Semi-analytical Numerical Method Research Articles

Nonlinear stochastic wave equations in 1D with fractional Laplacian, power-law nonlinearity and additive [formula omitted]-regular noise

A qualitative study of nonlinear, 1D stochastic fractional wave equations with dissipative nonlinearities of power-law form is conducted on (t,x)∈[0,+∞)×Dutt+σ2(−uxx)α−a1u+a2‖u‖L2(D)ρu−κut=b0∂W0∂t on (t,x)∈[0,+∞)×D with D=[0,L], where positive fractional α-powers of Laplace operator are allowed, perturbed by additive space–time random noise W0 with fairly general covariance operator Q with finite trace(Q)<+∞ (i.e. Q-regularity). The Q-regular space–time noise W0, which is white in time and generally spatially correlated, is supposed to have a Fourier expansion along the eigenfunctions of the Laplace operator under nonrandom, Dirichlet- and Neumann-type homogeneous boundary conditions. We focus on studying the generalized energy functional V incorporating the fractional diffusive energy part and the energy part which is due to the viscous damping force both in continuous and discrete time. In continuous time, the technique of Fourier expansions and appropriate truncation by finite-dimensional systems leads to the control on the generalized energy functional, and hence we may verify existence, uniqueness and boundedness of moments of Fourier series solutions u. The total mean energy E[E(t)] cannot grow more than linearly in time t (with or without damping). Moreover, under the absence of damping (i.e. κ=0), E[E(t)] is governed by a kind of trace formula in case of additive (state-independent), Q-regular space–time noise W0. For numerical computations and more adequate discretization, we suggest to take nonstandard, partial-implicit midpoint-type methods for the Fourier coefficients. These semi-analytical numerical methods (approximate Fourier series) possess the property of conserving the expected, total energy with random initial data under the absence of nonlinearities. Eventually, we estimate probabilities of large fluctuations of some interesting functionals.

Penyelesaian Sistem Persamaan Hukum Laju Reaksi dengan Metode Transformasi Differensial

This research is focused on solving the rate law equation by using the differential transformation method. The rate law equation describes the chemical reaction problem from the concentration of a reactant that produces a product. The differential transformation method is a semi-analytic numerical method that can provide approximate solutions in the form of a series because the method is obtained from the expansion of the Taylor series expansion. With the help of Maple software, a comparison of the solution plots of y_1 (t),y_2 (t) and y_3 (t), can be observed that the difference in computational results between the Runge-kutta method and the differential transformation depends on the order of k. The curve of the differential transformation method is getting closer to the curve of the Runge-Kutta method at a certain value of k for each y_1 (t),y_2 (t) and y_3 (t). The conclusion of this research is that the application of the differential transformation method has been successfully carried out in the case of a system of ordinary differential equations. For further research, the researcher suggests that the next research applies the method of differential transformation in cases and initial values that are more varied.

Physical modeling and geometric shape simulation for one-dimensional flexible objects with cylindrical surface constraints

This study develops forces equilibrium differential equations for the geometric modeling of 1D flexible objects with surface constraints. These second-order equations are an extension of the Cosserat elastic rod theory and include both bending and torsion. Variables were established for the centerline and attitude in the Cartesian coordinate system of the cross section. This paper specifically investigates the case of a 1D flexible object constrained by a cylindrical surface. To solve this problem, a novel hybrid semi-analytical numerical method is proposed. In this process, a Hamiltonian function and an initial integral operator are introduced in a cylindrical coordinate system. The analytical solution, decoupled in polar coordinates, is then derived. The improved finite difference method was then used to obtain three cylindrical coordinates, which ensured numerical stability and efficiency. The results of a geometric shape simulation with differing boundary conditions demonstrate that this proposed method is capable of real-time modeling. As such, this technique could be a promising new tool for use in graphics simulations of elongated structures, such as DNA molecules, drill pipes, and submarine cables.

