HAM-Based Adaptive Multiscale Meshless Method for Burgers Equation

An elegant and powerful technique is Homotopy Perturbation Method (HPM) to solve linear and nonlinear ordinary and partial differential equations. The method, which is a coupling of the traditional perturbation method and homotopy in topology, deforms continuously to a simple problem which can be solved easily. The method does not depend upon a small parameter in the equation. Using the initial conditions this method provides an analytical or exact solution. From the calculation and its graphical representation it is clear that how the solution of the original equation and its behavior depends on the initial conditions. Therefore there have been attempts to develop new techniques for obtaining analytical solutions which reasonably approximate the exact solutions. Many problems in natural and engineering sciences are modeled by nonlinear partial differential equations (NPDEs). The theory of nonlinear problem has recently undergone much study. Nonlinear phenomena have important applications in applied mathematics, physics, and issues related to engineering. In this paper we have applied this method to Burger’s equation and an example of highly nonlinear partial differential equation to get the most accurate solutions. The final results tell us that the proposed method is more efficient and easier to handle when is compared with the exact solutions or Adomian Decomposition Method (ADM).

Homotopy perturbation method (HPM) is a semi-analytical technique for solving linear as well as nonlinear ordinary/partial differential equations. The method may also be used to solve a system of coupled linear and nonlinear differential equations. This chapter first illustrates the basic idea of the HPM. It uses the present method to solve a nonlinear ordinary differential equation, a nonlinear partial differential equation, and a coupled nonlinear system of partial differential equations. It is worth mentioning that the HPM can also be used for handling linear ordinary and partial differential equations. The test problems demonstrated in the chapter confirm that the HPM is an efficient method for solving linear/nonlinear ordinary, partial, and coupled differential equations. The series usually converges with an increase in the number of terms, but one may not expect the compact form solution always.

A topic of interest in financial mathematics is the Black-Scholes model. However, the underlying asset price in the stock market may not be satisfied by this linear model, which was developed under a number of assumptions, including liquidity and the absence of transaction costs. The linear model has restricted its precision in actual market conditions. We study the transaction cost model for modelling illiquid markets from the extended nonlinear Black-Scholes model. Using a semi-discretization finite difference approach, the nonlinear partial differential equation is transformed into a nonlinear ordinary differential equation. Deep Learning (DL) is an advanced technique of machine learning solves the converted ordinary differential equation by fully connected neural network (FCNN). By modelling the complex and nonlinear relationships among market variables, DL models can generate option pricing forecasts that are more dependable and precise, not just for continuous data but also for discontinuous data (at jump point). Introduction: Nonlinear partial differential equation plays a crucial role in financial modelling, especially in the pricing of derivatives like options. The Black- Scholes model, introduced in 1973, continues to be one of the most widely used frameworks for pricing European options. However, this model is a linear one and offers an analytical solution, yet it is not appropriate for the complexities of real market assumptions that exhibit nonlinear effects. We examine the transaction cost model for modelling illiquid markets from the extended nonlinear Black-Scholes model created by Seelama et al. (2021). First using a semi-discretization finite difference approach, the nonlinear partial differential equation is transformed into a nonlinear ordinary differential equation. Solves the converted ordinary differential equation by Deep Learning (DL) based fully connected neural network (FCNN) algorithm. This algorithm is capable of handling the nonlinear behaviour of model and produce more accurate option value for European call. Objectives: Find the solution of more realistic nonlinear model of Black-Scholes equation include transaction cost in illiquid market with deep learning algorithm for a European call option. Methods: From extended nonlinear Black-Scholes model, the nonlinear model of transaction cost in illiquid market is considered for study. The nonlinear partial differential equation is converted into a nonlinear ordinary differential equation by semi-discretization finite difference method. DL is a sophisticated machine learning technique that solves transformed ordinary differential equations using fully connected neural network (FCNN) algorithm. DL algorithm uses a Python program. Results: For European call option, option values are predicted for different number of neurons with different loss functions like MSE and MAE at the time of maturity. Graphical representation shows the accuracy of the algorithm at continuous as well as at jump point (strike price). Conclusions: The method of solving a complex nonlinear partial differential equation by transforming it into a nonlinear ordinary differential equation is valuable. More precise pricing estimates for option values with nonlinear effects in financial data could be improved by deep learning.

