Semi-analytical Scheme Research Articles

Optimal control analysis applied to the transmission dynamics of Hepatitis B to primary liver cancer

Infection with hepatitis virus, especially hepatitis B virus causes an irritation to the liver. Ranked among the top ten diseases with high mortality rate, viral hepatitis poses a great health challenges worldwide, with threat to chronic infection, hepatitis-related liver cancer and cirrhosis. Hence, a mathematical model to study the dynamics of hepatitis B virus infection from progressing into primary liver cancer was developed for analysis. We aimed at obtaining the optimal control strategies needed to reduce the number of new cases of this disease, and also reducing the deterioration rate of people living with this disease from sliding into primary liver cancer. We introduced four distinct control variables at each point in the model, and assumed that all the controls are set of Lebesque Measurable functions. The Pontryagin’s maximum principle (PMP) is employed to establish the optimal effect of these controls on the disease under study. Existence of model solution was established using the appropriate theorem. The model with control strategies was analytically solved using PMP and numerically simulated for each compartments, to establish the effect of the control variables on the dynamics of transmission of this infection within the compartment, and their overall effect on the entire population. Differential transform method (DTM) was later adopted as a semi-analytic scheme to solve the developed model. The series solution of DTM was numerically plotted for each compartment and compared with Runge-Kutta order 4 (RK4) numerical scheme. Analysis of the model with control pinpoint the importance of sensitization and vaccination on the overall dynamics of the infection, while the numerical plot of DTM and RK4 established the efficacy of the adopted semi-analytic method (DTM) to accurately solve the system of equations of the model.

A semi-analytical transient undisturbed velocity correction scheme for wall-bounded two-way coupled Euler-Lagrange simulations

Force closure models employed in Euler-Lagrange (EL) point-particle simulations rely on the accurate estimation of the undisturbed fluid velocity at the particle center to evaluate the fluid forces on each particle. Due to the self-induced velocity disturbance of the particle in the fluid, two-way coupled EL simulations only have access to the disturbed velocity. The undisturbed velocity can be recovered if the particle-generated disturbance is estimated. In the present paper, we model the velocity disturbance generated by a regularized forcing near a planar wall, which, along with the temporal nature of the forcing, provides an estimate of the unsteady velocity disturbance of the particle near a planar wall. We use the analytical solution for a singular in-time transient Stokeslet near a planar wall (Felderhof [1]) and derive the corresponding time-persistent Stokeslets. The velocity disturbance due to a regularized forcing is then obtained numerically via a discrete convolution with the regularization kernel. The resulting Green's functions for parallel and perpendicular regularized forcing to the wall are stored as pre-computed temporal correction maps. By storing the time-dependent particle force on the fluid as fictitious particles, we estimate the unsteady velocity disturbance generated by the particle as a scalar product between the stored forces and the pre-computed Green's functions. Since the model depends on the analytical Green's function solution of the singular Stokeslet near a planar wall, the obtained velocity disturbance exactly satisfies the no-slip condition and does not require any fitted parameters to account for the rapid decay of the disturbance near the wall. The numerical evaluation of the convolution integral makes the present method suitable for arbitrary regularization kernels. Additionally, the generation of parallel and perpendicular correction maps enables to estimate the velocity disturbance due to particle motion in arbitrary directions relative to the local flow. The convergence of the method is studied on a fixed particle near a planar wall, and verification tests are performed in the Stokes regime on a settling particle parallel to a wall and a free-falling particle perpendicular to the wall.

