An Iterative Finite Difference Method for Solving Nonlinear Gordon-Type Problems

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.

On the combined Shooting-Pseudo-Arclength method for finding frequency response of nonlinear fractional-order differential equations

A frequency‐agile microstrip patch antenna printed on a biased nematic liquid crystal (N‐LC) substrate is accurately modelled. The dielectric properties of the N‐LC are controlled through the strength of the bias voltage. The model is based on the solution of Poisson's equation coupled to a nonlinear partial differential equation (PDE) describing the orientation of the directors in a non‐homogeneous electric field. The coupled problem was solved using an iterative finite difference (FD) scheme. The obtained numerical results are compared and verified against measurements already published in the literature.

In the current paper, a simplified model of Tower Cranes has been presented in order to investigate and analyze the nonlinear differential equation governing on the presented system in three different cases by Algebraic Method (AGM). Comparisons have been made between AGM and Numerical Solution, and these results have been indicated that this approach is very efficient and easy so it can be applied for other nonlinear equations. It is citable that there are some valuable advantages in this way of solving differential equations and also the answer of various sets of complicated differential equations can be achieved in this manner which in the other methods, so far, they have not had acceptable solutions. The simplification of the solution procedure in Algebraic Method and its application for solving a wide variety of differential equations not only in Vibrations but also in different fields of study such as fluid mechanics, chemical engineering, etc. make AGM be a powerful and useful role model for researchers in order to solve complicated nonlinear differential equations.

A possibility for predicting the time-dependent pattern of sickness of human beings, i.e., biological catastrophe, has been shown by proposal of a nonlinear ordinary differential equation, describing temporal features of six macroscopic molecular groups chemically interacting in living beings, (Naitoh in Jpn J Ind Appl Math 28:15–26, 2011; Naitoh and Inoue in J Artif Life Robot 18:127–132, 2013). Logically, we also find that, six is minimum and essential as the number of macroscopic molecular groups for describing living systems. Then, along with the number theory applied for the differential equation, we derive critical mathematical conditions for predicting the premonition just before sickness (discrepancy from a healthy condition), which agree with an important knowledge revealed by the linear analysis proposed by Chen (Dynamical network biomarkers for identifying early-warning signals of complex diseases, Beppu, Oita Japan, 2015). In the present report, we first show that computational several time-histories of sickness obtained by solving the nonlinear differential equation with various parameters describing polymorphism agree well with actual time-dependent patterns of sickness for human beings, which is further evidence of usefulness of the nonlinear differential equation and its critical mathematical conditions. Thus, to examine the boundary between biological and abiological chemical reaction systems, which is related to the origin of living systems, we next check whether or not the nonlinear equation can also predict such abiological catastrophic phenomenon as misfire in artifacts including mechanical combustion engines.

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

A high-order accurate flow solution method for quadrilateral elements is developed and tested. The method employs the flux reconstruction (FR) approach proposed by Huynh. Ideal orders of accuracy are achieved for several problems on structured-based grids while some deterioration is observed on purely unstructured grids. Cares for curved wall boundary are also shown. The localized artificial diffusivity (LAD) method is incorporated to capture discontinuities that is crucial for compressible flow simulations. The FR+LAD works very well for 1D test cases but multi-dimensional shock problems requires further modifications to confine the artificial diffusivity to the shock location. Hybrid scheme of the FR and finite difference method taking advantage of 1D-feature alleviates the instability. I. Introduction Expanding demands for the computational fluid dynamics (CFD) requires more reliable numerical schemes. Specifically, turbulent flow simulations, acoustic simulation, combustion simulations and their combined problems. The key words in such simulations may be high accuracy and high resolution. In structured grid CFD, a compact finite difference scheme satisfies both requirements at higher level than existing other methods. If we put more emphasis on easier handling of complex geometries and flow-data dependent grid generation (grid adaptation), then unstructured grid method should also be explored and it should depart from 2nd-order scheme with limiter. Recent efforts in this area are development of the discontinuous Galerkin (DG) method, 1,2 the spectral finite volume (SV) method 3 and the staggered-grid multi-domain method spectral finite difference (SG) method 4,5 and its evolutional method called a spectral difference method. 6 The common and maybe the only way for those unstructured methods is to add degrees of freedom in each cell to achieve high order since the wide stencil of the original grid is not possible due to unstructured nature. The ENO-type method that uses original degrees of freedom in the grids and extends adaptive stencils to the flow solutions is also explored but the difficulty of extending stencils due to undirectional nature of unstructured grids and the unsuitability for parallel computing prohibits wide applications. A flux reconstruction (FR) scheme 7,8 studied in the paper is the new one among the former group. It uses differential form of conservation laws like the SD method but it is more general and includes equivalent counter part of the schemes in the group. In this paper, the performance of the FR is tested by a newly developed Navier-Stokes solver. Realistic flow applications of the FR in the Navier-Stokes equations are seldom seen since it is relatively new scheme. The Localized artificial diffusivity (LAD) 9–13 is incorporated for the shock capturing in the FR. The last item is motivated by the challenge and success of combining the SD and LAD in the recent study. 11,12 Wang 14 developed the lifting collocation penalty (LCP) method to extend the FR to triangular or tetrahedral grids and applied the LCP-FR combined method for hybrid unstructured grids. In the present study, only quadrilateral cells are treated to directly apply the LAD method that was developed on structured grids. The shock capturing by the FR+LAD high order scheme is tried.

