Metaheuristic optimization algorithms for approximate solutions to ordinary differential equations

Solving ordinary differential equations (ODEs) is vital in diverse fields. However, it is difficult to obtain the exact analytical solutions of ODEs due to their changeable mathematical forms. Traditional numerical methods can find approximate solutions for specific ODEs. Unfortunately, they often suffer from ODEs' forms and characteristics. To approximate different types of ODEs, this paper proposes a generic method based on adaptive differential evolution. Besides, in order to further reduce the error of the obtained approximate solutions, an improved Fourier periodic expansion function is developed, which is then combined with the least square weight method to formulate the ODEs as an optimization problem. Since the proposed method is not limited to ODEs' forms and constraint conditions, it can be used to approximate any ODEs, including linear ODEs and nonlinear ODEs. The proposed method is evaluated on twenty popular test cases. The results indicate that the proposed method is able to accurately approximate different ODEs with better performance compared with other methods.

In this work, we take as our base scalar second-order ordinary differential equations (ODEs) which have seven equivalence classes with each class possessing three Lie point symmetries. We show how one can calculate the conditional symmetries of third-order non-linear ODEs subject to root second-order nonlinear ODEs which admit three point symmetries. Moreover, we show when scalar second-order ODEs taken as first integrals or conditional first integrals are inherited as Lie point symmetries and as conditional symmetries of the derived third-order ODE. Furthermore, the derived scalar nonlinear third-order ODEs without substitution are considered for their conditional symmetries subject to root second-order ODEs having three symmetries.

Micro-cantilever beams are currently employed as sensor in various fields. Of main applications, is using such beams in wind speed sensors. For this purpose, curved out of plane micro-cantilever beams are used. Uniform pressure on such beams causes a large deflection of beam. General mechanics of material theory deals with small deflection and thus cannot be used for explaining this deflection. Although there are a body of works on analysing of large deflection [1], nonlinear deflection, of curved beams [2], yet there is no research on large deflection of curved beam under horizontal uniform distributed force. Theoretically, the wind force is applying horizontally on curved micro-cantilever beam. Here, we neglect the effect of moving weather from beam sides. We first aim how to drive the governed equation. A curved beam does not have a calculable centroid. Also large deflection of beam changes its curvature which would change the centroid of beam consciously. The variation of centroid makes very though calculating the bending moment of each cross section in the beam. To address this issue, an integral equation will be used. The total force will be considered as a single force applied at the centroid. The second challenge is solving the governed nonlinear ordinary differential equation (ODE). Although there are several methods to solve analytically nonlinear ODE, Lie symmetry method, with all its complication, is a general method for this kind of equations. This approach covers all current methods in analytical solving nonlinear ODEs. In this method, an infinitesimal transformation should be calculated. All transformations under one parameter creates a group that called Lie group. A value of parameter which transfers the equation onto itself is called invariant of ODE. One can calculate canonical coordinates ODEs by the invariant. Solving the canonical coordinates ODEs yields to calculating the canonical coordinates. Canonical coordinate are used to reduce the order of nonlinear ODE [3]. By repeating this method one can solve high order ODEs. Our last question is how to do numerical solution of ODE. The possible answer will help to explain the phenomena of deflection clearly and compare the analytical solution with numerical results. Small dimensions of beam, small values of applied force from one side and Young modules value from the other side, will create a stiff ODE. Authors experience in this area shows that the best method to sole these kind of equations is LSODE. This method can be used in Maple. Here, primary calculations show that the governed equation is second order nonlinear ODE and we propose two possible invariants to solve ODE. Overall, the primary numerical solution has shown perfect match with the exact solution.

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.

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

Acoustic nonlinearities in a quasi 1-D duct with arbitrary mean properties and mean flow

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.

In this paper we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of integrals of motion. The proposed algorithm is simple, straightforward and efficient and helps to unearth several new types of linearizing transformations besides the known ones in the literature. To make our studies systematic we divide our analysis into two parts. In the first part we confine our investigations to the scalar ODEs and in the second part we focus our attention on a system of two coupled second-order ODEs. In the case of scalar ODEs, we consider second and third-order nonlinear ODEs in detail and discuss the method of deriving maximal number of linearizing transformations irrespective of whether it is local or nonlocal type and illustrate the underlying theory with suitable examples. As a by-product of this investigation we unearth a new type of linearizing transformation in third-order nonlinear ODEs. Finally the study is extended to the case of general scalar ODEs. We then move on to the study of two coupled second-order nonlinear ODEs in the next part and show that the algorithm brings out a wide variety of linearization transformations. The extraction of maximal number of linearizing transformations in every case is illustrated with suitable examples.

Numerical methods are crucial for solving ordinary differential equations (ODEs) that frequently arise in various fields of science and engineering. This study compares three numerical methods: the fourth-order Runge-Kutta method (RK4), the fourth-order Runge-Kutta Contra-harmonic Mean method (CoM4), and the fourth-order Adam-Bashforth-Moulton method (ABM4) in solving initial value problems of ODEs. Three IVPs of ODEs have been solved with varying step sizes using the three methods that have been proposed, and the solutions for each step size are examined. Numerical comparisons between RK4, CoM4, and ABM4 methods have been presented to solve three initial problems of ODE. Simulation results show that each method has advantages and limitations depending on the type of ODE being solved. We find that for very small step sizes, the numerical solutions agree the best with the exact solution. As such, all three proposed approaches are sufficient to solve the IVP ODE accurately and efficiently. Among the three proposed methods, we observe that the mean absolute error for the RK4 method is the smallest, followed by the ABM4 method.