Hybrid method for modelling anti-plane vibrations of layered waveguides with bonded composite joints

The use of mesh-based methods for modeling elongated composite structures with inhomogeneities leads to an increase in computational costs when discretizing the waveguide part of greater linear dimensions, while the semi-analytical numerical methods do not allow one to directly describe the structures with local inhomogeneities of an arbitrary shape. To compensate for the shortcomings of these two classes of numerical methods, we propose a hybrid scheme based on the spectral element method (SEM) and the semi-analytical finite element method (SAFEM) for studying the anti-plane vibrations of a composite structure in the frequency domain. Thus, for a waveguide, this scheme makes it possible to represent the solution via the SAFEM as a sum of modes or guided waves, and the adjacent regions are discretized using the SEM. The displacement and stress continuity conditions are imposed on the common boundary of two domains. To couple the solutions, we introduce an auxiliary function of displacement, which is approximated by applying the same basis functions as those used in SEM and SAFEM (Lagrange interpolation polynomials on the Gauss–Legendre–Lobatto nodal points). The unknown expansion coefficients of this function are determined by the Galerkin and collocation methods. It has been established that both methods provide the same accuracy. The results obtained by the hybrid approach employing the Galerkin and collocation methods are compared with the results calculated in the standard finite element software. It is shown that they are in good agreement. The presented hybrid approach can be straightforwardly extended to the case of in-plane motion, but it requires significant refinement for a three-dimensional case.

A Three-Coil Setup for Controlled Divergence in Magnetic Nozzle

Magnetic nozzles with their contactless feature have opened new grounds for long-distance space travels. For quicker mission completions along with improving system reliability, it becomes necessary to design robust systems that exhibit the ability to confine thrust. A semianalytical numerical method is proposed for rapid and assumption-less calculations of the magnetic field because of a thick coil with rectangular cross section. The method is extended for the proposal of a three-coil setup, where divergence angle of the magnetic field lines is readily controlled in-flight to provide the flexibility to modify the output magnetic thrust and the nozzle's efficiency.

The Method of Lines Analysis of TE Mode Propagation in Silica based Optical Directional Couplers

Optical directional couplers fabricated using planar light wave circuit (PLC) technology are versatile tools in integrated photonics devices. They have the advantages of small size, high consistency, ability for high volume production, and excellent possibility to be integrated with electronics circuits. Optical waveguide couplers are mainly utilized as power dividers, optical switches, and wavelength division multiplexers/de-multiplexers (WDM). A number of methods have been used to analyze directional couplers, such as coupled mode theory (CMT), the beam propagation method (BPM), the method of lines (MoL), finite-difference methods (FDM), and finite element methods (FEM). Among these numerical approaches, MoL is the simplest method to analyze mode propagation inside directional couplers because it has the advantages of very fast convergence and accurate solutions for one-dimensional structures. The objective of this study was to analyze the propagation of TE modes in optical directional couplers by using MoL. The parameters used, i.e. waveguide width, refractive index, and wavelength, were taken from the characteristics of silica-on-silicon directional couplers that were used in fabrication. MoL is considered a special finite-difference method, which discretizes a one- or two-dimensional wave equation in the transverse direction and uses an analytical solution for the propagation directions. Basically, MoL is a semi analytical numerical method with the advantages of numerical stability, computational efficiency, and calculation time reduction. Further, we explored the possibility of using directional couplers as optical switching devices.

Calculating the magnetic force of solenoid inductor wound by rectangular cross‐section wire

A new semi-analytical expression is presented based on two integrations for calculating the magnetic force of a solenoid inductor wound by rectangular cross-section wire. The authors simplified the solenoid inductor coil to a collection of circular coils, and the current density of the coils in the radial direction which was inversely proportional to its radius is considered. The obtained expressions can be implemented by adaptive Gaussian quadrature integration with MATLAB programming. In addition, they used the filament method and the finite element method to calculate the magnetic force on the solenoid inductor coil. The correctness of the semi-analytical expressions was verified by comparing the results of the semi-analytical numerical method with those of the filament method and the finite element method, respectively. The derived semi-analytical expressions of magnetic force allow a low computational time compared with the filament method and the finite element method to a specific accuracy. The calculation results show that: the solenoid inductor coil tends to compress in the axial direction, and the axial magnetic force on the outermost coils is the largest; the solenoid inductor coil tends to expand outward in the radial direction, and the radial magnetic force on the intermediate coil is the largest.

Analysis of the Rigorous Coupled Wave Approach for [formula omitted]-polarized light in gratings

We study the convergence properties of the two-dimensional Rigorous Coupled Wave Approach (RCWA) for p-polarized monochromatic incident light. The RCWA is a semi-analytical numerical method that is widely used to solve the boundary-value problem of scattering by a grating. The approach requires the expansion of all electromagnetic field phasors and the relative permittivity as Fourier series in the spatial variable along the direction of the periodicity of the grating. In the direction perpendicular to the grating periodicity, the domain is discretized into thin slices and the actual relative permittivity is replaced by an approximation. The approximate relative permittivity is chosen so that the solution of the Maxwell equations in each slice can be computed without further approximation. Thus, there is error due to the approximate relative permittivity as well as the truncation of the Fourier series. We show that the RCWA embodies a Galerkin scheme for a perturbed problem, and then we use tools from the Finite Element Method to show that the method converges with increasing number of retained Fourier modes and finer approximations of the relative permittivity. Numerical examples illustrate our analysis, and suggest further work.

