Large deformations of cord-reinforced multilayered shells

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.

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

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

In this paper, we present a modified and new version of spectral method which is based on minimization of obtained residual term in norm, where is a weight function with respect to Jacobi polynomials. Using this approach is efficient and effective rather than Tau and collocation methods. It reduces the nonlinear ordinary differential equations to the nonlinear programming problems which is an easy problem to solve. Hence, easy implementation of the method is the importance of our approach and some numerical test experiments show the accuracy and efficiency of this method. Key words: A new spectral method, nonlinear ordinary differential equations.

• Deriving equilibrium equations for cylindrical shells with viscoelastic functionally graded properties (VFG). • Viscoelastic properties are modeled using Zener three-parameter models for shear behavior, while assuming elastic behavior for bulk properties. • The loadings consist of axial and lateral pressures, either long or short duration, and continuous or piecewise distribution. The presented solution does not depend on the duration of loading. • No need for new modeling for sensitivity analysis and optimization. This manuscript presents an analytical method for determining the deformations of cylindrical shells with viscoelastic functionally graded (VFG) properties under axial and transverse pressures. The displacement field follows first-order shear deformation theory (FSDT), with a standard linear solid (SLS) model representing viscoelastic behavior. Property variations through the shell thickness are modeled using a continuous power function. Kinematics adhere to von Kármán relations, and Hooke's law serves as the constitutive equation. Equilibrium equations are derived via the virtual work principle, resulting in a system of coupled nonlinear partial differential equations. The Galerkin method is applied to remove the spatial part of the equations. The remaining ordinary differential equations are solved analytically using the multiple-scale method. The results indicate the creep phenomenon of the shell under constant loads. Sensitivity analysis shows that the power law index as well as the viscoelastic properties can affect the deformation rate. Additionally, for piecewise loading, recovery behavior is observed in the response. The analytical procedure presented can be an effective formulation for optimization through trial and error. The results are compared with the finite element (FE) method for special cases.

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.

Resonance method commonly used for Painleve classification of ordinary differential equations often detects negative resonances, other than a trivial one of –1 (which should always be present). However, it is asserted by some researchers that the nature of these negative resonances is not fully understood. The problem arises as how to use negative nontrivial resonances to obtain information about the analytical properties of solutions of non-linear differential equations, in particular, to construct rational solutions. In the present paper, the method for constructing rational solutions f certain classes of autonomous non-linear ordinary differential equations is presented.

Micro-cantilever beams are interested structures in MEMS because of their fabrication is very easy and its versatility. The importance of micro-cantilevers beam in MEMS has driven various investigations like static and dynamic performances under different loading such as potential fields. In this research the non-linear differential equation which models dynamics of a micro-cantilever beams vibration subjected to electrostatic field has been studied. The model which has one degree of freedom is used to calculate the pull-in voltage. This model adopted based on different method of calculating stiffness of micro-cantilever beam. The nonlinear ordinary differential equation which used to model the dynamics of the cantilever subjected to electric field close to snap on is highly stiff. Investigation on solving of nonlinear stiff ordinary equation showed that only Lsode algorithm yield to correct solution to the problem. Lsode is equipped with a robust adaptive time step selection mechanism that enables solutions to very stiff problems, as the one under discussion. The best match in the resonant frequency for equivalent stiffness based on four different models was considered. The stiffness model suitable for the best match in deflection is proved to be different from the model that yields. Pull-in voltage under electric field was studied. Pull-in voltage has been investigated from the analytical and numerical perspective. A complete parametric study of structural damping effect on large deflection of micro-cantilever beam was studied was done numerically in this work. Different kind of impulse voltages were considered and effect of them on pulling voltage numerically was studied. A cumbersome mathematical method, Lie symmetry, was used to drive a closed from of time response to step voltage for undamped system and pull in voltage of such system was calculated. Finally, a closed form driven from the nonlinear ODE for calculating pulling voltage was presented.

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.

Asymptotic to polynomials solutions for nonlinear differential equations

This study is focused to elaborate on the effect of heat source/sink on the flow of non-Newtonian Burger nanofluid toward the stretching sheet and cylinder. The current flow analysis is designed in the form of higher order nonlinear partial differential equations along with convective heat and zero mass flux conditions. Suitable similarity transformations are used for the conversion of higher order nonlinear partial differential equations into the nonlinear ordinary differential equations. For the computation of graphical and tabular results, the most powerful analytical technique, known as the homotopy analysis method, is applied to the resulting higher order nonlinear ordinary differential equations. The consequence of distinct flow parameters on the Burger nanofluid velocity, temperature, and concentration profiles are determined and debated in a graphical form. The key outcomes of this study are that the Burger nanofluid parameter and Deborah number have reduced the velocity of the Burger nanofluid for both the stretching sheet and cylinder. Also, it is attained that the Burger nanofluid temperature is elevated with the intensifying of thermal Biot number for both stretching sheet and cylinder. The Burger nanofluid concentration becomes higher with the escalating values of Brownian motion parameter and Lewis number for both stretching sheet and cylinder. The Nusselt number of the Burger nanofluid upsurges due to the increment of thermal Biot number for both stretching sheet and cylinder. Also, the different industrial and engineering applications of this study were obtained. The presented model can be used for a variety of industrial and engineering applications such as biotechnology, electrical engineering, cooling of devices, nuclear reactors, mechanical engineering, pharmaceutical science, bioscience, medicine, cancer treatment, industrial-grid engines, automobiles, and many others.

A nonlinear ordinary differential equation is presented. This equation is shown to occur as a particular integral of Einstein’s equations of gravitation, and may be used to generate a class of solutions which contains the Kerr, and the Tomimatsu–Sato metrics as particular cases.

The article proposes an adaptive algorithm that generates all object signals, including those for which measurements are not performed due to the difficulties associated with on-line measurements. The algorithm is modeled on the idea of the Kalman filter using its equation, however, the selection of gains is optimized in a different way, i.e. the constant values depend on the adopted ranges of adaptation errors. Moreover, the knowledge of the statistics of all noise signals is not imposed and there is no linearity constraint. This approach allowed to reduce the complexity of calculations. This algorithm can be used in real-time systems to generate signals of objects described by non-linear differential equations and it is universal, which allows it to be used for various objects. In the conducted research, on the example of a biochemically contaminated river, only easily measurable signals were used to generated the object signals, and in addition, in the case of absence some measurements, the functioning of the algorithm did not destabilize.

This tutorial is about new system analysis techniques that were developed in the past few years based on the sum of squares decomposition. We present stability and robust stability analysis tools for different classes of systems: systems described by nonlinear ordinary differential equations or differential algebraic equations, hybrid systems with nonlinear subsystems and/or nonlinear switching surfaces, and time-delay systems described by nonlinear functional differential equations. We also discuss how different analysis questions such as model validation and safety verification can be answered for uncertain nonlinear and hybrid systems.