The Benjamin–Bona–Mahony–Burgers (BBMB) equation is an important nonlinear partial differential equation in fluid mechanics, used to describe the propagation of long waves. Due to its nonlinearity, obtaining exact solutions is difficult, which motivates the use of efficient approximate methods.In this paper, the BBMB equation is solved using the Variational–Homotopy Perturbation Method (VHPM), which combines the Variational Iteration Method (VIM) and the Homotopy Perturbation Method (HPM), providing a rapidly convergent series solution without requiring a small perturbation parameter. The results obtained by VHPM are compared with those of VIM and HPM. The solution is expressed as a series and truncated at the second-order term due to the method’s fast convergence. Numerical examples are presented for both homogeneous and non-homogeneous cases, and the accuracy is evaluated using the error norms L_2 and L_∞.The results show that VHPM produces more accurate and stable solutions than the other methods. The reported error norms and graphical comparisons confirm the effectiveness and reliability of the proposed technique. Consequently, the VHPM is shown to be a powerful and efficient tool for solving the BBMB equation and similar nonlinear partial differential equations. Keywords: Benjamin, Bona, Mahony, Berger’s Equation, Variational, Homotopy Perturbation Method.

The Homotopy Analysis Method (HAM) has emerged as a powerful analytical technique for solving nonlinear ordinary and partial differential equations across diverse scientific and engineering fields. Unlike traditional perturbation methods, HAM offers flexibility in handling strongly nonlinear problems without requiring small parameters. This paper provides a comprehensive review of the theoretical foundation, computational implementations, and applications of HAM, its development, and its wide range of applications. The discussion includes the mathematical framework, convergence control, comparison with other methods, and a detailed survey of its applications in fluid mechanics, heat transfer, quantum mechanics, and beyond. Furthermore, we highlight recent advancements and potential future directions for HAM.

The magnetohydrodynamic (MHD) flow of a third grade fluid through a rectangular channel, considering the effect of aspect ratio, has been investigated. The flow considered is steady, laminar, incompressible and hydro-dynamically fully developed. The equation, describing the flow, is a highly non-linear partial differential equation (PDE) with remote possibility of having an exact solution and even numerical solution also is very difficult to obtain. A combination of the homotopy perturbation method (HPM) and integral method (IM) has been employed to solve the non-linear PDE which is scarce in open literature. The results of the present study are compared with the results obtained by the least square method (LSM) of the MHD third grade fluid flow through a rectangular channel, without the effect of aspect ratio and are found to be in close agreement. The results indicate that the flow field is significantly affected by the aspect ratio which should be considered for practical applications. In all the available literatures of the third grade fluid flow, the aspect ratio effect is neglected and this simplifying assumption reduces the highly complicated non-linear PDE to a non-linear ordinary differential equation (ODE). The novelty of the subject work lies in the inclusion of the effects of aspect ratio in the governing equation describing the flow of a third grade fluid through a channel and solving this by a combined analytical method (HPM and IM). Further, the effects of the Hartmann number and non-Newtonian third grade fluid parameter on the flow filed are discussed.

Non-linear vibration and instability of multi-phase composite plate subjected to non-uniform in-plane parametric excitation: Semi-analytical investigation

In this research paper, a semi-analytical method,i.e.,homotopy analysis method, is implemented for finding the solution of non-linear partial differential equations. Homotopy analysis method is very effective, and easy to evaluate as compared to other numerical methods. The results through HAM on illustrative examples are compared with two well-known methods, namely, Variational Iteration Method(VIM) and Adomian Decomposition Method(ADM). The comparisons of obtained solutions results in the high accuracy through HAM when compared to the other competing methods. Hence, the solution obtained using HAM has much faster convergence to the exact solution.KeywordsNon-linear partial differential equationNumerical resultsHomotopy analysis method