Magneto-Hybrid Nanofluids Flow via Mixed Convection past a Radiative Circular Cylinder

The goal of the current analysis is to scrutinize the magneto-mixed convective flow of aqueous-based hybrid-nanofluid comprising Alumina and Copper nanoparticles across a horizontal circular cylinder with convective boundary condition. The energy equation is modelled by interpolating the non-linear radiation phenomenon with the assisting and opposing flows. The original equations describing the magneto-hybrid nanofluid motion and energy are converted into non-dimensional equations and solved numerically using a new hybrid linearization-Chebyshev spectral method (HLCSM). HLCSM is a high order spectral semi-analytical numerical method that results in an analytical solution in η-direction and thereby the solution is valid in overall the η-domain, not only at the grid points. The impacts of diverse parameters on the allied apportionment are inspected, and the fallouts are described graphically in the investigation. The physical quantities of interest containing the drag coefficient and the heat transfer rate are predestined versus fundamental parameters, and their outcomes are elucidated. It is witnessed that both drag coefficient and Nusselt number have greater magnitude for Cu-water followed by hybrid nanofluid and Al2O3-water. Moreover, the value of the drag coefficient declines versus the enlarged solid volume fraction. To emphasize the originality of the current analysis, the outcomes are compared with quoted works, and excellent accord is achieved in this consideration.

Semi analytical cum numerical analysis of two-dimensional elasticity problem of plate with eccentric hole in bipolar coordinate system

A semi analytical cum numerical method for analyzing 2D elasticity problem of plate with eccentric hole in bipolar coordinate systems is given. Complete formulation of the plate problem is derived in bipolar coordinate system. A novel numerical integration technique is adopted to obtain the numerical results. Few parameters are identified that adversely affects the stresses and displacements of isotropic and FG plate with eccentric hole. Verification of the mathematical models with numerical results such as finite element models (COMSOL Multiphysics) and wide variety of applicability of the developed models for hypothetical/real field problems is seen from the present work.

Seismogram Synthesis for Multilayered Heterogeneous Media with Irregular Interfaces by the Global Generalized Reflection–Transmission Matrices Method

ABSTRACTWe present a semianalytical numerical method to simulate SH-wave propagation in heterogeneous multiple layers with smooth property variations. Wave propagation in such piecewise heterogeneous media will dominate the reflection and transmission at interfaces, whereas the scattered waves by volume heterogeneities inside each geological layer are superimposed on the boundary waves to cause fluctuations in amplitude and phase. The incident, boundary-scattering, and volume-scattering waves are accurately superposed through the generalized Lippmann–Schwinger integral (GLSI) equation. The multilayered heterogeneous media with irregular interfaces are sliced into horizontal and heterogeneous slabs, with each discretized into a series of volume elements. The traditional generalized reflection–transmission matrices (GRTMs) method is extended to solve the GLSI equation for each heterogeneous slab in the frequency–wavenumber domain by incorporating the boundary–volume integral equation numerical techniques. The global GRTM is obtained by integrating all the slab’s GRTMs in terms of the boundary condition across adjacent slabs. The wavefield at an arbitrary depth can be computed in a recursive relation from a source term. The accuracy of the method is tested by comparing it with other full-wave solutions to the SH problem. To show the applicability of the method, we calculate synthetic seismograms to demonstrate the significant impact of volume heterogeneities on seismic responses in layered heterogeneous media.

Mathematical Modeling of the Transmission Dynamics of Syphilis Disease Using Differential Transformation Method

In this work, we developed a mathematical model for the transmission dynamics of the Syphilis disease under some assumptions made. The method of differential transformation is employed to compute an approximation to the solution of the non-linear systems of differential equations for the transmission dynamic of the disease model. The differential transformation method is a semi-analytic numerical method or technique, which depends on Taylor series and has application in many areas including Biomathematics. The disease-free equilibrium of the syphilis model is analyzed for local asymptotic stability and the associated epidemic basic reproduction number R₀ is less than unity. It is also known that the global dynamics of the disease are completely determined by the basic reproduction number. Sensitivity analysis is performed on the model’s parameters to investigate the most sensitive parameters in the dynamics of the disease, for control and eradication.

