Approximate Analytical Technique Research Articles

An efficient approximate analytical technique for the fractional model describing the solid tumor invasion

An efficient approximate analytical technique for the fractional model describing the solid tumor invasion

Distinguishing Between Long-Transient and Asymptotic States in a Biological Aggregation Model

Aggregations are emergent features common to many biological systems. Mathematical models to understand their emergence are consequently widespread, with the aggregation–diffusion equation being a prime example. Here we study the aggregation–diffusion equation with linear diffusion in one spatial dimension. This equation is known to support solutions that involve both single and multiple aggregations. However, numerical evidence suggests that the latter, which we term ‘multi-peaked solutions’ may often be long-transient solutions rather than asymptotic steady states. We develop a novel technique for distinguishing between long transients and asymptotic steady states via an energy minimisation approach. The technique involves first approximating our study equation using a limiting process and a moment closure procedure. We then analyse local minimum energy states of this approximate system, hypothesising that these will correspond to asymptotic patterns in the aggregation–diffusion equation. Finally, we verify our hypotheses through numerical investigation, showing that our approximate analytic technique gives good predictions as to whether a state is asymptotic or transient. Overall, we find that almost all twin-peaked, and by extension multi-peaked, solutions are transient, except for some very special cases. We demonstrate numerically that these transients can be arbitrarily long-lived, depending on the parameters of the system.

Fractional Trigonometric Function Approximations in a Regular System

Introduction: Linear fractional differential equations of the commensurable order, specifically those related to fractional trigonometry, pose challenges in terms of analytical solutions. background: This study presents an approach to the solution of fractional differential equations with commensurate order, specifically those associated with fractional trigonometry. Methods: The purpose of this article is to present a novel approximate analytical technique to solve linear fractional differential equations of the commensurable order, which are related to fractional trigonometry. This method is used to obtain approximate solutions by forming linear combinations of appropriate fractional basic functions, for which Laplace transforms are irrational functions. This approximation technique enables us to represent these functions using timeinvariant linear system models. Also, the implementation of analog circuits of these basic fractional order systems can be obtained using approximation of rational functions. Results: The research demonstrates the efficacy and precision of this method through illustrative examples, showcasing its effectiveness in solving linear fractional systems. Conclusion: The newly introduced approximate analytical method presents a promising approach to solving linear fractional differential equations of the commensurable order, providing accurate solutions and possibly offering applications in various fields.

Efficient approximate analytical technique to solve nonlinear coupled Jaulent–Miodek system within a time-fractional order

<abstract><p>In this article, we considered the nonlinear time-fractional Jaulent–Miodek model (FJMM), which is applied to modeling many applications in basic sciences and engineering, especially physical phenomena such as plasma physics, fluid dynamics, electromagnetic waves in nonlinear media, and many other applications. The Caputo fractional derivative (CFD) was applied to express the fractional operator in the mathematical formalism of the FJMM. We implemented the modified generalized Mittag-Leffler method (MGMLFM) to show the analytical approximate solution of FJMM, which is represented by a set of coupled nonlinear fractional partial differential equations (FPDEs) with suitable initial conditions. The suggested method produced convergent series solutions with easily computable components. To demonstrate the accuracy and efficiency of the MGMLFM, a comparison was made between the solutions obtained by MGMLFM and the known exact solutions in some tables. Also, the absolute error was compared with the absolute error provided by some of the other famous methods found in the literature. Our findings confirmed that the presented method is easy, simple, reliable, competitive, and did not require complex calculations. Thus, it can be extensively applied to solve more linear and nonlinear FPDEs that have applications in various areas such as mathematics, engineering, and physics.</p></abstract>

Approximate analytical solutions of fractional order differential equations by conformable Sumudu decomposition method

This work offers a fast, approximate analytical technique for solving time fractional differential equations. CSTADM is a gentle combination of two powerful approaches to solve fractional PDE’s. The hybrid approach suggested here produces more accurate findings and convergent solution series faster. Also, the proposed approach is used to get the analytical solution of nonlinear fractional PDE’s. The fractional derivative’s type is assumed to be conformable. Compared to standard fractional derivative definitions, this approach is effective and straightforward in applications involving fractional differential equations with complicated solutions. Furthermore, the results show that the acquired results increase the precision and efficacy of the proposed technique.

