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Application of He's homotopy and perturbation method to solve heat transfer equations: A python approach
Abstract
In this research article, an attempt has been made to solve the linear/nonlinear and steady/unsteady heat transfer equations using the Homotopy and Perturbation method (HPM). Moreover, the implementation of HPM has been done by using SymPy, a library in python, to solve problems symbolically. Total three problems were dealt viz. steady-state conduction with heat generation, lumped capacitance analysis with a variable specific heat of the material, and heat transfer in uniform rectangular fin with radiation from the surface. In all the cases, the HPM has given excellent results compared to the analytical and numerical. Finally, the execution of SymPy has been explained, and a detailed procedure to implement HPM through python has been presented for all three cases.
... C. M. Mohana and B. Rushi Kumar 27,28 discussed the coupled nonlinear fluid flow problems using built-in MATLAB R2023b software using BVP-4c along with the shooting technique (an implicit Runge-Kutta method), which is a reliable technique where collocation polynomial ensures C 1 continuity of the solution and mesh selectivity in order to obtain the solution's accuracy and requires good initial guesses for successful computations other than any numerical techniques [29][30][31] . In our present investigation, we adapt the HPM semi-analytical approach 21,[33][34][35] , which suits perfectly to the current study as compared to the other asymptotic approximate techniques, and BVP-4c utilization, apart from any other numerical solver. ...
... The method often exhibits faster convergence with a reduced number of iterations when compared to other semi-analytical methods. The series solution generally exhibits rapid convergence towards the exact solution [33][34][35][36] . The Homotopy Perturbation Method (HPM) was selected for its distinct advantages over other approaches [22][23][24][25][26] , particularly in terms of computational efficiency and accuracy. ...
This study examines the semi-analytical expression of a hollow cylindrical fiber membrane bioreactor catalyzed by an immobilized enzyme (lipase), focusing on the dynamic kinetic resolution (DKR) of racemic (R)- and (S)-ibuprofen ester through the homotopy perturbation method (HPM). This method effectively solves the coupled non-linear system of second-order ordinary differential equations (ODEs), incorporating the mass transfer associated with the DKR reaction rate. The DKR rate equations are computed utilizing both enantiomers, which represent the enzymatic hydrolysis of the substrate within the barrier’s support matrix, where the racemization term is negligible. Additionally, the corresponding racemization of the unreacted substrate outside the membrane is considered, where the enzymatic hydrolysis term is negligible. The dimensionless steady-state solution for the dynamics of the mean integrated effectiveness factor (MIEF) related to convective hydrolysis-racemization phenomena quantifies the local effectiveness factor, which impacts the Thiele modulus and the bulk initial concentrations of a substance within the membrane layer. The Bodenstein number is a critical parameter that optimizes reaction conditions, enhances mixing efficiency, and mitigates mass transfer limitations. The normalized values from the sensitivity analysis are analyzed to identify the key parameters of the present system. The conversion efficiency and limitations of the proposed model utilizing HPM are presented. Our proposed solution compares the semi-analytical expression with the simulated numerical results of BVP-4c using the built-in solver in MATLAB R2023b software, demonstrating good agreement across all values of the reaction parameters. The agreement between semi-analytical and numerical computations demonstrates the validity of these approximation methods. The findings provide significant insights for enhancing membrane reactor design and improving the efficiency of industrial processes for racemic ibuprofen ester production.
... For the developed neural network that uses Smooth ReLU in the hidden layers, it is appropriate to use the He initialization method [58]. The 1st hidden layer weight ...
... For the developed neural network that uses Smooth ReLU in the hidden layers, it is appropriate to use the He initialization method [58]. The 1st hidden layer weight initialization is not required since this layer performs simple summation. ...
The article’s main provisions are the development and application of a neural network method for helicopter turboshaft engine thermogas-dynamic parameter integrating signals. This allows you to effectively correct sensor data in real time, ensuring high accuracy and reliability of readings. A neural network has been developed that integrates closed loops for the helicopter turboshaft engine parameters, which are regulated based on the filtering method. This made achieving almost 100% (0.995 or 99.5%) accuracy possible and reduced the loss function to 0.005 (0.5%) after 280 training epochs. An algorithm has been developed for neural network training based on the errors in backpropagation for closed loops, integrating the helicopter turboshaft engine parameters regulated based on the filtering method. It combines increasing the validation set accuracy and controlling overfitting, considering error dynamics, which preserves the model generalization ability. The adaptive training rate improves adaptation to the data changes and training conditions, improving performance. It has been mathematically proven that the helicopter turboshaft engine parameters regulating neural network closed-loop integration using the filtering method, in comparison with traditional filters (median-recursive, recursive and median), significantly improve efficiency. Moreover, that enables reduction of the errors of the 1st and 2nd types: 2.11 times compared to the median-recursive filter, 2.89 times compared to the recursive filter, and 4.18 times compared to the median filter. The achieved results significantly increase the helicopter turboshaft engine sensor readings accuracy (up to 99.5%) and reliability, ensuring aircraft efficient and safe operations thanks to improved filtering methods and neural network data integration. These advances open up new prospects for the aviation industry, improving operational efficiency and overall helicopter flight safety through advanced data processing technologies.
... Because performing the operation by hand is not an option, computer programmes must be created instead. Python programming is a popular choice for mathematical and scientific computation because of its simple syntax and ease of use [8][9][10][11][12][13][14][15]. Additionally, highly robust and extensive are the libraries for arrays (NumPy) and data visualisation (Matplotlib) [16][17][18][19][20]. ...
