Two-dimensional Equations Research Articles

Approximation of Two-Dimensional Time-Fractional Navier-Stokes Equations involving Atangana-Baleanu Derivative

Approximation of Two-Dimensional Time-Fractional Navier-Stokes Equations involving Atangana-Baleanu Derivative

Numerical Reconstruction of Time-Dependent Boundary Conditions to 2D Heat Equation on Disjoint Rectangles at Integral Observations

In this paper, two-dimensional (2D) heat equations on disjoint rectangles are considered. The solutions are connected by interface Robin’s-type internal conditions. The problem has external Dirichlet boundary conditions that, in the forward (direct) formulation, are given functions. In the inverse problem formulation, the Dirichlet conditions are unknown functions, and the aim is to be reconstructed upon integral observations. Well-posedness both for direct and inverse problems is established. Using the given 2D integrals of the unknown solution on each of the domains and the specific interface boundary conditions, we reduce the 2D inverse problem to a forward heat 1D one. The resulting 1D problem is solved using the explicit Saul’yev finite difference method. Numerical test examples are discussed to illustrate the efficiency of the approach.

SOLITON SOLUTIONS FOR THE TWO-DIMENSIONAL LOCAL FRACTIONAL BOUSSINESQ EQUATION

In this work we study the two-dimensional local fractional Boussinesq equation. Based on the basic definitions and properties of the local fractional derivatives and bilinear form, we studied the soliton solutions of non-differentiable type with the generalized functions defined on Cantor sets by using bilinear method. Meanwhile, we discuss the result when fractal dimension is 1, and compare it with the result when fractal dimension is [Formula: see text].

On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics

In recent years, various complex systems and real-world phenomena have been shown to include memory and hereditary properties that change with respect to time, space, or other variables. Consequently, fractional partial differential equations containing variable-order fractional operators have been extensively resorted for modeling such phenomena accurately. In this paper, we consider the two-dimensional fractional cable equation with the Caputo variable-order fractional derivative in the time direction, which is preferable for describing neuronal dynamics in biological systems. A point-wise scheme, namely, the Crank–Nicolson finite difference method, along with a group-wise scheme referred to as the explicit decoupled group method are proposed to solve the problem under consideration. The stability and convergence analyses of the numerical schemes are provided with complete details. To demonstrate the validity of the proposed methods, numerical simulations with results represented in tabular and graphical forms are given. A quantitative analysis based on the CPU timing, iteration counting, and maximum absolute error indicates that the explicit decoupled group method is more efficient than the Crank–Nicolson finite difference scheme for solving the variable-order fractional equation.

Gradient preserving Operator Inference: Data-driven reduced-order models for equations with gradient structure

Hamiltonian Operator Inference has been introduced in Sharma et al. (2022) to learn structure-preserving reduced-order models (ROMs) for Hamiltonian systems. This approach constructs a low-dimensional model using only data and knowledge of the Hamiltonian function. Such ROMs can keep the intrinsic structure of the system, allowing them to capture the physics described by the governing equations. In this work, we extend this approach to more general systems that are either conservative or dissipative in energy, and which possess a gradient structure. We derive the optimization problems for inferring structure-preserving ROMs that preserve the gradient structure. We further derive an a priori error estimate for the reduced-order approximation. To test the algorithms, we consider semi-discretized partial differential equations with gradient structure, such as the parameterized wave and Korteweg–de-Vries equations, and equations of three-dimensional linear elasticity in the conservative case and the one- and two-dimensional Allen-Cahn equations in the dissipative case. The numerical results illustrate the accuracy, structure-preservation properties, and predictive capabilities of the gradient-preserving Operator Inference ROMs.

Neural network design for non-Newtonian Fe3O4–blood nanofluid flow modulated by electroosmosis and peristalsis

In this study, we present a novel approach that utilizes the Levenberg–Marquardt algorithm (LMA) based on artificial neural networks (ANNs) to evaluate the flow characteristics of a thermally evolved blood-based nanofluid in the presence of peristalsis and electroosmosis. The Casson fluid model is employed to govern the non-Newtonian characteristics observed in the flow of blood. In addition, the thermal properties of the nanofluidic medium in contact with platelet magnetite nanomaterials are also studied in detail. Further, the effects of thermal radiation, thermal buoyancy force, magnetic field and Joule heating are also given due consideration. The mathematically formulated two-dimensional equations describing the flow of Casson liquid are brought into their dimensionless form under the lubrication theory. A dataset for the proposed ANN models is generated to explore various scenarios of the fluidic model by varying the pertinent parameters using NDSolve in Mathematica. The computational approach utilizing LMA is deployed across three distinct phases of performance assessment, distributing the data into training, testing and validation sets at the proportions of 80%, 10% and 10%, respectively. This implementation involves the utilization of 10 hidden neurons. The utilization of regression analysis for testing, mean-squared error calculation, error histograms and correlation assessment in numerical replications of the ANNs is also examined to verify their capability, accuracy, validity and effectiveness. This study is crucial for understanding the peristaltic blood transportation in small blood vessels of living organisms.