Performance analysis and accuracy of numerical methods for solving simultaneous nonlinear particulate processes with multi-dimensional extension

The simultaneous population balance equation (PBE) incorporating diverse particulate process aggregation, breakage, growth, nucleation and source is a long-standing and significant problem in the field of particulate sciences such as pharmaceutical industry, chemical engineering, astrophysics and biology. Owing to the complex and nonlinear nature of the governing equations, finding an analytical solution is quite challenging for empirical kernels. In this work, simultaneous PBEs are solved using sectional and semi-analytical techniques. The performance, adaptability and accuracy of both methods for solving simultaneous PBEs are discussed in detail. The iterative semi-analytical scheme for solving simultaneous PBE is introduced, and a detailed convergence analysis using Banach fixed point theory is also presented. Both the schemes are further extended to solve simultaneous PBE in multi-dimensions. Performance analysis is also executed for multi-dimensional models. This article is a comprehensive study on various solution techniques for solving simultaneous PBEs. Therefore, it will be beneficial for the researchers working in industry and computational fluid dynamics modelling to identify the best method according to their requirements.

Free vibration analysis of closed and open horizontal cylindrical shells utilizing coupled transfer function method and differential transform method

This article aims to analyze the free vibration of closed and open horizontal circular cylindrical shells based on the concept of state-space transformation in conjunction with the differential transform method (DTM). The governing equations of the cylinders were developed based upon Sanders’ thin shell theory. Then, the set of coupled governing differential equations transformed into nine first-order differential equations. Afterwards, the well-established semi-analytical scheme so-called DTM was employed to solve the differential equations. The numerical investigation of the present method was also comprehensively carried out by analyzing some closed and open cylindrical shells with different boundary and compatibility conditions. To ensure the correctness of the results, the outcomes were compared with experimental and other numerical results, especially with the well-known finite element method. Furthermore, some plots of two-dimensional and three-dimensional mode shapes for the first and second axial modes are depicted.

A Novel Semi-Analytical Scheme to Deal with Fractional Partial Differential Equations (PDEs) of Variable-Order

This article introduces a new numerical algorithm dedicated to solving the most general form of variable-order fractional partial differential models. Both the time and spatial order of derivatives are considered as non-constant values. A combination of the shifted Chebyshev polynomials is used to approximate the solution of such equations. The coefficients of this combination are considered a function of time, and they are obtained using the collocation method. The theoretical aspects of the method are investigated, and then by solving some problems, the efficiency of the method is presented.

Semi analytical scheme for the presentation of solution to Fractional Fokker–Planck Equation

In this work, we have also proposed new semi analytical scheme combining least-squares and asymptotic homotopy perturbation methods AHPM for the solution of Fractional Fokker–Planck Equation (FFPE). The Caputo version of derivative is used. The finding of this work is fast convergence of the AHPM to the solutions of linear and nonlinear applied problems. AHPM is a modified form of the HPM and is particularly useful for solving problems with small or large parameters. This can be seen as a clear advantage of this new semi analytical method over existing methods because its implementation and the calculation involved in it are much simpler and easier. The proposed method solved the problems without using Ji-Huan He and Adomian polynomials. Numerical solutions obtained in this way show that this approach is simple and computationally very attractive. An error estimate for the solution is also provided. Approximate solutions of AHPM are tabulated and graphically displayed to show that AHPM is effective and accurate.

A comprehensive analysis of uncertainties in warm rain parameterizations in climate models based on in situ measurements

Abstract Because of the coarse grid size of Earth system models (ESM), representing warm-rain processes in ESMs is a challenging task involving multiple sources of uncertainty. Previous studies evaluated warm-rain parameterizations mainly according to their performance in emulating collision-coalescence rates for local droplet populations over a short period of a few seconds. The representativeness of these local process rates comes into question when applied in ESMs for grid sizes on the order of 100 kilometers and time steps on the order of 20-30 minutes. We evaluate several widely used warm-rain parameterizations in ESM application scenarios. In the comparison of local and instantaneous autoconversion rates, the two parameterization schemes based on numerical fitting to stochastic collection equation (SCE) results perform best. However, because of Jessen’s inequality, their performance deteriorates when grid-mean, instead of locally-resolved, cloud properties are used in their simulations. In contrast, the effect of Jessen’s inequality partly cancels the overestimation problem of two semi-analytical schemes, leading to an improvement in the ESM-like comparison. In the assessment of uncertainty due to the large time step of ESMs, it is found that the rain-water tendency simulated by the SCE is roughly linear for time steps smaller than 10 minutes, but the nonlinearity effect becomes significant for larger time steps, leading to errors up to a factor of 4 for a time step of 20 minutes. After considering all uncertainties, the grid-mean and time-averaged rain-water tendency based on the parameterization schemes are mostly within a factor of 4 of the local benchmark results simulated by SCE.