This article explores the application of Monte Carlo simulation to model the two-dimensional Laplace equation which is commonly used in the steady-state heat conduction problems. By using statistical random walk principles, the study develops a Python-based algorithm to approximate solutions for the Laplace equation having fixed boundary (Dirichlet) conditions. The methodology involves formulating probability-based steps, discretizing the equation, and simulating particle paths to estimate temperatures at each grid point. A Python program has been developed to automate this process which has been tested using a sample problem. The results showed excellent agreement with analytical solutions, achieving 98% accuracy with 2000 random walks per node. The findings highlight the trade-off between increased accuracy and computational effort, as accuracy improves with a higher number of random walks. This approach and the provided Python code offer researchers a framework for applying Monte Carlo methods to similar problems and thus illustrate the adaptability of Python for complex simulations.
... Santos et al. [25] demonstrated that a general form of the gas-phase energy equation would be independently found from the species diffusion transport modeling with a simple expression of incorporating multi-species enthalpy diffusion terms. An attempt was made to analyze the linear/nonlinear and steady/unsteady heat transfer formulations through the homotopy and perturbation schemes by Dumka et al. [26]. ...
In the current research, a novel hybrid scheme is proposed to solve nonlinear equations arising in heat transfer through the combination of meta‐heuristic algorithms and interpolation methods. In order to define a proper objective function for minimization by particle swarm optimization (PSO), the constrained problem is converted into an unconstrained one through the penalty method. Furthermore, the moving least square (MLS) technique is implemented to interpolate and approximate the derivatives appeared in the equation. The main problem for challenging this combined scheme is nonlinear transient convective–radiative heat transfer in existence of a magnetic field. To study the efficiency of the MLS, the results would be contrasted with those extracted by a finite difference method (FDM) based PSO approach. Through five distinctive examples, the evolutionary diagrams as well as temperature distributions found by different methods are displayed, and the effects of the constant parameters are investigated. Besides, the simulations of this research work clearly depict good agreements of the numerical results obtained by the suggested idea with those reported in literature.
... Python programming is utilized to automate the process of charting and analyzing load calculations in consideration of the aforementioned factors, owing to its capabilities in complex and scientific computation [33][34][35][36]. Python's strengths lie in its large community support and extensive library offerings. ...
In this research article, Python programming has been used to model the dynamic wind loading on the structures by using the Gust factor approach. The evaluation has included the design wind velocity and pressure, the Gust factor, and wind loads, which are in sync with the stability criteria as outlined in the IS 875 (Part 3), 2015. One practical numerical problem has been addressed to assess the accuracy and reliability of the developed functions, by providing a detailed step-by-step explanation of the programming process and algorithm. The results thus obtained have demonstrated a very good agreement with the results which have been obtained by using the analytical approach. The observations have indicated that the windward loads exhibit higher magnitudes than those in the across-wind direction. Notably, it has also been observed that the wind loads decrease with the height due to reductions in the tributary area and the top mean floor height.
... This is where the Python programming language becomes highly valuable. Python is easy to grasp, thanks to its clear syntax and vibrant programming community [6][7][8][9][10]. Modelling problems in Python typically requires only a few lines of code, supported by powerful modules such as SymPy, NumPy, SciPy, and Matplotlib [11][12][13][14][15][16]. ...
This paper presents the development of a Python module designed for various Logic Gates. Three circuits were built using the functions defined in Python for the logic gates, and the outcomes were found to align completely with those reported in the literature. The circuits were tested with different inputs obtaining the expected outputs.
... While CFD models often use interface capturing methods like Volume of Fluid (VOF) [45], level Set [46], or front tracking [47], the model developed in this study uses a methodologically distinct approach tailored to our specific research questions concerning interfacial thermal dynamics. Dumka et al. [48] used Python for heat transfer issues, and for modeling thermodynamic gas power cycles [49]. The decision to focus on pure mathematical modeling by using python programming language was driven by the aim of exploring the near interface thermal dynamics in order to describe the liquid-vapor interface of an LH2 tank. ...
... The fundamental strengths of Python are its vast community and extensive library. The NumPy [12], SciPy [13], SymPy [14,15], and Matplotlib are some of the modules which are competent of performing scientific, numeric, symbolic computations as well as data visualization and interpretation. With a very small number of lines of code, the modules are extremely capable of carrying out any mathematical computation, including those using algebra, trigonometry, calculus, set theory, etc. ...
In this article, an attempt has been made to explain and model the Taylor table method in Python. A step-by-step algorithm has been developed, and the methodology has been presented for programming. The developed TT_method () function has been tested with the help of four problems, and accurate results have been obtained. The developed function can handle any number of stencils and is capable of producing the results instantaneously. This will eliminate the task of hand calculations and the use can directly focus on the problem solving rather than working hours to descretize the problem.
... The Python language also has great potential to solve and create various equations. Pankaj et al. (2022) have proposed the Python method of perturbation to solve heat transfer equations. Chukwuka et al. (2021) have proposed a Python optimization method for co-digestion. ...
In this paper, a fault diagnosis method based on Boolean matrix filtering and optimizing backpropagation (BP) neural networks is proposed for angle heads of computer numerical control (CNC) machine tools. The matrix filtering is first carried out with the fault case database and the fault cause symptom Boolean matrix according to the fault types and characteristics of the machine tool angle head. On the basis of the combination of multiple fault causes obtained from the initial filtering, the Euclidean distance method is used to narrow the results of fault cause filtering. The BP neural network model with weight vector is established and optimized to perform an accurate diagnosis. Finally, the fault diagnosis and management system of the angle head of a CNC machine tool, integrating with the Boolean matrix filtering method, Euclidean distance method, and BP neural network model, is developed and implemented with the Python language and QT development framework.
... Programming languages with simple syntax and straightforward implementation include Python [15]- [18]. This programming language has a large number of modules and very rich in the community support [19], [20]. NumPy [16], [21]- [24] which stand for 'Numerical Python' is a Python module dedicated to perform numerical computations. ...
