Approximate Solution Research Articles

An Approximate Analytical Solution of Obesity Epidemic Model by Homotopy Analysis Method

An Approximate Analytical Solution of Obesity Epidemic Model by Homotopy Analysis Method

Finite element method for fractional order parabolic obstacle problem with nonlinear source term

This paper considers the finite element method for a fractional order parabolic obstacle problem in time with a nonlinear source term. An error estimation of the spatially semi-discrete problem is discussed. A fully discrete problem is proposed by introducing the finite difference approximation for the time variable. The L2-unconditional stability and convergence of the approximate solution have been proven.

A new identification of Lagrange multipliers to study solutions of nonlinear Caputo–Fabrizio fractional problems

This research aims to provide a new identification of Lagrange multipliers using a novel hybrid method based on the Khalouta transform method and the variational iteration method and to study the solutions of nonlinear fractional problems. This method is called Khalouta variational iteration method (KHVIM). The fractional derivative is considered in Caputo–Fabrizio sense. The uniqueness and convergence results are investigated using Banach fixed point theorem. The results are obtained in the form of successive approximations corresponding to the proposed problem. To demonstrate the accuracy and effectiveness of the proposed method, three different nonlinear fractional partial differential equations are provided. Approximate solutions of the fractional equations were obtained. These solutions quickly converged to exact solutions with lower computational cost. Furthermore, the method used in this study is more generalized and allows our results to be more extensive and cover several new and existing fractional problems in the literature.

Approximate Solutions for Non-Linear Evolution Stochastic Equations with Variations of Drift Parameters

This paper considered system of stochastic differential equations with emphasis on variations of drift parameters as it affects financial markets. The Ito’s method was applied in solving the problems analytically which resulted to three different investment solutions accordingly. The appropriate conditions were accomplished which controlled various drift parameters in the assessment financial markets. Hence, the impressions on each solution of investors in financial markets were analyzed graphically. Finally, the influences of the relevant parameters of stochastic variables were effectively discussed all in this paper.

Exploring fractional-order new coupled Korteweg-de Vries system via improved Adomian decomposition method.

This paper aims to extend the applications of the projected fractional improved Adomian Decomposition method (fIADM) to the fractional order new coupled Korteweg-de Vries (cKdV) system. This technique is significantly recognized for its effectiveness in addressing nonlinearities and iteratively handling fractional derivatives. The approximate solutions of the fractional-order new cKdV system are obtained by employing the improved ADM in fractional form. These solutions play a crucial role in designing and optimizing systems in engineering applications where accurate modeling of wave phenomena is essential, including fluid dynamics, plasma physics, nonlinear optics, and other mathematical physics domains. The fractional order new cKdV system, integrating fractional calculus, enhances accuracy in modeling wave interactions compared to the classical cKdV system. Comparison with exact solutions demonstrates the high accuracy and ease of application of the projected method. This proposed technique proves influential in resolving fractional coupled systems encountered in various fields, including engineering and physics. Numerical results obtained using Mathematica software further verify and demonstrate its efficacy.

Implementation of an effective transform technique for convergence analysis of the fractional enzyme kinetic and blood alcohol level models

In the current study, we solve the nonlinear time fractional‐order enzyme kinematic system and the linear time fractional blood alcohol level using an effective method called the Adomian natural decomposition method (ANDM), which is a combination of the Adomian decomposition method (ADM) and the natural transform method (NTM). On the basis of the Banach fixed point theorem, we offer proofs for the existence and uniqueness theorems as they apply to nonlinear ordinary differential equations. Using the current method, approximate analytical solutions of nonlinear time fractional‐order enzyme kinematic system and exact solutions to the linear time fractional‐order blood alcohol level problem have been obtained. The outcomes indicate that the ANDM is highly efficient and useful. The method outlined in this paper is intended to be applied in subsequent work to handle comparable nonlinear problems connected to fractional calculus.

