HAAR WAVELETS BASED TIME DISCRETIZATION TECHNIQUE FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

<p>Physics-informed neural networks (PINNs) is an emerging category of neural networks which can be trained to solve supervised learning tasks while taking into consideration given laws of physics described by general nonlinear partial differential equations. PINNs demonstrate promising characteristics such as performance and accuracy using a minimal amount of data for training, utilized to accurately represent the physical properties of a system's dynamic environment. In this work, we employ the emerging paradigm of PINNs to demonstrate their potential in enhancing the security of intelligent cyber-physical systems. In particular, we present a proof-of-concept scenario using the use case of water distribution networks, which involves an attack on a controller in charge of regulating a liquid pump through liquid flow sensor measurements. PINNs are used to mitigate the effects of the attack while demonstrating the applicability and challenges of the approach.</p>

This paper investigates the dynamics of exact traveling-wave solutions for nonlinear spatial and temporal fractional partial differential equations with conformable order derivatives arising in nonlinear propagation waves of small amplitude including nonlinear fractional modified Benjamin–Bona–Mahony equation, fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation and fractional (2+1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation as well. By utilizing the Sine-Gordon expansion method (SGEM), new real- and complex-valued exact traveling-wave solutions are reported by preferring suitable values of physical free parameters. The nonlinear governing equations are reduced into auxiliary nonlinear ordinary differential equations with aid of fractional traveling-wave transformation, in which the fractional derivative is evaluated in a conformable sense. The productivity process of the proposed method for predicting the desirable solutions is also provided. Some of the obtained solutions are simulated graphically in 3D and contour plots. Meanwhile, the effects of the fractional parameter [Formula: see text] in the space and the time direction are illustrated in 2D plots to ensure the novelty, applicability and credibility of the SGEM. These results reveal that the suggested method is general and adequate for dealing with nonlinear models featuring fractional derivatives and can be employed to analyze wide classes of complex phenomena of partial differential equations occurring in engineering and nonlinear dynamics.

Solution of nonlinear partial differential equations from base equations

Moving boundary and boundary value problems occur in many physical and engineering processes involving heat transfer and phase changes, they range from icing problems, for example the Stefan problem, to the melting of metals and plastics using lasers. This book reviews and compares some well known and some recently developed methods to solve these problems. Recent advances in the finite difference solution of linear and non-linear partial differential equations are presented and algorithmic manipulations that enhance the computational efficiency are incorporated into the overall schemes. In this new approach, the the finite difference substitutions are made into the solutions of the partial differential equation rather than classically into the partial differential equation itself. This approach can be exploited for many types of partial differential equation and they may therefore be of interest to researchers studying a wide range of applications in physical problems such as diffusion and heat transfer equations, optoelectronic paraxial wave equations, the quantum mechanical wave equations, time dependent pulse propagation, and similar WKB-derived problemsM

The purpose of this paper is to provide sufficient conditions for the local and global existence of solutions for the general nonlinear distributed-order fractional differential equations in the time domain. Also, we provide sufficient conditions for the uniqueness of the solutions. Furthermore, we use operational matrices for the fractional integral operator of the second kind Chebyshev wavelets and shifted fractional-order Jacobi polynomials via Gauss–Legendre quadrature formula and collocation methods to reduce the proposed equations into systems of nonlinear equations. Also, error bounds and convergence of the presented methods are investigated. In addition, the presented methods are implemented for two test problems and some famous distributed-order models, such as the model that describes the motion of the oscillator, the distributed-order fractional relaxation equation, and the Bagley–Torvik equation, to demonstrate the desired efficiency and accuracy of the proposed approaches. Comparisons between the methods proposed in this paper and the existing methods are given, which show that our numerical schemes exhibit better performances than the existing ones.

The Benjamin-Bona-Mahony (BBM) equation is a nonlinear partial differential equation that describes the propagation of long waves in a shallow water channel. In this work, we present a comprehensive solution for the BBM equation using the Riccati-Bernoulli sub-ODE method. The method involves transforming the BBM equation into a Riccati equation, which is then further transformed into a Bernoulli equation. The Bernoulli equation is then solved analytically, and the solution is used to obtain the solution for the original BBM equation. Our results show that the Riccati-Bernoulli sub-ODE method provides an efficient and accurate solution for the BBM equation. The dfmethod can be extended to solve other nonlinear partial differential equations (NPDEs), making it a valuable tool for researchers in various fields.

Analytical solutions of the space–time fractional Kundu–Eckhaus equation by using modified extended direct algebraic method

: Complex physical phenomena involving chemically reacting systems or the transport of heat or fluids are often modeled by coupled systems of time-dependent, nonlinear partial differential equations. The difficulties in understanding the stability of the differential equation systems and in designing efficient, accurate numerical methods for their solution are widely recognized and were the focus of this research. We have worked on four general aspects of the analysis and numerical approximation of systems of partial differential equations. These areas of research are: (1) modeling aspects and stability analysis for nonlinear time-dependent partial differential equations; (2) use and analysis of finite element or finite difference methods to discretize coupled systems of nonlinear differential equations; (3) development of adaptive or local grid refinement capabilities to resolve local phenomena in large-scale applications; and (4) development of data structures, preconditioners, and efficient solution algorithms for large-scale problems on new computer architectures. Emphasis has been placed upon multiphase or multicomponent, transport-dominated flow processes with dynamic local phenomena. The research also involved a mix of analysis, algorithm development, and large-scale computation using newer computer architectures.