... Programming languages with simple syntax and straightforward implementation include Python [15]- [18]. This programming language has a large number of modules and very rich in the community support [19], [20]. NumPy [16], [21]- [24] which stand for 'Numerical Python' is a Python module dedicated to perform numerical computations. ...
In this article, a model of the double integration process has been attempted using Monte Carlo simulation. Both non-rectangular and rectangular domains are dealt separately. Python programming has been used to automate the entire random number-based integration process. Three example problems are used to test the newly created cods, and the results achieved are consistent with those reported in the literature.
... The choice of Python programming for mathematical and Advancement in Mechanical Engineering and Technology Volume 6 Issue 1 DOI: https://doi.org/10.5281/zenodo.7756159 scientific computation is eminent as its syntax is very easy and it is easy to implement [11][12][13][14][15][16][17][18][19][20][21][22]. Also, the libraries for arrays (NumPy) and data plotting (Matplotlib) are also very rich and robust [23][24][25]. ...
In this article an attempt has been made to model one-dimensional steady state heat conduction problem using Monte Carlo simulation (random walk). First, the differential equation has been discretized and the probabilities are set. Then an algorithm has been devised to apply the simulation and the detailed process has been described and is composed of a Python-program. A numerical problem was used to test the software, and it was found that the outcomes are extremely similar to those of the analytical results. It has been observed that more the number of random walks the more the accuracy of the results will be.
... Keeping all above factors in view, Python programming is being applied for automating the process of charting and analysing load computations, as it provides platforms for sophisticated and scientific computation [17]- [20]. The actual advantages of Python are its extensive community and comprehensive library. ...
In this research article an attempt has been made to analyse the design wind pressure on the rectangular buildings using Python programming. For this purpose, design wind pressure is calculated and compared using IS 875 (Part 3) 1987 and the revised code IS 875 (Part 3) 2015. The concept have been applied to three different building heights i.e., 20, 40, and 60 m having same plan 10×10 m 2. It has been observed that the impact of wind pressure on the building rises with the building height along with the fact that the design wind pressures obtained from revised code show more pressures in comparison to the old one. Therefore, the design based on the revised code will be more close to the reality as it incorporates risk factor, directionality factor, area averaging factor, and combination factors. Also, the module developed for wind loads can be readily used by the researchers/designers for the better understanding of programming and the design loads.
... The easiest language to pick up is Python as it have a powerful library for performing numeric or symbolic computations [8][9][10]. The symbolic calculations are done with the help of the SymPy [11][12][13][14] module, whereas for numerical computations, NumPy [15][16][17][18] is used. Matplotlib [19,20] is a powerful tool for data visualization. ...
In this research article an attempt has been made to solve thermal power plant problems using Python programming. The hand calculations and use of thermodynamic property tables make it difficult sometimes to arrive at the correct solutions. But due to the PYroMAT library one can easily search for property data and solve for the Rankine cycle. Also, the python makes it so easy to plot the Rankine cycle once data at each equilibrium point is obtained. Two touch numerical problems have been used to show the capability of PYroMAT library in solving for the power plant thermodynamics problems. The results obtained from the codes matches exactly with the literature.
... Here comes the importance of python language, although the python is slow but the optimal use of its modules like SciPy [9,10] and NumPy [11][12][13][14] can ease the task of code writing along with accuracy and precision. Moreover Matplotlib.pylab is very handy library for post processing of results [8,15,16]. In this article Jupyter notebook has been used to write Python codes along with data visualization. ...
In this article, an attempt has been made to explain and code the Thomas algorithm, also called TDMA (Tridiagonal matrix algorithm) in Python. A step-by-step algorithm has been developed, and the methodology has been presented for programming. The developed TDMA function has been tested with the help of two numerical problems, and accurate results have been obtained. This article will help the students with numerical computations while developing their codes for different numerical problems.
... Moreover, the computations will be very fast compared to the hand calculations, and will be accurate. To simplify the solution and implement the thermodynamic equations mentioned above, the Python programming language is most suited as its syntax is very easy and its modules are very powerful [3][4][5][6]. To solve problems symbolically, SymPy is a very powerful tool which is also written in Python [7][8][9][10]. ...
In this paper, an attempt has been made to develop a Python module for evaluating the first law of thermodynamics, which includes the process of work done and amount of heat gained or lost by the system and the amount of internal energy stored. The modules NumPy and Matplotlib were used to perform the stipulated task. In addition, the correctness of codes was checked against different numerical problems, and it has been observed that the program results match exactly with the results in the literature. As a result, the functions thus developed have shown high accuracy with the least effort and error in all the cases.
... Here comes the importance of Python language for programming. Python is effortless to understand, with lucid syntax and a rich programming community [3][4][5][6]. Modelling any problem in Python requires a few lines of code, backed up by modules like SymPy, NumPy, SciPy, Matplotlib etc. [7][8][9][10][11][12]. Several researchers have adopted this language for numerically and symbolically solving different engineering problems [13][14][15]. ...
In this paper, an attempt has been made to develop a Python module for different Logic Gates. The correctness of the codes has been checked against three numerical problems, and it has been observed that the program results match exactly with the results in the literature. As a result, the developed functions have shown high accuracy with the least effort and error in all the cases.
... Here comes the importance of Python programming, which is very easy to implement [2][3][4][5]. The strength of Python lies in its modules, specially NumPy, Sympy, and Matplotlib, which are capable of solving any problem numerically and symbolically and plotting the data with only a few lines of code [6][7][8][9][10]. ...