Two strains model of infectious diseases for mathematical analysis and simulations

ABSTRACT In this study, we study a two-strain nonlinear model for the transmission of COVID-19 with a vaccinated class. Here, it is remarkable that the model we consider contains two kinds of viruses known as Omicron and Delta variants denoted by A and B , respectively. Also, the uninfected population is denoted by S , the vaccinated class by V and the recovered individuals by R . In the presented study, we consider the proposed model under conformable fractional order derivatives. The fundamental reproductive number and equilibrium points are computed. Moreover, we determine the existence and uniqueness of the solution to the suggested model using fixed-point theory. Furthermore, we provide a suitable methodology by applying the Euler numerical method to calculate the approximate solution of each compartment of the proposed model. Additionally, using MATLAB-16, we simulate the given results graphically for a variety of fractional orders using some real values of the parameters and initial conditions.

Variable order fractional diabetes models: numerical treatment

ABSTRACT In this paper, we present two numerical approximation solutions for the updated nonlinear variable order fractional differential equations of diabetes disease. The variable order fractional differential operator is based on the Caputo fractional derivative concept. The first used method is developed using the nonstandard finite difference process, such that a new quick approximation is applied to tackle the nonlinear terms. By applying some reasonable assumptions to the given data, we can prove that this scheme maintains the positivity and boundedness of the discretized solutions. The second method is based on approximation with shifted Jacobi polynomials. The properties of Jacobi polynomials, together with the shifted Jacobi-Gauss-Lobatto nodes were utilized to reduce the proposed system into a set of algebraic nonlinear equations. Furthermore, some numerical experiments are conducted to compare the performance of the introduced schemes. Our numerical simulations show that the nonstandard finite difference approach requires less computational time compared to the Jacobi-Gauss-Lobatto spectral collocation. While the last one is more accurate when used for solving the variable order fractional diabetes model.

Analysis of a fractional-order model for dengue transmission dynamics with quarantine and vaccination measures

A comprehensive mathematical model is proposed to study two strains of dengue virus with saturated incidence rates and quarantine measures. Imperfect dengue vaccination is also assumed in the model. Existence, uniqueness and stability of the proposed model are proved using the results from fixed point and degree theory. Additionally, well constructed Lyapunov function candidates are also applied to prove the global stability of infection-free equilibria. It is also demonstrated that the model is generalized Ulam-Hyers stable under some appropriate conditions. The model is fitted to the real data of dengue epidemic taken from the city of Espirito Santo in Brazil. For the approximate solution of the model, a non-standard finite difference(NSFD) approach is applied. Sensitivity analysis is also carried out to show the influence of different parameters involved in the model. The behaviour of the NSFD is also assessed under different denominator functions and it is observed that the choice of the denominator function could influence the solution trajectories. Different scenario analysis are also assessed when the reproduction number is below or above one. Furthermore, simulations are also presented to assess the epidemiological impact of dengue vaccination and quarantine measures for infected individuals.

Modal stability analysis of toroidal pipe flow approaching zero curvature

The present study investigates the modal stability of the steady incompressible flow inside a toroidal pipe for values of the curvature $\delta$ (ratio between pipe and torus radii) approaching zero, i.e. the limit of a straight pipe. The global neutral stability curve for $10^{-7} \leq \delta \leq ~10^{-2}$ is traced using a continuation algorithm. Two different families of unstable eigenmodes are identified. For curvatures below $1.5 \times 10^{-6}$ , the critical Reynolds number ${{Re}}_{cr}$ is proportional to $\delta ^{-1/2}$ . Hence, the critical Dean number is constant, ${{De}}_{cr} = 2\,{{Re}}_{cr}\,\sqrt {\delta } \approx 113$ . This behaviour confirms that the Hagen–Poiseuille flow is stable to infinitesimal perturbations for any Reynolds number and suggests that a continuous transition from the curved to the straight pipe takes place as far as it regards the stability properties. For low values of the curvature, an approximate self-similar solution for the steady base flow can be obtained at a fixed Dean number. Exploiting the proposed semi-analytic scaling in the stability analysis provides satisfactory results.

Numerical study of nonlinear time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising in propagation of waves

In this manuscript, the numerical approximation of the solution for time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) nonlinear partial differential equation is described involving the Caputo fractional derivative with the help of two novel techniques namely, Sumudu Residual Power Series (SRPS) method and Homotopy Perturbation Sumudu Transform (HPST) method. CDGSK equation describe the propagation of capillary waves and shallow water waves. The results obtained through numerical approximation shows that both SRPS and HPST under consideration are reliable and efficient. The present framework precisely reflects the behavior of the obtained result for various fractional orders. Additionally, we conduct a comparison between the exact solution and an approximate series solution to affirm the efficacy of the proposed techniques. This analysis highlights the efficiency and precision of both techniques in solving nonlinear fractional differential equations. All the numerical simulations are demonstrated graphically using Maple and Matlab.