Operator norm-based determination of failure probability of nonlinear oscillators with fractional derivative elements subject to imprecise stationary Gaussian loads

An approximate analytical technique is developed for bounding the first-passage probability of lightly damped nonlinear and hysteretic oscillators endowed with fractional derivative elements and subjected to imprecise stationary Gaussian loads. In particular, the statistical linearization and stochastic averaging methodologies are integrated with an operator norm-based approach to formulate a numerically efficient proxy for the first-passage probability. This proxy is employed to determine the realizations of the interval-valued parameters of the excitation model that yield the extrema of the failure probability function. Ultimately, each failure probability bound is determined in a fully decoupled manner by solving a standard optimization problem followed by a single evaluation of the first-passage probability. The proposed approximate technique can be construed as an extension of a recently developed operator norm scheme to account for oscillators with fractional derivative elements. In addition, it can readily treat a wide range of nonlinear and hysteretic behaviors. To illustrate the applicability and effectiveness of the proposed technique, a hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative elements subject to imprecise stationary Gaussian loads are considered as numerical examples.

Analytical and Approximate Solutions for Fractional Systems of Nonlinear Differential Equations

This paper studies the epidemic model of rational order via an efficient genius modern analytical approximate technique named Natural Transform Adomian Decomposition Method (NTADM). It is based on Caputo fractional derivative. To demonstrate the effectiveness of the present method, the results are displayed in graphs. Accordingly, the NTADM can be very easily applied to other nonlinear models.

Analysis of the Axial Vibration of Non-Uniform and Functionally Graded Rods via an Analytical-Based Numerical Approach

In this study, an analytical-based numerical approach was proposed for the analysis of the free axial vibration of homogeneous and functionally graded rods with varying cross-sectional areas. The proposed approach is based on analytical approximation techniques, such as the Adomian decomposition method, variational iteration method, and homotopy perturbation method. However, the governing equations of the problems solved in this study were variable coefficient differential equations. These equations provide analytical solutions for strictly limited cases. Analytical approximation methods easily handle problems with uniform material properties and constant cross-sections, whereas with varying cross-sectional areas, the analytical integration process becomes a difficult task for the software. If the rod’s material is functionally graded with varying cross-sectional areas, the analytical integration process becomes a cumbersome task. The proposed approach eliminates all difficulties and requires computation within several seconds. The application of this method is straightforward, and the results obtained in this study are in excellent agreement with the solutions provided in the literature.

Innovative Approach to Evaluating the Approximate Solutions Concerning the Fractional Sharma-Tasso-Oliver Equation Comparison

In this research, we introduce an improved analytical approximation technique for addressing the time-fractional Sharma-Tasso-Olever problem. To manage nonlinear fractional differential equations that emerge in numerous physical phenomena, we establish an alternative basis for the Laplace Residual Power Series approach (LRPSA). The generalized Taylor series equation and residual functions form the foundation of this strategy. The proposed solution yields positive outcomes. The dependability, efficiency, and simplicity of the suggested method are showcased across all categories of fractional nonlinear problems encountered in technological and scientific domains. Two examples are given to illustrate the effectiveness of the proposed approach in solving various kinds of fractional ordinary differential equations. A comparison with other techniques such as RPS, VIM, HPM reveals that our method produces favourable and efficient results.

Scientific exploring of Marangoni convection in stagnation point flow of blood-based carbon nanotubes nanofluid over an unsteady stretching surface

This study examines Marangoni convection in blood-based carbon nanotubes nanofluid's stagnation point flow over a time-dependent stretching surface. This study is inspired by the emerging importance of nanofluids in a variety of scientific and technical fields due to their unique and varied uses and effective thermal activities. Some examples of potential applications of these fluids include cancer treatment, magnetic refrigeration, drug delivery, and magnetic resonance imaging. Two types of nanoparticles are considered name is single wall carbon nanotube and multi wall carbon nanotube blood is takin as base fluid. Nonlinear partial differential equations are used to simulate the specified flow issue using momentum and energy conservation principles. Using a similarity transformation, the resultant is transformed into nonlinear with reduced dimensions. The relations for velocity profile and temperature distribution are calculated from the developed nonlinear ordinary differential equation by using an approximate analytical technique called the homotopy asymptotic method. Subsequently, these equations are implemented and executed within Mathematica software. The investigation focuses on significant outcomes such as momentum filed, energy filed, Skin friction coefficients, and Nusselt number. Graphs are used to interpret the effects of several factors, including the Marangoni parameter, nanoparticle volume friction, stretching parameter, slip parameter, and Prandtl number. The behavior of Nusselt's number and skin friction coefficient was also checked with the help of graphs and tables. The convergence of the solutions is checked with the help of auxiliary functions as well.