In this paper, an attempt has been made to develop a Python module for evaluating energy in transit (viz., heat and work). The work considered in this article is non-dissipative displacement work. Functions for constant pressure, volume, and temperature were developed, along with adiabatic and polytropic processes. Also, the sensible and latent heat were also transformed into functions. The modules NumPy and Matplotlib were used to perform the stipulated task. The correctness of codes was checked against four different numerical problems, and it has been observed that the program results match exactly with the results in the literature. As a result, the developed functions thus developed have shown high accuracy with the least effort and error in all the cases.
... The rich library and huge community are the real strengths of Python. The NumPy [11][12][13][14], SciPy [15], SymPy [16,17], and Matplotlib are some of the libraries which are capable of numeric, scientific, symbolic computations along with data visualization and interpretation.The modules are very much capable of performing any mathematical computation whether it is algebra, trigonometry, calculus, set theory etc. with minimal lines of codes. Though it is computer tool but, now a days researchers are using Python in every stream of science to automate the research processes [18]. ...
To remove the errors associated with the hand calculations involved in load calculations involved in power plant a python-based function approach has been introduced in this research article. Different factors have been automated by developing Python modules. Also, modules have been devised to generate load and load duration curves. Two problems have been used to check the accuracy of developed functions. The results are very encouraging and are in well agreement with the hand calculations.
... Python has also been used by Marowka [39] for parallel computation. For solving heat transfer problems, a Python implemented He's Homotopy and Perturbation approach has been adopted by Dumka et al. [40]. Dumka et al. [41] have also used Python to model thermodynamic gas power cycles. ...
Buckingham's Pi theorem plays an important role in engineering, applied mathematics, and physics for dimensional analysis. From the given variables, it will be utilised to evaluate the set of dimensionless parameters. It indicates that the validity of the physical law is independent of the specific unit system, and it can be expressed as an identity incorporating only dimensionless variables associated with the law. A python-based function has been developed and reported in this manuscript, which can be utilised to evaluate Pi terms (commonly called π terms) for any fluid flow problem. Smaller moderation in the written function can make it capable of solving any fundamental dimensions. Different fluid mechanics problems are utilised to test and validate the reported function, five of which are presented in this manuscript along with code. Obtained results are in good agreement with the theoretically obtained results.
... Saarela and Jauhiainen [25] have reported comparative analysis based on python for the analysis of different feature importance measures. For obtaining solutions of linear and nonlinear heat transfer differential equations Dumka et al. [26] have reported the python based He's Homotopy and Perturbation method. Pawar et al. [27] have reported the use of Python based approach to handle fluid flow problems. ...
In this research article, an attempt has been made to model gas power cycles (Carnot, Otto, Diesel, Dual, Atkinson, Lenoir, Stirling, and Ericsson). The thermodynamic modelling of the processes has been done in the python. A detailed algorithm for each cycle is developed and implemented via python programs. The pressure and volume at each state for different processes were obtained and plotted in p-v diagrams. The results of the program developed have shown good agreement with the literature. The developed code can also be extended for any other thermodynamic cycle.
In this research article, a Python-based machine learning model prediction study was conducted based on the study results obtained from sudden expansion tubes containing different expansion angles, dimpled fin structures and nanofluids, whose thermo-hydraulic performance was previously examined. In the study, Artificial Neural Network and Ridge regression models were used to make predictions on the average Nusselt number (Nu), average Darcy friction factor (f) and performance evaluation criteria (PEC). Physical variations of the sudden expansion tube were taken into account and a detailed comparison of the results was made. A superior average Nu was acquired as 172.45 %, 22.05 %, 17.18 %, 13.65 %, and 7.76 % compared to Ag-MgO/H2O, Al2O3/H2O (blade), CoFe2O4/H2O, Al2O3/H2O (cylindrical), and Al2O3/H2O (platelet), respectively. The highest Performance Evaluation Criteria (PEC) for Re= 2000 based on Al2O3/H2O (platelet) shows an increase of 4.84 %, 12.08 %, 11.76 %, 66.05 %, and 148.94 % compared to Al2O3/H2O (cylindrical), Al2O3/H2O (blade), CoFe2O4/H2O, Fe3O4/H2O, and Ag-MgO/H2O, respectively. From the results obtained, it was determined that Python-based Machine Learning approach which facilitates custom optimizations showed a significant performance with small margins of error in predicting the heat transfer parameters. The lowest error rates of machine learning and polynomial ridge regression models ranged from 0.2 % to 5.4 % for the unseen test set and the application of Python-based algorithms provided considerable savings in calculation time compared to conventional methods. On the other hand, using machine learning models with feature engineering has been found to increase model performance by at least 30 %. In these years when studies on the predictions of thermo-hydraulic studies are very rare in the literature, this study is intended to facilitate scientists, engineers and academicians who will further study on this subject.
An effective method for resolving non-linear partial differential equations with fractional derivatives is the New Sumudu Transform Iterative Method (NSTIM). It excels at solving difficult mathematical puzzles and offers insightful information about the behaviour of time-fractional Fisher equations. The method, which makes use of Caputo's sense derivatives and Wolfram in Mathematica, is reliable, simple to use, and gives a visual depiction of the solution. The analytical findings demonstrate that the proposed approach is effective and simple in generating precise solutions for the time-fractional Fisher equations. The results are made more reliable and applicable by including Caputo's sense derivatives. Mathematical modelling relies on the effectiveness and simplicity of the NSTIM approach to solve time-fractional Fisher equations since it enables precise solutions without the use of a lot of processing power. The NSTIM approach is a useful tool for researchers in a variety of domains since it also offers a flexible framework that is easily adaptable to other fractional differential equations. It now becomes possible to examine the dynamics and behaviour of complex systems governed by time-fractional Fisher equations with efficiency and reliability, opening up new research avenues. The ability to solve time-fractional Fisher equations efficiently and reliably using the NSTIM approach has significant implications for various fields such as population dynamics, mathematical biology, and epidemiology. Researchers can now analyze the spread of diseases or study the population dynamics of species with higher accuracy and less computational effort. This advancement in solving fractional differential equations paves the way for deeper insights into the behavior and patterns of complex systems, ultimately advancing scientific understanding and offering new possibilities for practical applications.