Sinusoidal Natural Convective Flow of Non-Newtonian Nanoliquid Over a Radiative Vertical Plate in a Saturated Porous Medium

The present study explores the free convective of a micropolar nanoliquid flow through a vertical radiative plate in a saturated porous medium. Surface temperature based on streamwise sinusoidal is addressed. The basic PDEs governing the problem are converted into a dimensionless PDE system with proper transformations. The obtained PDE system is solved by applying a new hybrid linearization spectral collocation method (HLSCM). HLSCM is a semi-analytical numerical high order method that results in analytical approximate solutions in $\eta $ -orientation, and so the solutions are defined overall the $\eta $ range, not only at grid points. Plots are diagramed to investigate the impacts pertinent emerging parameters on the velocity and temperature distributions. Along with those, the heat transfer rate and drag coefficient are exhibited and debated in detail. It is manifested that the increment in the amplitude of wall temperature and micropolar parameter produces a sufficient increment of the drag coefficient and Nusselt number. Moreover, it is also uncovered that magnifying in temperature ratio elucidates a significant rise in both the drag coefficient and heat transport through a porous medium.

New numerical iterative method for nonlinear wave equations

&lt;sec&gt;Nonlinear acoustics is an important branch of acoustics and has important applications in some areas, such as high-intensity focused ultrasound, ultrasonic suspension, acoustic cavitation, acoustic harmonic imaging, and parametric emission array. The solving of nonlinear equations in these fields is very important. &lt;/sec&gt;&lt;sec&gt;Regarding the solution of the wave equation of a nonlinear acoustic system, the methods used at this stage generally include complete numerical calculation method, strict analytical method, and perturbation method. 1) For the complete numerical calculation method, it covers the finite element method and the finite difference method. The physical meaning of the solution obtained by this kind of method is not clear, and it is difficult to reveal the physical nature of nonlinear event. And in many cases it will lead to the numerical divergence problems, and it is not suitable for all nonlinear problems. 2) For the strict analytical method, it can only deal with nonlinear acoustic problems of very few systems, such as the propagation of nonlinear acoustic waves in an ideal fluid. 3) For perturbation method, its advantage is that the method is simple and the physical meaning of the solution is clear, but it is only suitable for dealing with nonlinear effects at low sound intensity. And it takes into consideration only the effect of low-order harmonics on higher-order harmonics, with ignoring its reaction, so it does not satisfy the law of conservation of energy.&lt;/sec&gt;&lt;sec&gt;In this paper, we propose a new, semi-analytical numerical iterative method of solving nonlinear wave equations. It is a form of expanding the sound field into a Fourier series in the frequency domain, realizing the separation of time variables from space coordinates. Then, according to the specific requirements for the calculation accuracy, the high frequency harmonics are cut off to solve the equation. Compared with the results from the complete numerical methods (such as finite element method and finite difference method), the solution from this iterative method has a very clear physical meaning. That is, its solution is a combination of harmonics of all orders. Compared with the perturbation method, it can consider the interaction of various harmonics and can satisfy the law of conservation of energy (provided that the system has no dissipation). It is used to study the propagation properties of nonlinear acoustic waves in liquids. The results show that the iterative method is also applicable in the range of sound intensity where the perturbation method is applicable. In a wide range of sound intensity where the perturbation method is unapplicable, the iterative method is still applicable and satisfies the law of conservation of energy (provided that the system has no dissipation). It is unapplicable only if the sound intensity is extremely loud and strong. And when more high-order harmonics are involved, the calculation time by using the numerical method proposed in this paper does not increase sharply.&lt;/sec&gt;

The Impact of Sinusoidal Surface Temperature on the Natural Convective Flow of a Ferrofluid along a Vertical Plate

The spotlight of this investigation is primarily the effectiveness of the magnetic field on the natural convective for a Fe3O4 ferrofluid flow over a vertical radiate plate using streamwise sinusoidal variation in surface temperature. The energy equation is reduplicated by interpolating the non-linear radiation effectiveness. The original equations describing the ferrofluid motion and energy are converted into non-dimensional equations and solved numerically using a new hybrid linearization-differential quadrature method (HLDQM). HLDQM is a high order semi-analytical numerical method that results in analytical solutions in η -direction, and so the solutions are valid overall in the η domain, not only at grid points. The dimensionless velocity and temperature curves are elaborated. Furthermore, the engineering curiosity of the drag coefficient and local Nusselt number are debated and sketched in view of various emerging parameters. The analyzed numerical results display that applying the magnetic field to the ferroliquid generates a dragging force that diminishes the ferrofluid velocity, whereas it is found to boost the temperature curves. Furthermore, the drag coefficient sufficiently minifies, while an evolution in the heat transfer rate occurs as nanoparticle volume fraction builds. Additionally, the augmentation in temperature ratio parameter signifies a considerable growth in the drag coefficient and Nusselt number. The current theoretical investigation may be beneficial in manufacturing processes, development of transport of energy, and heat resources.