The numerical solution of a ordinary differential equation (ODE) basically describes any physical phenomenon of interest. Traditionally, the numerical solution of ODE is solved by a discretization process using the classical finite difference approximation. Recently, many studies have used deep neural networks (DNN) to approximate ODE solutions without discretization. However, without the optimization process, DNN cannot run independently. In this work, we proposed an alternative method that combines DNN with a genetic algorithm (GA) as an optimization method. GA initiates a population of DNN trainable parameters, making the DNN training process more streamlined on a single-generation GA. We focused on DNN-GA hybrid solution for linear and nonlinear ordinary differential equations (ODE‘s). Our method is achieved by solving several test problems and demonstrating a comparison with stochastic gradient descent (SGD) optimizers. In most ODE problems, using the same number of iterations and architecture, DNN-GA produces a better loss function than using DNN-SGD.

The optimisation of complex engineering design problems is highly challenging due to the consideration of various design variables. To obtain acceptable near-optimal solutions within reasonable computation time, metaheuristics can be employed for such problems. However, a plethora of novel metaheuristic algorithms are developed and constantly improved and hence it is important to evaluate the applicability of the novel optimisation strategies and compare their performance using real-world engineering design problems. Therefore, in this paper, eight recent population-based metaheuristic optimisation algorithms—African Vultures Optimisation Algorithm (AVOA), Crystal Structure Algorithm (CryStAl), Human-Behaviour Based Optimisation (HBBO), Gradient-Based Optimiser (GBO), Gorilla Troops Optimiser (GTO), Runge–Kutta optimiser (RUN), Social Network Search (SNS) and Sparrow Search Algorithm (SSA)—are applied to five different mechanical component design problems and their performance on such problems are compared. The results show that the SNS algorithm is consistent, robust and provides better quality solutions at a relatively fast computation time for the considered design problems. GTO and GBO also show comparable performance across the considered problems and AVOA is the most efficient in terms of computation time.

In these recent years, several metaheuristic optimization algorithms have been developed to solve complex real-world problems. However, according to the no free lunch theorem, there is no metaheuristic algorithm which has the best suited for solving all optimization problems. Therefore, developing new metaheuristic algorithms is an open problem. Two major categories of metaheuristics are swarm intelligence and evolutionary algorithms. Both of them have their own advantages. In this paper, we present a naturally swarm and evolutionary algorithm in which its particles have memory, collaboration, information sharing, and competition inherited from one specific analogy. So, it has the advantages of both swarm and evolutionary algorithms without complications of hybridizing them. We formulate a new metaheuristic optimization algorithm called open source development model algorithm (ODMA) inspired by open source development model and communities, in such a way that each potential solution is considered as a software, and by evolution of the softwares, better solutions of the function that should be optimized are searched. The main operations of the algorithm include employing features and methods of leading softwares, evolution of leading softwares based on their history and forking from leading softwares. After a detailed formulation and explanation of its implementation, the algorithm is evaluated by $$17$$17 well-known benchmark functions, and the results are compared with PSO, ICA, and GA. The results show that ODMA outperforms GA, PSO, and ICA in terms of finding the global optimum and convergence.

Several optimization problems from various types of applications have been efficiently resolved using available meta-heuristic algorithms such as Particle Swarm Optimization and Genetic Algorithm. Recently, many meta-heuristic optimization techniques have been extensively reported in the literature. Nevertheless, there is still room for new optimization techniques and strategies since, according to the literature, there is no meta-heuristic optimization algorithm that may be considered as the best choice to cope with all modern optimization problems. This paper introduces a novel meta-heuristic optimization algorithm named Dynamic Group-based Optimization Algorithm (DGCO). The proposed algorithm is inspired by the cooperative behavior adopted by swarm individuals to achieve their global goals. DGCO has been validated and tested against twenty-three mathematical optimization problems, and the results have been verified by a comparative study with respect to state-of-the-art optimization algorithms that are already available. The results have shown the high exploration capabilities of DGCO as well as its ability to avoid local optima. Moreover, the performance of DGCO has also been verified against five constrained engineering design problems. The results demonstrate the competitive performance and capabilities of DGCO with respect to well-known state-of-the-art meta-heuristic optimization algorithms. Finally, a sensitivity analysis is performed to study the effect of different parameters on the performance of the DGCO algorithm.

This textbook provides a broad introduction to continuous and discrete dynamical systems. With its hands-on approach, the text leads the reader from basic theory to recently published research material in nonlinear ordinary differential equations, nonlinear optics, multifractals, neural networks, and binary oscillator computing. Dynamical Systems with Applications Using Python takes advantage of Python’s extensive visualization, simulation, and algorithmic tools to study those topics in nonlinear dynamical systems through numerical algorithms and generated diagrams. After a tutorial introduction to Python, the first part of the book deals with continuous systems using differential equations, including both ordinary and delay differential equations. The second part of the book deals with discrete dynamical systems and progresses to the study of both continuous and discrete systems in contexts like chaos control and synchronization, neural networks, and binary oscillator computing. These later sections are useful reference material for undergraduate student projects. The book is rounded off with example coursework to challenge students’ programming abilities and Python-based exam questions. This book will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a range of disciplines, such as biology, chemistry, computing, economics, and physics. Since it provides a survey of dynamical systems, a familiarity with linear algebra, real and complex analysis, calculus, and ordinary differential equations is necessary, and knowledge of a programming language like C or Java is beneficial but not essential.

Step forward on nonlinear differential equations with the Atangana–Baleanu derivative: Inequalities, existence, uniqueness and method