Finding Solutions of the Navier‐Stokes Equations through Quantum Computing—Recent Progress, a Generalization, and Next Steps Forward

Solution of nonlinear partial differential equations from base equations

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.

: 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.

Diverse fields of modern science, including mathematical modeling and physics, plasma physics, atmospheric sciences, marine sciences, hydrodynamics, nonlinear mechanics, and other complex nonlinear physical phenomena, are expressed through Nonlinear Partial Differential Equations (NLPDEs). Exact solutions play an important role in understanding the behavior of solitary wave solutions and the dynamical properties of significant outcomes for higher-dimensional NLPDEs. As an integrable extension of the nonlinear Schrödinger equation, the nonlocal Kundu-Eckhaus model comprehends higher-order nonlinearities and nonlocal effects. The stability properties of dark, bright, periodic multiple, and singular soliton solutions to the nonlocal Kundu-Eckhaus (KE) equation within the Parity-Time (PT) symmetry model are studied here, whereas past studies have neglected the effects of nonlocal interactions and PT symmetry. is a well-established method, particularly for depicting wave dynamics in nonlinear optical and quantum systems. We attain exact traveling wave solutions evolved in hyperbolic and trigonometric forms, representing several classes of solitons by applying this method. The findings elucidate that the obtained soliton solutions are stable under small perturbations, highlighting their robustness and the persistence of localized waveforms in nonlocal circumstances. We plotted 3D, 2D, and contour graphs for simulating our findings using MATLAB. This research sheds light on the mysterious understanding of nonlocal nonlinear wave behaviors and offers analytical methodologies for future scrutiny in fields such as optical physics and complex nonlinear systems.

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).

Quantum computing is an extremely promising research combining theoretical and experimental quantum physics, mathematics, quantum information theory and computer science. Classical simulation of quantum computations will cover part of the gap between the theoretical mathematical formulation of quantum mechanics and the realization of quantum computers. One of the most important problems in “quantum computer science” is the development of new symbolic languages for quantum computing and the adaptation of existing symbolic languages for classical computing to quantum algorithms. The present paper is devoted to the adaptation of the Mathematica symbolic language to known quantum algorithms and corresponding simulation on the classical computer. Concretely we shall represent in the Mathematica symbolic language Simon’s algorithm, the Deutsch-Josza algorithm, Grover’s algorithm, Shor’s algorithm and quantum error-correcting codes. We shall see that the same framework can be used for all these algorithms. This framework will contain the characteristic property of the symbolic language representation of quantum computing and it will be a straightforward matter to include this framework in future algorithms.KeywordsDeutsch-Josza algorithmGrover’s algorithmQuantum computingQuantum error-correctingShor’s algorithmSimon’s algorithmSimulation of quantum algorithms

Continuum models for the spatial dynamics of growing cell populations have been widely used to investigate the mechanisms underpinning tissue development and tumour invasion. These models consist of nonlinear partial differential equations that describe the evolution of cellular densities in response to pressure gradients generated by population growth. Little prior work has explored the relation between such continuum models and related single-cell-based models. We present here a simple stochastic individual-based model for the spatial dynamics of multicellular systems whereby cells undergo pressure-driven movement and pressure-dependent proliferation. We show that nonlinear partial differential equations commonly used to model the spatial dynamics of growing cell populations can be formally derived from the branching random walk that underlies our discrete model. Moreover, we carry out a systematic comparison between the individual-based model and its continuum counterparts, both in the case of one single cell population and in the case of multiple cell populations with different biophysical properties. The outcomes of our comparative study demonstrate that the results of computational simulations of the individual-based model faithfully mirror the qualitative and quantitative properties of the solutions to the corresponding nonlinear partial differential equations. Ultimately, these results illustrate how the simple rules governing the dynamics of single cells in our individual-based model can lead to the emergence of complex spatial patterns of population growth observed in continuum models.

In a range of nonlinear fields, for example molecular biology, physics in plasma, quantum mechanics, elastic media, nonlinear optics, the surface of water waves, and others, many complicated nonlinear behaviors can be pronounced using nonlinear fractional partial differential equations. The nonlinear traveling waves, including nonlinear space-time fractional order differential equations such as the time fractional Sharma-Tasso-Olver (STO) and the space-time fractional Kowerteg-de Vries-Burgers (KdV-Burgers) equations, are analysis through a resourceful approach, namely the extended tanh-function by the definition of the conformable derivative. The stated equations are prevalently used in marine and coastal infrastructure, such as fluid flow, plasma engineering, fiber optics, fusion and fission phenomena, and so on. By renovating fractional order differential equations to ordinary differential equations, fractional order differential transformation simplifies the solution process. Several types of solutions, such as soliton types, bell types, kink types, and some other kinds of results, are developed and portrayed for definite values of free parameters that are plotted in terms of 3D, contour, and spherical shapes. The method is an algebraic approach that is both direct and efficient for dealing with nonlinear equations and yield a distinct category of solutions known as solitons. The suggested method is being used in this study to generate attractive, and effective results that are flexible, easy, and quicker to simulate using general mathematical techniques, namely Maple. It is worth declaring that all resultant solutions are verified for accuracy by exchanging them directly within the original equation through the software package and originating them accurately.MSC: 35C25, 35C07, 35C08, 35Q20, 76B25

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.

Low rank approximation in simulations of quantum algorithms

The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations.

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.

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

Digital quantum simulators provide a diversified tool for solving the evolution of quantum systems with complicated Hamiltonians and hold great potential for a wide range of applications. Although much attention is paid to the unitary evolution of closed quantum systems, dissipation and noise are vital in understanding the dynamics of practical quantum systems. In this work, we experimentally demonstrate a digital simulation of an open quantum system in a controllable Markovian environment with the assistance of a single ancillary qubit. By Trotterizing the quantum Liouvillians, the continuous evolution of an open quantum system is effectively realized, and its application in error mitigation is demonstrated by adjusting the simulated noise intensities. High-order Trotter for open quantum dynamics is also experimentally investigated and shows higher accuracy. Our results represent a significant step toward hardware-efficient simulation of open quantum systems and error mitigation in quantum algorithms in noisy intermediate-scale quantum systems.

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.