The static analysis of arch dam-foundation system by using scaled boundary finite element method

The scaled boundary finite element (SBFEM) as a semi-analytical numerical method has been used in many fields. SBFEM has the following advantages: reducing the spatial dimension by one, rigorously modelling the unbounded domain and involving no fundamental solution. The computational model based on SBFEM will be conducive to the reliability and efficiency of safety evaluation of arch dam. However, owing to the requirement of scaling inherently, the SBFEM cannot be immediately applied in the analysis of arch dam which belongs to irregular shell structure. This limitation can be overcome by employing the sub-structure method to subdivide the arch dam into several sub-structures which satisfy the scaling requirement. An example of static analysis of the arch dam-foundation system is presented to verify the effectiveness of the SBFEM sub-structure method. The result is helpful to establish the calculating model of arch dam-reservoir-foundation system based on scaled boundary finite element method.

Assessment of induced voltages in common and differential‐mode for a PV module due to nearby lightning strikes

Nearby lightning strikes are prone to induce overvoltage transients in photovoltaic (PV) modules and in their power conditioning circuitry, which can permanently damage the PV system. Therefore, it becomes important to establish a method for accurate assessment of such transients. To this aim, the authors propose a three-dimensional (3D) semi-analytical numerical method to study the electromagnetic transients caused in PV modules by nearby lightning strikes. The approach bases on a semi-analytical expression of the magnetic vector potential generated by a geometrically complex lightning channel. The proposed method is able to calculate the transient overvoltage in a PV module, both in common and differential-mode, taking also into account capacitive and inductive couplings between the internal circuit and the PV metallic frame. Both modes are required to design the surge protective devices (SPDs) in PV power systems. Comparing to the models in literature, the proposed approach explicitly considers the complex geometry of the lightning channel. Statistical analysis allows assessing the impact of channel geometry by randomly generating a number of likely lightning paths. Results show that the lightning-induced overvoltage in a PV module is highly dependent on factors such as distance to the lightning channel and lightning channel geometry.

A semianalytical method for fuzzy integro-differential equations under generalized Seikkala derivative

In this article, a semianalytical numerical method has been presented to solve fuzzy integro-differential equation which may be linear or nonlinear under multi-point or mixed boundary conditions. A convergence analysis of the proposed method has been studied to emphasize its reliability in general. In order to show the effectiveness of this method, some illustrative examples are given. We have shown that with a small number of obtained approximating terms, we achieve a high accuracy level of the obtained results. Comparisons have been made between the solutions of our method and some existing methods. Moreover, proper graphs are provided to show that increasing the number of approximating terms yields a significant decrease in the error of the approximate solution.

A semi-analytic collocation method for space fractional parabolic PDE

ABSTRACTThis paper has presented a new semi-analytic numerical method for solving multi-point problems for nonlinear singular ordinary differential equations (ODEs) of a high order. The method consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters. These free parameters are determined by the use of the collocation procedure. Some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results. In particular, the singular multi-point boundary value problems (BVPs) with equations of the fourth, fifth and sixth orders are considered. Numerical results show that the proposed method is of high accuracy and is efficient for solving a wide class of the nonlinear singular ODEs. The algorithm of the method is very simple and does not require any specific technique in handling the singularity at the endpoints of the solution domain.

Numerical Modelling of Surface Water Wave Interaction with a Moving Wall

In the present manuscript, we consider the practical problem of wave interaction with a vertical wall. However, the novelty here consists in the fact that the wall can move horizontally due to a system of springs. The water wave evolution is described with the free surface potential flow model. Then, a semi-analytical numerical method is presented. It is based on a mapping technique and a finite difference scheme in the transformed domain. The idea is to pose the equations on a fixed domain. This method is thoroughly tested and validated in our study. By choosing specific values of spring parameters, this system can be used to damp (or in other words to extract the energy of) incident water waves.