Analysis of a New NFV Scheme Preserving DMP for Two-Dimensional Sub-diffusion Equation on Distorted Meshes

Analysis of a New NFV Scheme Preserving DMP for Two-Dimensional Sub-diffusion Equation on Distorted Meshes

A numerical method for two-dimensional transient nonlinear convection-diffusion equations

In this research, we have developed the half-boundary method (HBM) for nonlinear convection-diffusion equations (CDEs), which hold significant importance in nuclear power engineering. The HBM employs a variable relationship between the nodes within the computational domain and the nodes located on half of the boundaries. This approach offers notable benefits, includingthe reduction ofthe maximum matrix order and the optimization the maximum memory storage for calculations. Moreover, the HBM is an efficient and streamlined approach to directly handle Neumann boundary conditions, thanks to the utilization of mixed variables. We primarily investigate the memory usage and accuracy of the proposed algorithm in the unsteady-state CDEs, in context of material nonlinear CDEs, the Burgers’ equation and the system of Burgers' equations. The numerical results obtained demonstrate the method’s potential in simulating flow and heat transfer phenomena, particularly in situations where convection is dominant.

Generalized Finite Integration Method with Laplace transform for European option pricing under Black–Scholes and Heston models

In this paper, we combine a recently developed Generalized Finite Integration Method (GFIM) with Laplace transform technique for pricing options under the Black Scholes model and Heston model respectively. Instead of using traditional time-stepping process, we first perform Laplace transform on the governing equation and boundary conditions to remove the temporal derivatives. The Generalized Finite Integration Method is then exploited to handle the spatial differential operators in the transformed space. From numerical Laplace inversion algorithm, we restore the required time-dependent option price. For verification, several option pricing models governed by one-dimensional Black–Scholes equation and two-dimensional extended Heston equation are constructed to demonstrate the efficiency and feasibility of the proposed approach.

Coherence measurements of polaritons in thermal equilibrium reveal a power law for two-dimensional condensates.

We have created a spatially homogeneous polariton condensate in thermal equilibrium, up to very high condensate fraction. Under these conditions, we have measured the coherence as a function of momentum and determined the total coherent fraction of this boson system from very low density up to density well above the condensation transition. These measurements reveal a consistent power law for the coherent fraction as a function of the total density over nearly three orders of its magnitude. The same power law is seen in numerical simulations solving the two-dimensional Gross-Pitaevskii equation for the equilibrium coherence.

LAPLACE DECOMPOSITION METHOD FOR SOLVING THE TWO-DIMENSIONAL DIFFUSION PROBLEM IN FRACTAL HEAT TRANSFER

In this paper, the Local Fractional Laplace Decomposition Method (LFLDM) is used for solving a type of Two-Dimensional Fractional Diffusion Equation (TDFDE). In this method, first we apply the Laplace transform and its inverse to the main equation, and then the Adomian decomposition is used to obtain approximate/analytical solution. The accuracy and applicability of the LFLDM is shown through two examples. The LFLDM results are in good agreement with the exact solution of the problems.

A MEEVC discretization for two-dimensional incompressible Navier-Stokes equations with general boundary conditions

In this work, we present a mass, energy, enstrophy and vorticity conserving (MEEVC) mixed finite element discretization for two-dimensional incompressible Navier-Stokes equations as an alternative to the original MEEVC scheme proposed in [A. Palha and M. Gerritsma, J. Comput. Phys., 2017]. The present method can incorporate general boundary conditions. Conservation properties are proven. Supportive numerical experiments with both exact and inexact quadrature are provided.

Compact difference scheme for the two-dimensional semilinear wave equation

In this article, we investigate the use of a high-order compact finite difference method for solving a two-dimensional damped semilinear wave equation. We use a new iterative methods that employs compact finite difference operators to approximate the second-order spatial derivatives and achieve fourth-order convergence. We establish the stability and convergence of the compact finite difference scheme in the sense of the discrete H1-norm. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed scheme.

High order well-balanced asymptotic preserving IMEX RKDG schemes for the two-dimensional nonlinear shallow water equations

In this paper, a high order well-balanced asymptotic preserving scheme is presented for the two-dimensional nonlinear shallow water equations over variable bottom topography in all Froude number regimes. To obtain the well-balanced property, the system is first reformulated as a new form by introducing an auxiliary parameter. The flux is then split into a linear stiff part to be treated implicitly and a nonlinear non-stiff part to be treated explicitly, and the source term is treated explicitly. An implicit-explicit Runge-Kutta discontinuous Galerkin scheme is designed for solving the equations. The proposed scheme can be proved to be well-balanced, asymptotic preserving and asymptotically accurate. Finally, several numerical tests are carried out to validate the performance of our proposed scheme.

Angular and modal equipartitioning of elastic waves in scattering media: An illustration based on energy transport.