The transport of fluids through pipes is very common in core engineering practice. The main issue that came up when the pipe flow networks were its analysis part. The Hardy Cross approach is very accurate and reliable for solving these issues, but because it is iterative, the likelihood of errors increases as the number of circuit loops grows. Therefore, in this research, a piece of effort has been made to automate the Hardy Cross technique using Python programming (as it is user-friendly and has a large library backup) to remove the errors that come with using hand calculations. The built program has been applied to four different pipe flow network problems, and the outcomes are the same as those presented in the literature.
Nomenclature: 2 A Cross-sectional area of pipe (m) c Fanning's friction factor f d Pipe diameter (m) D Hydraulic diameter (m) h L Pipe length (m) 3 Q Volume flow rate (m /s) R Reynolds number e v Average flow velocity f Darcy's Friction factor 3 ρ Fluid density (kg/m) Δp Pressure drop (Pa) ϵ Wall roughness (m)
This paper applies the solution of ODE's encounter in fluid mechanics by augmenting it with symbolic Python. The implementation procedure has been illustrated for two types of parallel flows, i.e., Couette and Hagen Poiseuille flow. The suggestive results thus obtained are plotted and presented using Matplotlib. The manuscript will help beginners of fluid mechanics solve the fluid flow problems computationally and produce the results in a better way.
In this research article an attempt has been made to examine the performance of Finite difference method (FDM) and Least square method (LSM) on the solution of first order ordinary differential equations (ODE). Both FDM and LSM are applied on the test problem and the results thus obtained are compared with the exact solution. It has been observed that for third degree basis function the results of LSM very close to the analytical result. On further increasing the degree the improvement in the result is very meagre and N=3 can be considered as the optimum solution for LSM. It has also been observed that the FDM is very sensitive to the number of grid points and deviates from the exact results by a substantive amount for lower number of nodes. Whereas the LSM is independent of the number of nodes.
As the UK battery modelling community grows, there is a clear need for software that uses modern software engineering techniques to facilitate cross-institutional collaboration and democratise research progress. The Python package PyBaMM aims to provide a flexible platform for implementation and comparison of new models and numerical methods. This is achieved by implementing models as expression trees and processing them in a modular fashion through a pipeline. Comprehensive testing provides robustness to changes and hence eases the implementation of model extensions. PyBaMM is open source and available on GitHub. For more information visit www.pybamm.org.
In this paper, to improve the vibrational response of microstructures, the impact of the nonlinear modal analysis of axially functionally graded (AFG) truncated conical micro-scale tube including the thermal loading for the different type of cross sections such as uniform section, linear tapered section, convex section, the exponential section are studied that are applicable for various application, for example, the micro-thermal fins, macro-/micro-fluid-flow diffuser, fluid-flow nozzle, fluid-flow throat, micro-sensor, etc. The nonlinear equations are obtained applying Hamilton’s principles based on the modified couple stress to determine the size effect and Euler–Bernoulli beam theory considering the von-Kármán’s nonlinear strain. The material combination varies along the tube’s length, denouncing the AFG tube made by metal and ceramic phases. The nonlinear equations are solved by applying a couple of homotopy perturbation methods (HPM) to calculating the nonlinear results and the generalized differential quadrature method (GDQM) to providing the initial conditions. The linear and nonlinear results presented the effect of various cross sections and other parameters on the micro-tube frequency that are valuable to design and manufacture the micro-electro-mechanical systems (MEMS).
A modification of the homotopy perturbation method is suggested with three effective expansions to solve a nonlinear oscillator with damping terms to expand the solution, the frequency and the amplitude. The Duffing equation with linear damping is used as an example to illustrate the simple solution process and effective results. The analysis exhibits that the amplitude behaves as an exponential decay with the damping parameter. This scheme yields a more effective result for the nonlinear oscillators and overcomes the shortcoming in some problems.
This research work is going to apply the homotopy perturbation method to solve the problem of flowing Newtonian fluid on a flat plate. For this purpose, initially, the problem, including the governing equations and boundary conditions, is defined, and after that, the considered assumptions to solve the defined problem are introduced. Next, the working principle of the homotopy perturbation method is described, and then, the way to obtain the analytical solution using the homotopy perturbation method is presented, and finally, the accuracy of the proposed analytical solution in comparison to the numerical approach is compared for validation. Both momentum and energy equations are solved. The maple software program is utilized for carrying out the mathematical calculations, while the validation is done using the profiles for stream function, velocity distribution, stress, and dimensionless temperature as the key indicators related to the solution. The conducted comparison shows that the analytical solution provided by the homotopy perturbation method is able to predict all the important performance criteria for the problem very well, and therefore, the homotopy perturbation method has a strong potential to be employed for providing the analytical solution for such problems.
Explainable artificial intelligence is an emerging research direction helping the user or developer of machine learning models understand why models behave the way they do. The most popular explanation technique is feature importance. However, there are several different approaches how feature importances are being measured, most notably global and local. In this study we compare different feature importance measures using both linear (logistic regression with L1 penalization) and non-linear (random forest) methods and local interpretable model-agnostic explanations on top of them. These methods are applied to two datasets from the medical domain, the openly available breast cancer data from the UCI Archive and a recently collected running injury data. Our results show that the most important features differ depending on the technique. We argue that a combination of several explanation techniques could provide more reliable and trustworthy results. In particular, local explanations should be used in the most critical cases such as false negatives.