We investigate the existence and Borel summability of formal power series in one variable with holomorphic coefficients, solutions of nonlinear partial differential equations with shrinkings having non-Kowalewski type in the complex domain. Sufficient conditions are given in terms of the shape of the functional partial differential equations and initial conditions. The existence and asymptotic behavior of local sectorial holomorphic solutions of these nonlinear functional partial differential equations are obtained as a by-product. We apply the results to the study of asymptotic the behavior of solutions of some advanced-argument partial differential equations in a neighborhood of infinity.

Recent advancement in the field of nonlinear analysis and fractional calculus help to address the rising challenges in the solution of nonlinear fractional partial differential equations. This paper presents a hybrid technique, a combination of Tarig transform and Projected Differential Transform Method (TPDTM) to solve nonlinear fractional partial differential equations. The effectiveness of the method is examined by solving three numerical examples that arise in the field of heat transfer analysis. In this proposed scheme, the solution is obtained as a convergent series and the result is used to analyze the hyper diffusive process with pre local information regarding the heat transfer for different values of fractional order. In order to validate the results, a comparative study has been carried out with the solution obtained from the two methods, the Laplace Adomian Decomposition Method (LADM) and Homotophy Pertubation Method (HPM) and the result thus observed coincided with each other. Inspite of the uniformity between the solutions, the proposed hybrid technique had to overcome the complexity of manupulation of Adomian polynomials and evaluation of integrals in LADM and HPM respectively. The methodology and the results presented in this paper clearly reveals the computational efficiency of the present method. Due to its computational efficiency, the TPDTM has the potential to be used as a novel tool not only for solving nonlinear fractional differential equations but also for analysing the prelocal information of the system.

Analog computing (AC) was the predominant form of computing up to the end of World War II. The invention of digital computers (DCs) followed by developments in transistors and thereafter integrated circuits (IC), has led to exponential growth in DCs over the last few decades, making ACs a largely forgotten concept. However, as described by the impending slow-down of Moore’s law, the performance of DCs is no longer improving exponentially, as DCs are approaching clock speed, power dissipation, and transistor density limits. This research explores the possibility of employing AC concepts, albeit using modern IC technologies at radio frequency (RF) bandwidths, to obtain additional performance from existing IC platforms. Combining analog circuits with modern digital processors to perform arithmetic operations would make the computation potentially faster and more energy-efficient. Two AC techniques are explored for computing the approximate solutions of linear and nonlinear partial differential equations (PDEs), and they were verified by designing ACs for solving Maxwell's and wave equations. The designs were simulated in Cadence Spectre for different boundary conditions. The accuracies of the ACs were compared with finite-deference time-domain (FDTD) reference techniques. The objective of this dissertation is to design software-defined ACs with complementary digital logic to perform approximate computations at speeds that are several orders of magnitude greater than competing methods. ACs trade accuracy of the computation for reduced power and increased throughput. Recent examples of ACs are accurate but have less than 25 kHz of analog bandwidth (Fcompute) for continuous-time (CT) operations. In this dissertation, a special-purpose AC, which has Fcompute = 30 MHz (an equivalent update rate of 625 MHz) at a power consumption of 200 mW, is presented. The proposed AC employes 180 nm CMOS technology and evaluates the approximate CT solution of the 1-D wave equation in space and time. The AC is 100x, 26x, 2.8x faster when compared to the MATLAB- and C-based FDTD solvers running on a computer, and systolic digital implementation of FDTD on a Xilinx RF-SoC ZCU1275 at 900 mW (x15 improvement in power-normalized performance compared to RF-SoC), respectively.

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 the past few years, there has been a growing interest in investigating the search for solitary wave solutions in the realm of nonlinear partial differential equations. This endeavor represents a captivating and challenging area of research. The primary objective of this study is to investigate the nonlinear Zakharov system through a comprehensive analysis that integrates logarithmic transformations and the symbolic structures of exponential functions. The nonlinear Zakharov system holds remarkable importance as a fundamental cornerstone in the field of plasma physics, derived from the profound intellect of esteemed mathematicians and physicists. By examining a diverse range of solution forms including trigonometric, hyperbolic, and rational expressions, this research delves into the complexities of plasma behavior. Notably, the introduction of arbitrarily selected constants imbues these solutions with a multifaceted and dynamic nature. In support of the findings, the article presents several numerical simulations that align with the derived solutions. The utilized methods stand out for being simple, reliable, and capableof creating fresh solutions for nonlinear partial differential equations in the realm of mathematical physics. Other significance of this research lies in its introduction of innovative methodologies previously unexplored in the study of this particular model, thereby broadening the scientific toolkit. Moreover, the versatility of these techniques offers the potential for seamless adaptation in addressing various other nonlinear partial differential equations.

Deep learning based solution of nonlinear partial differential equations arising in the process of arterial blood flow

In recent years, there have been intensive efforts to establish linearised oscillation results for onedimensional delay, neutral delay and advanced impulsive differential equations. An impressive number of these efforts have yielded fruitful results in many analytical and applied areas. This is particularly obvious in the areas of applied disciplines such as the linear delay impulsive differential equations. However, there still remains a lot more to be explored in this direction, especially, in the area of non-linear autonomous differential equations. In this paper, we are proposing the development of linearised oscillation techniques for some general non-linear autonomous impulsive differential equations with several delays.