To satisfy the power needs of human beings without affecting the environment, scientists have been working on the efficient practices of renewable energy sources such as solar, water, and wind. As a source of energy resources, solar radiation is preferred because of its unlimited availability and low ecological impact. Thus, this article examines the heat transfer of unsteady electrically conducting viscous nanofluid flowing over a stretchable cylindrical surface in the presence of solar radiation. The entropy generation is also analyzed with the velocity slip and convective heat transfer boundary conditions. Considering Beer’s law for representing solar radiation, the horizontal cylindrical surface for nanofluid flow, and solving the model by the homotopy analysis method (HAM) can be the novelty of this study. The governing nonlinear partial differential equations (PDEs) are transformed into systems of higher‐order nonlinear ordinary differential equations (ODEs) using appropriate similarity transformations. These ODEs are then solved via the HAM, applying the BVPh2.0 package on Mathematica 12.1. Comparisons with previously published studies confirm the validity of the method and highlight its consistency. The results reveal that the presence of a magnetic field interaction slows down the flow while increasing both local skin friction and the temperature of the nanofluid. Solar radiation and the Eckert and Biot numbers enhance the nanofluid’s temperature, whereas the Prandtl number and the unsteady parameter do not. Variations in temperature and velocity slip are found to reduce entropy generation. When the magnetic field interaction increased by 0.2, the local Nusselt number decreased by 0.1%, whereas the local skin friction rose by 6%. However, both the local Nusselt number and skin friction rose by 0.3% when the curvature parameter increased by 0.2. The findings demonstrate that the flow of nanofluid can be used to transfer heat from solar, which has practical applications in cooking, water heating, and electricity generation. Therefore, the global demand for energy can be partly met by harnessing solar energy effectively.

PurposeThe purpose of this paper is to use the generalized differential transform method (GDTM) and homotopy perturbation method (HPM) for solving time‐fractional Burgers and coupled Burgers equations. The fractional derivatives are described in the Caputo sense.Design/methodology/approachIn these schemes, the solutions takes the form of a convergent series. In GDTM, the differential equation and related initial conditions are transformed into a recurrence relation that finally leads to the solution of a system of algebraic equations as coefficients of a power series solution. HPM requires a homotopy with an embedding parameter which is considered as a small parameter.FindingsThe paper extends the application and numerical comparison of the GDTM and HPM to obtain analytic and approximate solutions to the time‐fractional Burgers and coupled Burgers equations.Research limitations/implicationsBurgers and coupled Burgers equations with time‐fractional derivative used.Practical implicationsThe implications include traffic flow, acoustic transmission, shocks, boundary layer, the steepening of the waves and fluids, thermal radiation, chemical reaction, gas dynamics and many other phenomena.Originality/valueThe numerical results demonstrate the significant features, efficiency and reliability of the two approaches. The results show that HPM is more promising, convenient, and computationally attractive than GDTM.

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

The nonlinear evolution equations with finite memory have a wide range of applications in science and engineering. The Burgers equation with finite memory transport (time-delayed) describes convection-diffusion processes. In this paper, we establish the new solitary wave solutions for the time-delayed Burgers equation. The extended tanh method and the exp-function method have been employed to reveal these new solutions. Further, we have calculated the numerical solutions of the time-delayed Burgers equation with initial conditions by using the homotopy perturbation method (HPM). Our results show that the extended tanh and exp-function methods are very effective in finding exact solutions of the considered problem and HPM is very powerful in finding numerical solutions with good accuracy for nonlinear partial differential equations without any need of transformation or perturbation

Radial basis functions based meshfree schemes for the simulation of non-linear extended Fisher–Kolmogorov model

In the present paper explores, the Burgers' Equation which is the considerable partial differential equation that emerges in nonlinear science. Also, Homotopy Analysis Method (HAM) has been implemented to Burgers' equation with given initial conditions. The efficieny of the proposed method is analyzed by using some illustrative problems. We are compared approximate solutions acquired via HAM with the exact solutions. As a result of comparisons, it is demonstrated that the gained solutions are in excellent agreement. Additionally, 2D-3D graphs and tables of attained results are drawn by means of a ready-made package program. The recent obtained results denote that HAM is extremely efficient technique. Nonlinear partial differential equations can be solved with the help of presented method.

This paper expands the ideas of the spectral homotopy analysis method to apply them, for the first time, on non-linear partial differential equations. The spectral homotopy analysis method (SHAM) is a numerical version of the homotopy analysis method (HAM) which has only been previously used to solve non-linear ordinary differential equations. In this work, the modified version of the SHAM is used to solve a partial differential equation (PDE) that models the problem of unsteady boundary layer flow caused by an impulsively stretching plate. The robustness of the SHAM approach is demonstrated by its flexibility to allow linear operators that are partial derivatives with variable coefficients. This is seen to significantly improve the convergence and accuracy of the method. To validate accuracy of the the present SHAM results, the governing PDEs are also solved using a novel local linearisation technique coupled with an implicit finite difference approach. The two approaches are compared in terms of accuracy, speed of convergence and computational efficiency.