Analysis of the human liver model through semi-analytical and numerical techniques with non-singular kernel

This work consists of the study of the time-fractional human liver model with the Caputo–Fabrizio fractional derivative. The existence and uniqueness of the proposed model are shown using fixed point theory. Also, the stability of the considered model is shown using the Ulam Hyres theorem and the Lyapunov function. The solution of the proposed model is obtained using a semi-analytical and numerical scheme. The series solution obtained from the semi-analytical method gives the proper result at any time, similarly, the numerical scheme gives the solution for a long time. The obtained numerical results are compared with real clinical data and earlier published work and found to be very close to real data than earlier published work. Results in the graphs and tables show that the proposed fractional-order model is superior to the traditional model.

Thermal analysis of rectangular moving fins with temperature‐variant properties by employing the Galerkin scheme

AbstractIn the present study, the thermal behavior of a longitudinal fin considering three types of heat transfer mechanisms (conduction, convection, and radiation) is investigated. For this research, thermal conductivity, heat source, and heat transfer coefficient are assumed nonindependent. A semianalytical scheme called the Galerkin Method is utilized for solving the dimensionless governing equation. The impacts of important physical variables like Peclet number, gradient of thermal conductivity, thermo‐geometric parameter, and radiation–conduction parameter on temperature profiles are analyzed comprehensively. The obtained results indicate that raising the thermo‐geometric parameter from 0 to 2 leads to a 32% reduction in the temperature profile. Also from the results, it can be found that a 28% increment in the temperature is observed by changing the gradient of thermal conductivity from 0 to 2.

On the stability analysis for a semi-analytical scheme for solving the fractional order blood ethanol concentration system using LVIM

On the stability analysis for a semi-analytical scheme for solving the fractional order blood ethanol concentration system using LVIM

A consistent second order ISPH for free surface flow

The Incompressible Smoothed Particle Hydrodynamics (ISPH) is now a popular numerical method for modelling free surface flows, in particular the breaking waves and violent wave-structures interaction. The ISPH requires the projection approach, leading to solving a pressure Poisson's equation (PPE). Although the accuracy and convergence of the numerical scheme to discretise the Laplacian operator involved in PPE is critical for securing a satisfactory solution of the PPE, the overall performance of the ISPH is also influenced by other key numerical implementations, including (1) estimation of the viscous terms; (2) calculation of the velocity divergence; (3) discretisation of the boundary conditions for the PPE; and (4) evaluation of the pressure gradient. In our previous paper [29], the quadratic semi-analytical finite difference interpolation scheme (QSFDI), which has a leading truncation error at third order derivatives, has been adopted to discretise the Laplacian operator. In this paper, the QSFDI will be adopted, not only for discretising the Laplacian operator, but also for approximating viscous terms, velocity divergence, boundary conditions and pressure gradient. The performance of the newly formulated consistent second order ISPH is assessed by various cases including the oscillating liquid drop, the wave propagation, and the liquid sloshing. The results do not only demonstrate a second order convergence over a limited range of conditions and a higher computational efficiency, i.e., requiring less computational time to achieve the same accuracy, but also show a better mass/energy conservation property and capacity of reproducing a smooth pressure field, than other ISPH models considered in this study.

