a model. This study shows that the solutions obtained by employing the suggested computational methods are in good agreement with each other. The computational procedures reveal that the HASTM solution generates a more general solution as compared to RPSM and HPSTM and incorporates their results as a special case. The numerical results presented in the form of graphs authenticate the accuracy of computational schemes. Hence, the implemented methods are very appropriate and relevant to handle nonlinear fractional models. In addition, the effect of variation of fractional order of a time derivative and time [Formula: see text] on populations of phytoplankton, toxic–phytoplankton and zooplankton has also been studied through graphical presentations. Moreover, the uniqueness and convergence analyses of HASTM solution have also been discussed in view of the Banach fixed-point theory.

Series Solutions of High-Dimensional Fractional Differential Equations

In the present paper, the series solutions and the approximate solutions of the time–space fractional differential equations are obtained using two different analytical methods. One is the homotopy perturbation Sumudu transform method (HPSTM), and another is the variational iteration Laplace transform method (VILTM). It is observed that the approximate solutions are very close to the exact solutions. The solutions obtained are very useful and significant to analyze many phenomena, and the solutions have not been reported in previous literature. The salient feature of this work is the graphical presentations of the third approximate solutions for different values of order α.

Solutions of fractional order differential equations modeling temperature distribution in convective straight fins design

In this paper, the problem of temperature distribution for convective straight fins with constant and temperature-dependent thermal conductivity is solved by using artificial neural networks trained by the biogeography-based heterogeneous cuckoo search (BHCS) algorithm. We have solved the integer and noninteger order energy balance equation in order to analyse the temperature distribution in convective straight fins. We have compared our results with homotopy perturbation method (HPM), variational iteration method (VIM), and homotopy perturbation Sumudu transform method (HPSTM). The results show that the ANN–BHCS algorithm gives better results than other analytical techniques. We have further checked the efficiency of the ANN–BHCS algorithm by using the performance metrics MAD, TIC, and ENSE. We have calculated the values of MAD, TIC, and ENSE for case 1 of the problem, and histograms of these metrics show the efficiency of our algorithm.

Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator

<p style='text-indent:20px;'>In this paper, an effective analytical scheme based on Sumudu transform known as homotopy perturbation Sumudu transform method (HPSTM) is employed to find numerical solutions of time fractional Schrödinger equations with harmonic oscillator.These nonlinear time fractional Schrödinger equations describe the various phenomena in physics such as motion of quantum oscillator, lattice vibration, propagation of electromagnetic waves, fluid flow, etc. The main objective of this study is to show the effectiveness of HPSTM, which do not require small parameters and avoid linearization and physically unrealistic assumptions. The results reveal that proposed scheme is a powerful tool for study large class of problems. This study shows that the results obtained by the HPSTM are accurate and effective for analysis the nonlinear behaviour of complex systems and efficient over other available analytical schemes.</p>

An efficient numerical technique for a biological population model of fractional order

In the present paper, a biological population model of fractional order (FBPM) with one carrying capacity has been examined with the help of reproducing kernel Hilbert space method (RKHSM). This important fractional model arises in many applications in computational biology. It is worth noting that, the considered FBPM is used to provide the changes that is made on the densities of the predator and prey populations by the fractional derivative . The technique employed to construct new numerical solutions for the FBPM which is considered of a system of two nonlinear fractional ordinary differential equations (FODEs). In the proposed investigation, the utilised fractional derivative is the Caputo derivative. The most valuable advantages of the RKHSM is that it is easily and fast implemented method. The solution methodology is based on the use of two important Hilbert spaces, as well as on the construction of a normal basis through the use of Gram-Schmidt orthogonalization process. We illustrate the high competency and capacity of the suggested approach through the convergence analysis. The computational results, which are compared with the homotopy perturbation Sumudu transform method (HPSTM), clearly show: On the one hand, the effect of the fractional derivative in the obtained outcomes, and on the other hand, the great agreement between the mentioned methods, also the superior performance of the RKHSM. The numerical computational are presented in illustrated graphically to show the variations of the predator and prey populations for various fractional order derivatives and with respect to time.

Approximate solution for fractional attractor one-dimensional Keller-Segel equations using homotopy perturbation sumudu transform method

AbstractIn this paper, homotopy perturbation sumudu transform method (HPSTM) is proposed to solve fractional attractor one-dimensional Keller-Segel equations. The HPSTM is a combined form of homotopy perturbation method (HPM) and sumudu transform using He’s polynomials. The result shows that the HPSTM is very efficient and simple technique for solving nonlinear partial differential equations. Test examples are considered to illustrate the present scheme.

Numerical computation of fractional Kersten-Krasil’shchik coupled KdV-mKdV system occurring in multi-component plasmas

In this paper, we study the nonlinear behaviour of multi-component plasma. For this an efficient technique, called Homotopy perturbation Sumudu transform method (HPSTM) is introduced. The power of method is represented by solving the time fractional Kersten-Krasiloshchik coupled KdV-mKdV nonlinear system. This coupled nonlinear system usually arises as a description of waves in multi-component plasmas, traffic flow, electric circuits, electrodynamics and elastic media, shallow water waves etc. The prime purpose of this study is to provide a new class of technique, which need not to use small parameters for finding approximate solution of fractional coupled systems and eliminate linearization and unrealistic factors. Numerical solutions represent that proposed technique is efficient, reliable, and easy to use to large variety of physical systems. This study shows that numerical solutions gained by HPSTM are very accurate and effective for analysis the nonlinear behaviour of system. This study also states that HPSTM is much easier, more convenient and efficient than other available analytical methods.