A Modified Perturbation Method for Global Dynamic analysis of Generalized Mixed Rayleigh–Liénard Oscillator with Cubic and Quintic Nonlinearities

Abstract Deriving analytical relationship between the system parameters and amplitude of the limit cycle is a meaningful and challenging task. Currently, numerous existing analytical approximate methods struggle to achieve this goal when expressions of restoring force or nonlinear damping is complicated. To overcome this shortcoming, this study proposes a modified generalized harmonic function perturbation method. Using the proposed method, a generalized mixed Rayleigh–Liénard oscillator with cubic and quintic nonlinearities was investigated. The analytical relationships between the system parameters and amplitude of the limit cycle, as well as the expression of its characteristic quantity, were derived. By employing these analytical relationships, the existence, stability, number, position, and amplitude of each limit cycle are quantitatively analysed. The homoclinic and heteroclinic bifurcations were also predicted using the above analytical relationships. Additionally, analytical approximate solutions for this oscillator were calculated using the proposed method. All results obtained in this study were subsequently confirmed numerically to demonstrate their feasibility and validity. Consequently, the proposed method can be considered an effective supplement to perturbation-based methods. This also implies that the work presented in this paper has a certain theoretical significance and application value in the research area of quantitative analysis methods for strongly nonlinear oscillators.

A Modified Three-Term Conjugate Descent Derivative-Free Method for Constrained Nonlinear Monotone Equations and Signal Reconstruction Problems

Iterative methods for solving constraint nonlinear monotone equations have been developed and improved by many researchers. The aim of this research is to present a modified three-term conjugate descent (TTCD) derivative-free method for constrained nonlinear monotone equations. The proposed algorithm requires low storage memory; therefore, it has the capability to solve large-scale nonlinear equations. The algorithm generates a descent and bounded search direction dk at every iteration independent of the line search. The method is shown to be globally convergent under monotonicity and Lipschitz continuity conditions. Numerical results show that the suggested method can serve as an alternative to find the approximate solutions of nonlinear monotone equations. Furthermore, the method is promising for the reconstruction of sparse signal problems.

Approximate bound state solutions of the Dunkl-Schrödinger equation for a hyperbolic double-well interaction

Abstract We construct approximate solutions to the stationary, one-dimensional Schrödinger equation for a hyperbolic double-well potential within the Dunkl formalism. Our approximation is applied to an inverse quadratic term contributed by the Dunkl formalism in the effective potential. The solutions we obtain are given in terms of confluent Heun functions. We establish parity of these solutions, discuss their elementary cases, and present an example of a system admitting bound states.

Prediction of Resonant Frequency of an Equilateral Triangular Microstrip Antenna Using Conformal Mapping Technique

In this manuscript, Conformal Mapping Technique (CMT) is applied to predict the resonant frequency of an Equilateral Triangular Microstrip Antenna (ETMA). The ETMA is mapped into a simple circular geometry with the aid of CMT. The wave equation is solved in the transformed domain using Galerkin technique to obtain the approximate solution of eigenfunction and eigenvalue. Twenty-three measured data are collected from the open literature to validate our theory. Our theoretical results are also compared with several CAD models to show the effectiveness of the present theory. It is found that our proposed theory can predict the resonant frequency of the ETMA with an error of less than 1.5%, whereas those CAD models can produce a large error (sometimes, 10-19% error). Internal field patterns, input impedance, far-field radiation patterns, quality factor, bandwidth gain, and directivity of the ETMA for the dominant mode are also investigated using CMT.

LOCAL TIME FRACTIONAL REDUCED DIFFERENTIAL TRANSFORM METHOD FOR SOLVING LOCAL TIME FRACTIONAL TELEGRAPH EQUATIONS

In this paper, we seek to find solutions of the local time fractional Telegraph equation (LTFTE) by employing the local time fractional reduced differential transform method (LTFRDTM). This method produces a numerical approximate solution having the form of an infinite series that converges to a closed form solution in many cases. We apply LTFRDTM on four different LTFTEs to examine the efficiency of the proposed method. The yielded results established the effectiveness of LTFRDTM as a reliable and solid approach for obtaining solutions of LTFTEs. The solutions coincided with the exact solution in the ordinary case when [Formula: see text]. It also required minimal amount of computational work and saved a lot of time.