Elzaki projected differential transform method for multi-dimensional aggregation and combined aggregation-breakage equations

Numerous real-world fields, including planetary science, bio-pharmaceutical, chemical study, food processing industry, and many more are profoundly impacted by population balance equations. Model complexity limits the analytical investigations to a few aggregation-breakage parameters, although various numerical and semi-analytical schemes are available. This article proposes a new semi-analytical approach, the Elzaki integral transform as a pre-treatment to reinforce domain decomposition for better accuracy and convergence, in conjunction with the projected differential transform method for finding the closed form or approximated series solutions for non-linear aggregation, aggregation-breakage, and multi-dimensional aggregation equations. The technique’s key benefit over traditional numerical methods is its ability to solve linear or non-linear differential equations directly without discretization or linearization. Theoretical convergence analysis and error estimates of series solutions for both one and two dimensional aggregation models are of particular interest. Finally, several numerical examples of aggregation, aggregation-breakage, and two dimensional aggregation equations are taken to validate the accuracy of the proposed method by comparing numerical simulations with exact solutions. Interestingly, we have obtained closed form solutions for the pure aggregation equation when considering constant and product aggregation kernels. Additionally, error tables and graphs help to highlight the method’s innovative nature.

Semi-analytical scheme with its stability analysis for solving the fractional-order predator-prey equations by using Laplace-VIM

Semi-analytical scheme with its stability analysis for solving the fractional-order predator-prey equations by using Laplace-VIM

Numerical Simulation of Constrained Flows through Porous Media Employing Glimm’s Scheme

This work uses a mixture theory approach to describe kinematically constrained flows through porous media using an adequate constitutive relation for pressure that preserves the problem hyperbolicity even when the flow becomes saturated. This feature allows using the same mathematical tool for handling unsaturated and saturated flows. The mechanical model can represent the saturated–unsaturated transition and vice-versa. The constitutive relation for pressure is a continuous and differentiable function of saturation: an increasing function with a strictly convex, increasing, and positive first derivative. This significant characteristic permits the fluid to establish a tiny controlled supersaturation of the porous matrix. The associated Riemann problem’s complete solution is addressed in detail, with explicit expressions for the Riemann invariants. Glimm’s semi-analytical scheme advances from a given instant to a subsequent one, employing the associated Riemann problem solution for each two consecutive time steps. The simulations employ a variation in Glimm’s scheme, which uses the mean of four independent sequences for each considered time, ensuring computational solutions with reliable positions of rarefaction and shock waves. The results permit verifying this significant characteristic.

Analyzing the convergence of a semi-numerical-analytical scheme for non-linear fractional PDEs

The purpose of this work is to develop a semi-analytical numerical scheme for solving fractional order non-linear partial differential equations (FOPDEs), particularly inhomogeneous FOPDEs, expressed in terms of the Caputo-Fabrizio fractional order derivative operator. To achieve this goal, we examine several fractional versions of nonlinear model equations from the literature. We then present the proposed scheme, discussing its stability and convergence properties. We show that the proposed scheme is efficient and accurate, and we provide numerical examples to illustrate its performance. Our findings demonstrate that the scheme has significant potential for solving a wide range of complex FOPDEs. Overall, this work contributes to the advancement of numerical techniques for solving fractional order non-linear partial differential equations and lays a foundation for further research in this area.

Analyzing pulse behavior in optical fiber: Novel solitary wave solutions of the perturbed Chen–Lee–Liu equation

This study explores the novel solitary wave solutions of the perturbed Chen–Lee–Liu (CLL) equation, aiming to elucidate the physical and dynamic behaviors of pulses in optical fiber. The perturbed CLL equation is derived from the well-known Schrödinger equation and serves as an iconic model. Two analytical techniques are employed to obtain these novel solitary wave solutions. Subsequently, these solutions are subjected to objective analysis using a widely recognized semianalytical scheme to comprehend their underlying mechanisms. Multiple graphs with diverse styles are utilized to illustrate the analysis of pulse waves in optical fiber and assess the accuracy of the analysis. The scientific novelty of this research lies in providing a comprehensive explanation through a comparative analysis of our recently published results in related research papers.

