Explicit Solution for Time-Fractional Batch Reactor System

2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Muhammad Jamil ◽  
Asmat Ara ◽  
Subir Das
Keyword(s):  
Homotopy Perturbation Method ◽  
Analytical Procedure ◽  
Numerical Solutions ◽  
Approximate Analytical Solution ◽  
Batch Reactor ◽  
Reactor System ◽  
Time Interval ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions ◽  
Good Agreement

In this paper, the new homotopy perturbation method (NHPM) has been successively applied for finding approximate analytical solutions of the fractional order batch reactor system. An approximate analytical solution for the concentration of reactants and products that is valid for a time interval. The approximate analytical procedure is depends only on two components. The behavior of the solution and effects of different parameters and fractional index are shown graphically. Numerical solutions of ordinary batch reactor system verify our approximate solution with good agreement.

scholarly journals A Novel Homotopy Perturbation Algorithm Using Laplace Transform for Conformable Partial Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Sajad Iqbal ◽  
Mohammed K. A. Kaabar ◽  
Francisco Martínez
Keyword(s):  
Partial Differential Equations ◽  
Analytical Solution ◽  
Differential Equations ◽  
Laplace Transform ◽  
Homotopy Perturbation Method ◽  
Approximate Analytical Solution ◽  
Partial Differential ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions ◽  
Different Types

In this article, the approximate analytical solutions of four different types of conformable partial differential equations are investigated. First, the conformable Laplace transform homotopy perturbation method is reformulated. Then, the approximate analytical solution of four types of conformable partial differential equations is presented via the proposed technique. To check the accuracy of the proposed technique, the numerical and exact solutions are compared with each other. From this comparison, we conclude that the proposed technique is very efficient and easy to apply to various types of conformable partial differential equations.

Free and Forced Vibrations of the Strongly Nonlinear Cubic-Quintic Duffing Oscillators

2017 ◽  
Vol 72 (1) ◽  
pp. 59-69 ◽  
Author(s):  
M.M. Fatih Karahan ◽  
Mehmet Pakdemirli
Keyword(s):  
Numerical Simulations ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Free And Forced Vibrations ◽  
Approximate Analytical Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

AbstractStrongly nonlinear cubic-quintic Duffing oscillatoris considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical solutions.

Application of Homotopy Perturbation Method for Harmonically Forced Duffing Systems

2011 ◽  
Vol 110-116 ◽  
pp. 2277-2283 ◽  
Author(s):  
Xiang Meng Zhang ◽  
Ben Li Wang ◽  
Xian Ren Kong ◽  
A Yang Xiao
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Primary Resonance ◽  
Effective Technique ◽  
Duffing System ◽  
First Order ◽  
Homotopy Perturbation ◽  
Analytical Approximations ◽  
Case Based

In this paper, He’s homotopy perturbation method (HPM) is applied to solve harmonically forced Duffing systems. Non-resonance of an undamped Duffing system and the primary resonance of a damped Duffing system are studied. In the former case, the first-order analytical approximations to the system’s natural frequency and periodic solution are derived by HPM, which agree well with the numerical solutions. In the latter case, based on HPM, the first-order approximate solution and the frequency-amplitude curves of the system are acquired. The results reveal that HPM is an effective technique to the forced Duffing systems.

scholarly journals Homotopy Perturbation Method

2017 ◽  
pp. 24-61
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Perturbation Method ◽  
Exact Solutions ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Homotopy Perturbation ◽  
Wide Range ◽  
Transfer Problems

The homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.

Optimal control homotopy perturbation method for cancer model

2019 ◽  
Vol 12 (03) ◽  
pp. 1950027 ◽  
Author(s):  
Malihe Najafi ◽  
Hadi Basirzadeh
Keyword(s):  
Mathematical Modeling ◽  
Optimal Control ◽  
Perturbation Method ◽  
Cancer Immunotherapy ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Cancer Model ◽  
Homotopy Perturbation

In this paper, we introduced the optimal control homotopy perturbation method (OCHPM) by using the homotopy perturbation method (HPM). Every one, by using of the proposed method, can obtain numerical solutions of mathematical modeling for cancer-immunotherapy. In this paper, in order to prove the preciseness and efficiency of the OCHPM method, we compared the obtained numerical solutions with HPM. The results obtained showed that the OCHPM method is powerful to generate the numerical solutions for some therapeutic models.

scholarly journals An approximate analytical technique for solving second order strongly nonlinear generalized duffing equation with small damping

2015 ◽  
Vol 39 (1) ◽  
pp. 103-114
Author(s):  
M Alhaz Uddin ◽  
M Wali Ullah ◽  
Rehana Sultana Bipasha
Keyword(s):  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Second Order ◽  
Extended Form ◽  
Analytical Technique ◽  
Strongly Nonlinear ◽  
Homotopy Perturbation ◽  
Kbm Method ◽  
Method Accuracy ◽  
Academy Of Sciences

In this paper, He’s homotopy perturbation method has been extended for obtaining the analytical approximate solution of second order strongly nonlinear generalized duffing oscillators with damping based on the extended form of the Krylov-Bogoliubov-Mitropolskii (KBM) method. Accuracy and validity of the solutions obtained by the presented method are compared with the corresponding numerical solutions obtained by the well-known fourth order Rangue-Kutta method. The method has been illustrated by examples.Journal of Bangladesh Academy of Sciences, Vol. 39, No. 1, 103-114, 2015

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