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

In the paper, two new analytic methods using the decomposition and linearization technique on nonlinear differential/integral equations are developed, namely, the decomposition–linearization–sequential method (DLSM) and the linearized homotopy perturbation method (LHPM). The DLSM is realized by an integrating factor and the integral of certain function obtained at the previous step for obtaining a sequential analytic solution of nonlinear differential equation, which provides quite accurate analytic solution. Some first- and second-order nonlinear differential equations display the fast convergence and accuracy of the DLSM. An analytic approximation for the Volterra differential–integral equation model of the population growth of a species is obtained by using the LHPM. In addition, the LHPM is also applied to the first-, second-, and third-order nonlinear ordinary differential equations. To reduce the cost of computation of He’s homotopy perturbation method and enhance the accuracy for solving cubically nonlinear jerk equations, the LHPM is implemented by invoking a linearization technique in advance is developed. A generalization of the LHPM to the nth-order nonlinear differential equation is involved, which can greatly simplify the work to find an analytic solution by solving a set of second-order linear differential equations. A remarkable feature of those new analytic methods is that just a few steps and lower-order approximations are sufficient for producing reasonably accurate analytic solutions. For all examples, the second-order analytic solution x2(t) is found to be a good approximation of the real solution. The accuracy of the obtained approximate solutions are identified by the fourth-order Runge–Kutta method. The major objection is to unify the analytic solution methods of different nonlinear differential equations by simply solving a set of first-order or second-order linear differential equations. It is clear that the new technique considerably saves computational costs and converges faster than other analytical solution techniques existing in the literature, including the Picard iteration method. Moreover, the accuracy of the obtained analytic solution is raised.

Nonlinear differential equations arise widely in applied mathematics, physics, and engineering, yet many conventional analytical and numerical methods remain limited in their ability to handle strong nonlinearities efficiently and accurately. This paper presents a novel computational framework based on the Modified Laplace–Adomian Polynomial Method (LAPM) for solving nonlinear differential equations. The proposed method integrates the Laplace transform with an enhanced form of the Adomian Decomposition Method, enabling complex nonlinear terms to be decomposed into rapidly convergent Adomian polynomials. This integration simplifies the solution procedure, reduces computational complexity, and preserves high accuracy. The performance of LAPM was validated using several benchmark nonlinear and linear differential equations, and the results demonstrated superior convergence speed, precision, and stability compared with traditional methods. The study concludes that the Modified Laplace–Adomian Polynomial Method is a reliable and efficient approach for solving a broad class of nonlinear differential equations. This work contributes to the advancement of computational methods by offering a robust alternative for the analysis of differential equation models encountered in mathematics, physics, and engineering.

Effect of mold heat transfer on the curing reaction of epoxy molding compound during transfer molding

: 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.

Solving time-fractional nonlinear equations, such as the Newell-Whitehead-Segel (N-W-S) and Burger's equations, is inherently complex due to the intricacies of fractional calculus and the limitations of current numerical and analytical techniques. This study introduces the Tarig Projected Differential Transform Method (TPDTM), a hybrid approach that offers a novel solution to time-fractional linear and nonlinear partial differential equations (PDEs) without the need for linearization, perturbation, or variable discretization. TPDTM stands out as a simple yet powerful method, offering remarkable accuracy, computational efficiency, and ease of implementation. In comparative analysis with the Finite Difference Method (FDM) and Laplace Adomian Decomposition Method (LADM), TPDTM demonstrates superior performance, particularly in its handling of Adomian polynomials, all while preserving stability and precision. With its rapid convergence and versatility, TPDTM proves to be a robust tool for tackling complex fractional PDEs, making it highly valuable for applied mathematics and physics. Looking forward, expanding TPDTM's application to coupled fractional systems and multi-scale problems will not only enhance its theoretical depth but also open exciting new possibilities for breakthroughs in engineering, physics, and computational modeling.

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