In this research work, we propose an unprecedented hybrid algorithm which involves the coupling of a new integral transform viz. the Elzaki transform and the well known Homotopy perturbation method called the Elzaki homotopy transformation perturbation method (EHTPM) to solve for the exact solution of three distinct types of fisher’s equation viz. the fisher’s equation of two cases, the sixth order fisher’s equation, and the nonlinear diffusion equation of the Fisher type. These equations are prominent in mathematical biology and highly applicable in genetic propagation, population dynamics, stochastic processes, combustion theory, as well as a prototype model for a spreading flame and so on. The efficacy and authenticity of this method was established via convergence and error analysis, and shows that the solutions obtained from EHTPM are unique and convergent. The results of EHTPM when compared with the exact results, homotopy results and results from other existing literature via table of comparison, 3D plots and the convergence plots validates that EHTPM is an efficient and reliable tool of providing exact solutions to a wider class of nonlinear partial differential equations in a simple and straight –forward manner, with no discretization, linearization, computation of Adomian polynomials, and devoid of errors.
A brief introduction to the development of the homotopy perturbation method is given, and the main milestones are elucidated with more than 90 references. This paper further improves the method by constructing a homotopy equation with one or more auxiliary parameters embedding in the linear term with a clear advantage in accelerating and controlling the approximation convergence speed. Moreover, a revision of a recent amplitude-period approximation formula is presented providing an answer to an open problem related to the optimal approximation along with a new universal formula. Duffing equation is used as an example to illustrate the solution process for the homotopy perturbation method, and only one or few iterations are needed in practical applications, making the method much attractive. From the side of amplitude-period formulation, the nonlinear pendulum, the Duffing equation and an oscillator with discontinuity are analyzed providing an asymptotic exact equivalence for bigger parameter values in the case of Duffing’s system. This mini review gives a tutorial guideline for practical applications of the homotopy perturbation method, the references are not exhaustive.
Python is gaining popularity in academia as the preferred language to teach novices serial programming. The syntax of Python is clean, easy, and simple to understand. At the same time, it is a high-level programming language that supports multi programming paradigms such as imperative, functional, and object-oriented. Therefore, by default, it is almost obvious to believe that Python is also the appropriate language for teaching parallel programming paradigms. This paper presents an in-depth study that examines to what extent Python language is suitable for teaching parallel programming to inexperienced students. The findings show that Python has stumbling blocks that prevent it from preserving its advantages when shifting from serial programming to parallel programming. Therefore, choosing Python as the first language for teaching parallel programming calls for strong justifications, especially when better solutions exist in the community.
SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.
In this paper, we present Perturbation Method (PM) to solve nonlinear problems. As case study PM is employed to obtain approximate solutions for differential equations related with heat transfer phenomena. Comparing figures between approximate and exact solutions, show the effectiveness of the method.
This paper investigates the numerical solution of a fractionally damped dynamic system. A single degree of freedom spring-mass mechanical system with fractional damping of order 1/2 is considered for the analysis. Homotopy perturbation method (HPM) is used to compute the dynamic responses of the system subjected to unit step and unit impulse loads. Obtained results are depicted in terms of plots. Comparisons are made with the analytic solutions obtained by using fractional green function of Podlubny (1999) [5] and numerical solution of [20] and [25] in the special cases, to show the effectiveness and validation of the present analysis.
The laminar fully developed nanofluid flow and heat transfer in a vertical channel are investigated. By means of a new set of similarity variables, the governing equations are reduced to a set of three coupled equations with an unknown constant, which are solved along with the corresponding boundary conditions and the mass flux conservation relation by the homotopy perturbation method (HPM). We have tried to show reliability and performance of the present method compared with the numerical method (Runge-Kutta fourth-rate) to solve this problem. The effects of the Grashof number (Gr), Prandtl number (Pr) and Reynolds number (Re) on the nanofluid flows are then investigated successively. The effects of the Brownian motion parameter (Nb), the thermophoresis parameter (Nt), and the Lewis number (Le) on the temperature and nanoparticle concentration distributions are discussed. The current analysis shows that the nanoparticles can improve the heat transfer characteristics significantly for this flow problem.
In this paper, authors demonstrate the efficiency of optimal homotopy asymptotic method (OHAM). This is done by solving nonlinear Volterra integral equation of first kind. OHAM is applied to Volterra integral equations which involves exponential, trigonometric function as their kernels. It is observed that solution obtained by OHAM is more accurate than existing techniques, which proves its validity and stability for solving Volterra integral equation of first kind.
In this paper, we present a reliable algorithm based on the homotopy analysis transform method (HATM) to solve the linear and nonlinear Klein–Gordon equations. The Klein–Gordon equation is the equation of motion of a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. It describes the quantum amplitude for finding a point particle in various places, the relativistic wave function, but the particle propagates both forwards and backwards in time. The HATM is a combined form of the Laplace transform method and homotopy analysis method. The method provides the solution in the form of a rapidly convergent series. Some numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. The results show that the HATM is very efficient, simple and can be applied to other nonlinear problems.
Replacing symbols with random variables makes it possible to naturally add statistical operations to complex physical models. Three examples of symbolic statistical modeling are considered here, using new features from the popular SymPy project.
In the present paper, a conjugate forced convection heat transfer from a good conducting plate with temperature-dependent thermal conductivity is studied. The semi-analytical solution for the non-linear integro-differential equation occurring in the problem is handled easily and accurately by implementing Differential Transform Method (DTM). The horizontal plate is heated with uniform heat flux at the lower surface while being cooled at the upper surface under laminar forced convection flow. A numerical approach is also performed via finite volume method to examine the validity of the results obtained by DTM. The results of DTM show closer agreement to the results of numerical method than the results obtained by perturbation method existed in the literature. It is concluded that for a good conducting plate with a finite thickness the distribution of the conjugate heat flux at the upper surface is significantly affected by the plate thickness. Moreover, we conclude that in the conjugate heat transfer case the temperature distribution of the plate is flatter than the one in the non-conjugate case.