Semi-analytic solutions to edge singularities of three-dimensional axisymmetric bodies

Axisymmetric geometries, such as cylindrical elements, are widely used in offshore structures. However, the presence of sharp edges in these geometries introduces challenges in numerical simulations due to singularities. To address this issue, one possible solution is to represent the singularities using analytic eigenfunctions. This approach can provide insights into the essence of the problem and has successfully applied to two-dimensional (2D) corner problems. However, finding appropriate eigenfunctions for the three-dimensional (3D) edges remains an open challenge. This paper proposes a semi-analytic scheme for 3D axisymmetric problems utilizing a scaled boundary finite element method (SBFEM). A dimensional reduction is introduced to the 3D Laplace equation, and a 3D edge is handled on the generatrix plane while governed by a complicated equation. The algorithm for resolving the SBFEM fundamental space is improved, and the singularities are approximated using a fractional-order basis. The effectiveness of the proposed method is demonstrated through its application to solve the radiation problem of a heaving cylinder. The method accurately captures the singular velocity field at the edge tip, ensuring that the boundary condition on the body surface is strictly satisfied in the neighborhood of the singularity. Accuracy of the mean drift force is ensured by performing direct pressure integrations over the body surface using a near-field formulation, which becomes as accurate as the middle-field formulation.

An extended variational iteration method for fractional BVPs encountered in engineering applications

PurposeThe purpose of this paper is to find approximate solutions for a general class of fractional order boundary value problems that arise in engineering applications.Design/methodology/approachA newly developed semi-analytical scheme will be applied to find approximate solutions for fractional order boundary value problems. The technique is regarded as an extension of the well-established variation iteration method, which was originally proposed for initial value problems, to cover a class of boundary value problems.FindingsIt has been demonstrated that the method yields approximations that are extremely accurate and have uniform distributions of error throughout their domain. The numerical examples confirm the method’s validity and relatively fast convergence.Originality/valueThe generalized variational iteration method that is presented in this study is a novel strategy that can handle fractional boundary value problem more effectively than the classical variational iteration method, which was designed for initial value problems.

Broad-Angle Multichannel Metagrating Diffusers

We present a semianalytical scheme for the design of broad-angle multichannel metagratings (MGs), sparse periodic arrangements of loaded conducting strips (meta-atoms), embedded in a multilayer printed circuit board (PCB) configuration. By judicious choice of periodicity and angles of incidence, scattering off such an MG can be described via a multiport network, where the input and output ports correspond to different illumination and reflection directions associated with the same set of propagating Floquet–Bloch (FB) modes. Since each of these possible scattering scenarios can be modeled analytically, constraints can be conveniently applied on the modal reflection coefficients (scattering matrix entries) to yield a diffusive response, which, when resolved, produce the required MG geometry. We show that by demanding a symmetric MG configuration, the number of independent S parameters can be dramatically reduced, enabling satisfaction of multiple such constraints using a single sparse MG. Without any full-wave optimization, this procedure results in a fabrication-ready layout of a multichannel MG, enabling retroreflection suppression and diffusive scattering from numerous angles of incidence simultaneously. This concept, verified experimentally via a five-channel MG prototype, offers an innovative solution to both monostatic and bistatic radar cross Section (RCS) reduction, avoiding design and implementation challenges associated with dense metasurfaces used for this purpose.

Analysis of 3D transient heat conduction in functionally graded materials using a local semi-analytical space-time collocation scheme

A local semi-analytical space-time collocation scheme, combined the localized space-time method of fundamental solutions (LSTMFS) and transformed function, is proposed for simulating 3D transient heat transfer in functionally graded materials. In this computation, the governing equation of the transient heat transfer in functionally graded materials is reduced to a standard diffusion equation by using transformed function. The transformed equation has a fundamental solution satisfying the governing equation, and thus the LSTMFS is used in estimating the solution of the degraded diffusion equation. As a local semi-analytical meshless method, the proposed LSTMFS has the advantages of a simple mathematical expression, high precision, and easy implementation. In comparison with the conventional space-time method of fundamental solutions, the developed approach forms a sparse linear system and is more suitable for high-dimensional problems and large-scale complex structures. Three benchmark numerical examples are provided to confirm the accuracy and feasibility of the proposed scheme.