We illustrate the angular and modal equipartitioning of elastic waves in scattering media using two-dimensional elastic radiative transfer equations . To solve these equations, we decompose the P and S specific intensities into direct and scattered components. We handle the direct component analytically, and derive integral equations for the scattered components of the P and S specific intensities. We construct a time-stepping algorithm with which we evolve the scattered components of the specific intensities numerically in time. We handle the advection of P and S energy analytically at the computational grid points and use numerical interpolation to deal with advection terms which do not lie on the grid points. We test this algorithm for a pure P source and a double couple, which radiates both P and S energy. We compare our numerical solutions against known approximations and find good agreement. We use this algorithm to illustrate the local behavior of equipartitioning over wave modes and angular directions. We find that both types of equipartitioning are a function of space and time, depending on the extent of scattering. This local behavior must be taken into account when studying diffusion and equipartitioning of elastic waves.

Solving the general form of the fractional Black–Scholes with two assets through Reconstruction Variational Iteration Method

Solving the general form of the fractional Black–Scholes with two assets through Reconstruction Variational Iteration Method

Global well-posedness for 2D non-resistive MHD equations in half-space

This paper focuses on the initial boundary value problem of two-dimensional non-resistive MHD equations in a half space. We prove that the non-resistive MHD equations have a unique global strong solution around the equilibrium state (0, e1) for Dirichlet boundary condition of velocity and modified Neumann boundary condition of magnetic.

Modelling and Impact Analysis of Combined Dam Failure of Small Tandem Reservoir Groups in Hilly Areas

Numerous small reservoirs in hilly areas play a significant role in flood control and profit-making in the basin, but there is also a risk of dam failure when encountering special circumstances such as excessive flooding, which may cause large losses downstream. In this paper, we take the small reservoirs in Shandong Province as the research object, analyze the hydraulic topology of three types of tandem reservoirs, generalize the shape of the breach by using the transient lateral local one-break to the end, calculate the dam-break flooding process by using the quadratic parabolic method, and simulate the dam-break by using the MIKE21 model based on the principle of the two-dimensional incompressible Reynolds mean stress equation for the incompressible fluid, and analyze the peak flow under various combinations of dam-break flow by setting various combinations of the number of dam-breaks and the timing of dam-break simulation. By setting various combinations of dam failure numbers and timing, we analyzed the changing rules of peak dam failure flow, inundation area and other elements under various combinations, and sought the most unfavorable combinations and parameters such as the timing of dam failure, and the results of the analyses can provide technical support for the assessment of the risk of dam failure in the tandem reservoirs in the hilly area and the risk correspondence.

A network model for can-annular combustors using a multidimensional transfer matrix based on the mode matching method

Heavy-duty land-based gas turbines with can-annular combustors typically exhibit a set of thermoacoustic modes with similar frequencies, due to their rotational symmetry and the weak coupling between neighboring cans. The Bloch boundary condition is often used to simplify these combustors. Because the widely used (quasi-)one-dimensional model cannot naturally consider high-order modal components in the three-dimensional acoustic field, it is valid only when the frequency is sufficiently low and the length of the can-to-can interaction gap is sufficiently short. In this paper, we present a theoretical model using the mode-matching method to consider high-order modal components. A reflection coefficient matrix is used to accurately describe the acoustic characteristics of the gap area. This reflection coefficient matrix is found to depend on the nondimensional cross-sectional area ratio, the Helmholtz number based on the length of the transition duct, and the Bloch number. To facilitate a full low-order network modelling of the high-frequency thermoacoustic modes in can-annular combustors, a flame transfer matrix is also used to describe the modal coupling effect across the flame. Based on this model, the thermoacoustic problem of a four-flame can-annular combustor is studied as an example. It is shown that the frequency and growth rate of the thermoacoustic modes can be predicted accurately using the reflection coefficient matrix and flame transfer matrix. The effects of high-order modal components across the flame and in the gap areas are studied separately. The results show that the model containing both of the two coupling effects has the widest range of validity. These couplings can be ignored if only low-frequency plane acoustic waves are considered but need to be carefully considered for higher frequency modes. Moreover, when multiple flames are considered, asymmetric flame distributions are found to more easily lead to modal coupling compared to symmetric flame distributions. All results are validated by comparing with the numerical solutions of the full two-dimensional Helmholtz equation using the finite element method.

Suppression of the vortex-induced vibration of the cylinder by the baffle plates at different submerged depths

In this paper, the suppression effect of baffle on VIV of cylinder at different depths is studied. According to two-dimensional transient model and RANS equation, VIV responses of a cylinder with two baffle plates at a fixed distance and a fixed Angle at different depths are numerically simulated. The submersion depth radio h* is mainly 1.2, 1.5, 2.5 and 5. The experimental data and predicting results are compared to display the accuracy of numerical method. When h* is 2.5, the baffle plates have the best suppression effect on VIV. Compared to the bare cylinder, the amplitude is decreased by 69%. Meanwhile, the vibration of the cylinder at different depths shows instability, and the motion response has great jitter. The lower baffle plate is subject to a unidirectional force when h* = 1.5 and 1.2.