The aim of this article is to introduce a new approximate method, namely homotopy perturbation transform method (HPTM) which is a combination of homotopy perturbation method (HPM) and Laplace transform method (LTM) to provide an analytical approximate solution to time-fractional Cauchy-reaction diffusion equation. Reaction diffusion equation is widely used as models for spatial effects in ecology, biology and engineering sciences. A good agreement between the obtained solution and some well-known results has been demonstrated. The numerical solutions obtained by proposed method indicate that the approach is easy to implement and accurate. Some numerical illustrations are given. These results reveal that the proposed method is very effective and simple to perform for engineering sciences problems.
Based on the homotopy perturbation method (HPM), a general analytical approach for obtaining approximate series solutions to Volterra integro-differential equations of fractional order is proposed. The approximate solutions are calculated in the form of a convergent series with easily computable components. In this paper, the uniqueness of the obtained solution and the convergence properties of the approach are studied. Some examples are presented, to verify convergence, and illustrating the efficiency and simplicity of the approach.
Although attempts have been made to solve time-dependent differential equations using homotopy perturbation method (HPM), none of the researchers have provided a universal homotopy equation. In this paper, going one step forward, we intend to make some guidelines for beginners who want to use the homotopy perturbation technique for solving their equations. These guidelines are based on the L part of the homotopy equation and the initial guess. Afterwards, for solving time-dependent differential equations, we suggest a universal L and v0 in the homotopy equation. Examples assuring the efficiency and convenience of the suggested homotopy equation are comparatively presented.
This paper is an elementary introduction to the concepts of the homotopy perturbation method. Particular attention is paid to giving an intuitive grasp for the solution procedure throughout the paper.
D. K. Biss (Topology and its Applications 124 (2002) 355-371) introduced the
topological fundamental group and presented some interesting basic properties
of the notion. In this article we intend to extend the above notion to homotopy
groups and try to prove some similar basic properties of the topological
homotopy groups. We also study more on the topology of the topological homotopy
groups in order to find necessary and sufficient conditions for which the
topology is discrete. Moreover, we show that studying topological homotopy
groups may be more useful than topological fundamental groups.
This paper focuses on finding the solution of some nonlinear partial differential equations with initial and boundary conditions. This is achieved using the homotopy perturbation method. The solutions obtained are said to be analytic approximate in nature. The applications basically are on inhomogeneous partial differential equations.
Purpose
This study aims that very lately, Mohand transform is introduced to solve the ordinary and partial differential equations (PDEs). In this paper, the authors modify this transformation and associate it with a further analytical method called homotopy perturbation method (HPM) for the fractional view of Newell–Whitehead–Segel equation (NWSE). As Mohand transform is restricted to linear obstacles only, as a consequence, HPM is used to crack the nonlinear terms arising in the illustrated problems. The fractional derivatives are taken into the Caputo sense.
Design/methodology/approach
The specific objective of this study is to examine the problem which performs an efficient role in the form of stripe orders of two dimensional systems. The authors achieve the multiple behaviors and properties of fractional NWSE with different positive integers.
Findings
The main finding of this paper is to analyze the fractional view of NWSE. The obtain results perform very good in agreement with exact solution. The authors show that this strategy is absolutely very easy and smooth and have no assumption for the constriction of this approach.
Research limitations/implications
This paper invokes these two main inspirations: first, Mohand transform is associated with HPM, secondly, fractional view of NWSE with different positive integers.
Practical implications
In this paper, the graph of approximate solution has the excellent promise with the graphs of exact solutions.
Social implications
This paper presents valuable technique for handling the fractional PDEs without involving any restrictions or hypothesis.
Originality/value
The authors discuss the fractional view of NWSE by a Mohand transform. The work of the present paper is original and advanced. Significantly, to the best of the authors’ knowledge, no such work has yet been published in the literature.
In this paper, two strategies are proposed to simplify the dynamics and design of the controller for a variable speed wind turbine (VSWT) with a fractional order doubly fed induction generator (DFIG) in region 2. In the first strategy, the homotopy singular perturbation method (HSPM) is presented using fractional order singular perturbation method (SPM) and modified homotopy perturbation method (HPM). By exerting this method, the fractional order DFIG based VSWT is divided into two lower order subsystems, including nonlinear integer order subsystem and linear fractional order subsystem. In the second strategy, a nonlinear optimal second order fast terminal sliding mode controller (SOFTSMC) using the Lyapunov-based approach with negative definite solution is designed for nonlinear integer order subsystem. Also, a fractional optimal SOFTSMC is proposed by defining the subsystem error dynamics for linear fractional order subsystem. Finally, the control inputs designed for these subsystems are combined and applied to the original system. The main advantage of the HSPM is that both the integer and fractional modelings are used in the design process. The resulting fractional and nonlinear optimal SOFTSMC can be used effectively to achieve the maximum power extraction from the wind, reduce mechanical loads, attenuate the chattering, finite time convergence by applying a small control input, minimize the control input, ensure the better tracking, and tackle the effects of parametric uncertainties, unmodeled dynamics, and external disturbances. Stability of the original system is evaluated by investigating the Lyapunov stability theorem for each subsystem. The proposed controller designed by the HSPM is compared to some existing control laws, and their performance evaluation are reported in Table 1. The results of Table 1 guarantee the effectiveness of the proposed approaches.