SPLINE QUASI-INTERPOLATION NUMERICAL METHODS FOR INTEGRO-DIFFERENTIAL EQUATIONS WITH WEAKLY SINGULAR KERNELS

In this work, we introduce a numerical approach that utilizes spline quasi-interpolation operators over a bounded interval. This method is designed to provide a numerical solution for a class of Fredholm integro-differential equations with weakly singular kernels. We outline the computational components involved in determining the approximate solution and provide theoretical findings regarding the convergence rate. This convergence rate is analyzed in relation to both the degree of the quasi-interpolant and the grading exponent of the graded grid partition. Finally, we present numerical experiments that validate the theoretical findings.

Feasibility of integrated numerical and Adaptive Neuro-Fuzzy Inference System models in predicting the thermophysical properties of unsteady coupled micropolar–Casson hybrid nanofluids

This article aims to probe postulated phenomena using a paired micropolar and Casson hybrid fluid over a rotating disk. The intension to design a numerical technique (Runge–Kutta fourth-order along with shooting technique) integrated Adaptive Neuro-Fuzzy Inference System (ANFIS) envisioned with a thermal and exponentially space-dependent heat source, nonlinear thermal radiation and entropy production is developed in this phase of work. To ensure that, a nonlinear partial differential equation set of equations has been transformed into an ordinary differential equation by using the proper self-similarity variables. The model's research results, with a few notable outliers, are mostly consistent with those from prior research that was merged into the dataset used to train the ANFIS model. With the impact of active factors, the results are esthetically exhibited for numerous profiles. This displays that with the rise in magnetic field and radiation, the velocity and temperature profiles increase sharply, resulting in a contradiction phenomenon with the decreasing electric field inputs. Also, tilting of vortex viscosity, spin gradient viscosity and microinertia density on the various microrotation components displays inclination. Moreover, ANFIS training was exploited to analyze the approximate solutions for specific scenarios, and the developed ANFIS was evaluated against a testing dataset to emphasize its performance. Due to their longer render, the nanoparticles exploited here are deemed suitable for use in bone implants, iodinated agents for blood imaging and red blood cell stimulation. Thus, the results of this study may be applied to therapeutic anemia therapies.

Dynamic programming with meta-reinforcement learning: a novel approach for multi-objective optimization

AbstractMulti-objective optimization (MOO) endeavors to identify optimal solutions from a finite array of possibilities. In recent years, deep reinforcement learning (RL) has exhibited promise through its well-crafted heuristics in tackling NP-hard combinatorial optimization (CO) problems. Nonetheless, current methodologies grapple with two key challenges: (1) They primarily concentrate on single-objective optimization quandaries, rendering them less adaptable to the more prevalent MOO scenarios encountered in real-world applications. (2) These approaches furnish an approximate solution by imbibing heuristics, lacking a systematic means to enhance or substantiate optimality. Given these challenges, this study introduces an overarching hybrid strategy, dynamic programming with meta-reinforcement learning (DPML), to resolve MOO predicaments. The approach melds meta-learning into an RL framework, addressing multiple subproblems inherent to MOO. Furthermore, the precision of solutions is elevated by endowing exact dynamic programming with the prowess of meta-graph neural networks. Empirical results substantiate the supremacy of our methodology over previous RL and heuristics approaches, bridging the chasm between theoretical underpinnings and real-world applicability within this domain.

A new method for determining crack propagation

This paper presents a new method for obtaining approximate closed-form solutions for crack propagation problems. In general, it is not possible to obtain explicit solutions of the parameters and elementary functions of these problems, due to their terms complexity, present in their definitions. In this way, we used numerical methods to obtain approximate numerical solutions. The method presented in this article provides a proposal, which, when defined by differential equations, is applicable and extensible to other problems. The approximate solutions are obtained as an exact solution of a modified initial value problem. The performance is evaluated in relation to problems with known exact solution.