Numerical study of fractional model of multi-dimensional dispersive partial differential equation

This article is devoted to a newly introduced numerical method for time-fractional dispersive partial differential equation in a multi-dimensional space. The time-fractional dispersive partial differential equation plays a great role in solving the problems arising in ocean science and engineering. The numerical technique comprises of Sumudu transform, homotopy perturbation scheme and He's polynomial, namely homotopy perturbation Sumudu transform method (HPSTM) is efficiently used to examine time-fractional dispersive partial differential equation of third order in multi-dimensional space. The approximate analytic solution of the time-fractional dispersive partial differential equation of third-order in multi-dimensional space obtained by HPSTM is compared with exact solution as well as the solution obtained by using Adomain decomposition method. The results derived with the aid of two techniques are in a good agreement and consequently these techniques may be considered as an alternative and efficient approach for solving fractional partial differential equations. Several test problems are experimented to confirm the accuracy and efficiency of the proposed methods.

An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma

In this paper, we present a coupling of homotopy perturbation technique and sumudu transform known as homotopy perturbation sumudu transform method (HPSTM). We show applicability of this method by solving fractional equal width (EW) equation, fractional modified equal width (MEW) equation and variant of fractional modified equal width (VMEW) equation. The fractional equal width equations play a key role in describing hydro-magnetic waves in cold plasma. Our aim is to study the nonlinear behavior of plasma system and highlight the important points. We examine the ability of HPSTM to study the fractional nonlinear systems and show its supremacy over other available numerical techniques. The other key point of this investigation is to examine two important fractional equations with different nonlinearity. The HPSTM gives excellent accuracy in analogous with the numerical solution. The numerical solutions indicate that the HPSTM is a powerful technique for studying the nonlinear behavior of plasma system very precisely and accurately.

An efficient analytical technique for fractional partial differential equations occurring in ion acoustic waves in plasma

In this work, we apply an efficient analytical algorithm namely homotopy perturbation Sumudu transform method (HPSTM) to find the exact and approximate solutions of linear and nonlinear time-fractional regularized long wave (RLW) equations. The RLW equations describe the nature of ion acoustic waves in plasma and shallow water waves in oceans. The derived results are very significant and imperative for explaining various physical phenomenons. The suggested method basically demonstrates how two efficient techniques, the Sumudu transform scheme and the homotopy perturbation technique can be integrated and applied to find exact and approximate solutions of linear and nonlinear time-fractional RLW equations. The nonlinear expressions can be simply managed by application of He's polynomials. The result shows that the HPSTM is very powerful, efficient, and simple and it eliminates the round-off errors. It has been observed that the proposed technique can be widely employed to examine other real world problems.

Analytic study for fractional coupled Burger’s equations via Sumudu transform method

AbstractIn this work, we aim to apply a reliable analytic algorithm based on homotopy perturbation Sumudu transform method (HPSTM) to examine the nonlinear time-fractional coupled Burger’s equations. The approximate analytical solution and some numerical examples show the accuracy and efficiency of the proposed method, which is simple and accurate in comparison to the Adomain decomposition method (ADM), homotopy perturbation method (HPM) and generalized differential transform method (GDTM).

Solving The Generalized Time Space Fractional Schrödinger Equation By Homotopy Perturbation Sumudu Transform Method

Solving The Generalized Time Space Fractional Schrödinger Equation By Homotopy Perturbation Sumudu Transform Method

Application of Homotopy Perturbation Sumudu Transform Method for Solving Nonlinear Fractional Heat-like Equations

In this paper, we propose an algorithm to find approximate solutions of the proposed system of the fractional heat-like equations. The proposed algorithm basically illustrates how the two powerful algorithms, the homotopy perturbation method (HPM) and the Sumudu transform method (STM) can be combined and used to get exact solutions of fractional partial differential equations. We also present some examples to illustrate the accuracy and the effectiveness of this algorithm.

Homotopy Series Solutions to Time-Space Fractional Coupled Systems

We apply the homotopy perturbation Sumudu transform method (HPSTM) to the time-space fractional coupled systems in the sense of Riemann-Liouville fractional integral and Caputo derivative. The HPSTM is a combination of Sumudu transform and homotopy perturbation method, which can be easily handled with nonlinear coupled system. We apply the method to the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The simplicity and validity of the method can be shown by the applications and the numerical results.

A computational approach for fractional convection-diffusion equation via integral transforms

Abstract In this paper, two efficient analytic techniques namely the homotopy analysis transform method (HATM) and homotopy perturbation Sumudu transform method (HPSTM) are implemented to give a series solution of fractional convection-diffusion equation which describes the flow of heat. These proposed techniques introduce significance in the field over the existing techniques that make them computationally very attractive for applications. Numerical solutions clearly demonstrate the reliability and efficiency of HATM and HPSTM to solve strongly nonlinear fractional problems.