Current research into algorithmic explanation methods for predictive models can be divided into two main approaches: gradient-based approaches limited to neural networks and more general perturbation-based approaches which can be used with arbitrary prediction models. We present an overview of perturbation-based approaches, with focus on the most popular methods (EXPLAIN, IME, LIME). These methods support explanation of individual predictions but can also visualize the model as a whole. We describe their working principles, how they handle computational complexity, their visualizations as well as their advantages and disadvantages. We illustrate practical issues and challenges in applying the explanation methodology in a business context on a practical use case of B2B sales forecasting in a company. We demonstrate how explanations can be used as a what-if analysis tool to answer relevant business questions.
A simple homotopy is constructed, by the modified Lindstedt-Poincare method(He, J.H. International Journal of Non-Linear Mechanics, 37, 2002, 309-314 ), the solution and the coefficient of linear term are expanded into series of the embedding parameter. Only one iteration leads to accurate solution.
The linear stability analysis is normally governed by equations that constitute an eigenvalue (characteristic value) problem. In this paper, for the first time, two semi analytical algorithms, (1) Differential Transform Method (DTM) and (2) Adomian Decomposition Method (ADM) are examined for solving a characteristic value problem occurring in linear stability analysis. In this paper, the characteristic value problem of Couette Taylor flow is selected because its simple geometry continues to be a paradigm for theoretical studies of hydrodynamic stability. The results show that DTM handles the solution conveniently and accurately. However, this paper limits the use of ADM for solving characteristic value problems. The results indicate that the present algorithm based on DTM could be used as a promising method for solving characteristic value problems.
The article makes the case that Python is the next wave in Earth sciences computing for one simple reason: Python enables users to do more and better science. The article also describes how these features provide abilities in scientific computing that are currently less likely to be available with existing tools, and highlight the growing support for Python in the Earth sciences as well as events at the upcoming 2013 AMS Annual Meeting that will cover Python in the Earth sciences. A number of other languages have some of Python's features: Fortran 90, for instance, also supports array syntax. Python's unique strengths are the interconnectedness and comprehensiveness of its tool suite and the ease with which one can apply innovations from other communities and disciplines. The Earth sciences community has begun to recognize the unique benefits of Python, and as a result its population of users is growing.
In this paper, a coupling method of a homotopy technique and a perturbation technique is proposed to solve non-linear problems. In contrast to the traditional perturbation methods, the proposed method does not require a small parameter in the equation. In this method, according to the homotopy technique, a homotopy with an imbedding parameter p∈[0,1] is constructed, and the imbedding parameter is considered as a “small parameter”. So the proposed method can take full advantage of the traditional perturbation methods. Some examples are given. The results reveal that the new method is very effective and simple.
A new algorithm is proposed based on semi-analytical methods to solve the conjugate heat transfer problems. In this respect, a problem of conjugate forced-convective flow over a heat-conducting plate is modeled and the integro-differential equation occurring in the problem is solved by two lately-proposed approaches, Adomian decomposition method and differential transform method. The solution of the governing integro-differential equation for temperature distribution of the plate is handled more easily and accurately by implementing Adomian decomposition method/differential transform method rather than other traditional methods such as perturbation method. A numerical approach is also performed via finite volume method to examine the validity of the results for temperature distribution of the plate obtained by Adomian decomposition method/differential transform method. It is shown that the expressions for the temperature distribution in the plate obtained from the two methods, Adomian decomposition method and differential transform method, are the same and show closer agreement to the results calculated from numerical work in comparison with the expression obtained by perturbation method existed in the literature.
The application of perturbation expansion techniques to the analysis of
heat transfer in structures and materials is discussed. Among the
specific techniques described are: regular perturbation expansions;
singular perturbation expansions; the strained coordinates method; and
matched asymptotic expansions. Consideration is also given to: extension
analysis; the improvement of perturbation series; and Pritulo's (1962)
algebraic strained coordinates technique. The techniques are applied to
a series of example problems which are drawn from heat transfer theory.
An extensive bibliography of heat transfer literature devoted to
perturbation analysis is provided.
In this article, the problem of Burgers equation is presented and 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. Comparison is made between the HPM and Exact solutions. 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. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010
In this article, the homotopy perturbation method [He JH. Homotopy perturbation technique. Comput Meth Appl Mech Eng 1999;178:257–62; He JH. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int J Non-Linear Mech 2000;35(1):37–43; He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004;156:527–39; He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003;135:73–79; He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004;151:287–92; He JH. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons & Fractals 2005;26:695–700] is applied to solve linear and nonlinear systems of integro-differential equations. Some nonlinear examples are presented to illustrate the ability of the method for such system. Examples for linear system are so easy that has been ignored.
In this paper, iterated He’s homotopy perturbation method is proposed to solving quadratic Riccati differential equation. Comparisons are made between Adomian’s decomposition method (ADM) and the exact solution and the proposed method. The results reveal that the method is very effective and simple.
The homotopy perturbation technique does not depend upon a small parameter in the equation. By the homotopy technique in topology, a homotopy is constructed with an imbedding parameter p∈[0,1], which is considered as a “small parameter”. Some examples are given. The approximations obtained by the proposed method are uniformly valid not only for small parameters, but also for very large parameters.
In this Letter, homotopy perturbation method (HPM), which does not need small parameters in the equations, is compared with the perturbation and numerical methods in the heat transfer field. The perturbation method depends on small parameter assumption, and the obtained results, in most cases, end up with a non-physical result, the numerical method leads to inaccurate results when the equation is intensively dependent on time, while He's homotopy perturbation method (HPM) overcomes completely the above shortcomings, revealing that the HPM is very convenient and effective. Comparing different methods shows that, when the effect of the nonlinear term is negligible, homotopy perturbation method and the common perturbation method have got nearly the same answers but when the nonlinear term in the heat equation is more effective, there will be a considerable difference between the results. As the homotopy perturbation method does not need a small parameter, the answer will be nearer to the exact solution and also to the numerical one.
Semi-analytical treatments of conjugate heat transfer
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