Variational Iteration Approach for Solving Fractional Integro-Differential Equations with Conformable Differintegrals

Fractional integro-differential equations involving conformable differintegrals have promising applications but lack solution methods. This work develops a variational iteration approach for such equations. Conformable fractional derivatives and integrals that generalize integer-order equivalents are defined. Existence and uniqueness results are established for a class of nonlinear fractional integro-differential equations. These exist under less restrictive smoothness assumptions than integer-order cases. The variational iteration method (VIM) is then applied, providing an analytical approximate technique for integro-differential problems of fractional order. Convergence analysis demonstrates the efficiency and accuracy of the VIM solutions. Several test cases validate the VIM, matching analytical solutions available for simpler fractional differential sub-cases. The proposed technique advances available methods for this emerging class of fractional integro-differential equations. Significantly, it enables application of such models by allowing accurate solution of associated mathematical system representations. Extensions to include more singular equation classes, comparisons with other methods, and real-world applications are suggested as future work.
 The abstract summarizes the key points: significance of problem, fractional models used, existence/uniqueness proofs, VIM approach and analysis, test case validations, implications for applications

Radiative heat transfer on the peristaltic flow of an electrically conducting nanofluid through wavy walls of a tapered channel

Present analysis deals with the characteristics of radiative heat on the peristaltic flow of nanofluid through wavy walls of a tapered channel. Peristaltic flows within the pumping process arise in the region having lower to higher pressure region. Assumption of velocity slip along with the convective boundary condition energizes the thermal system as well as the flow phenomena. However, the combined effect of Brownian motion and thermophorsis due to the cross-diffusion effect affects the flow properties. One of the significant applications of the flow is that the blood pumping in the human body. It acts as a vehicle through the liquid that past through the wavy channels walls contacting liquids expands in its length due to the dynamical rush. However, solution of the transformed governing flow phenomena is obtained by employing approximate analytical technique known as Variation Parameter Method (VPM). The characteristics of the parameters involved in the system are presented via graphs. Present outcome warrants that, the significance of magnetic strength and flow through the permeable medium may favours to enhance the pumping procedure as the pressure gradient lower down in the non-Darcy medium found to be one of the important observations. Further, significant enhancement in the fluid temperature and concentration profile is exhibited due to the consideration of the convective conditions i.e. the augmentation in the thermal and solutalBiot numbers respectively.

Comparative analysis of analytical and numerical approximations for the flow and heat transfer in mixed convection stagnation point flow of Casson fluid

A mathematical description of non-Newtonian mixed convective Casson fluid stagnation point flow and transfer of heat exists in terms of partial differential equations. We considered the same to study it further under the effects of unsteadiness and varied film thickness parameters. For inclusion of these parameters in the flow model study we modified the available similarity transformations. The governing equations with three independent variables are converted into ordinary differential equations by employing the modified invertible transformations. For the mass and heat transfer in the mixed convection stagnation point unsteady flow of Casson fluid over a stretching sheet, a detailed comparative analysis is carried out in this paper, of the analytical and numerical approximation techniques. The Homotopy Analysis Method is applied for the analytical solutions while the Runge-Kutta with a Shooting Method (RKF45) and Finite Difference Method are used for obtaining the numerical solutions. With these solution schemes we present an analysis of velocity and temperature profiles under the effects of embedded parameters such as the Casson fluid parameter, unsteadiness parameter, mixed convection parameter, Prandtl number, Eckert number, and stretching ratio. The results are presented in both graphical and tabulated forms and they illustrate the dependence of mass and heat transfer characteristics of Casson fluid upon the embedded parameters. Further, we have shown an agreement between the analytical and the approximate solutions for the considered flow and heat transfer which reflects a validation of the results presented here.

A Hybrid Analytical Approximate Technique for Solving Transient Natural Convection in a Horizontal Concentric Cylinders

A Hybrid Analytical Approximate Technique for Solving Transient Natural Convection in a Horizontal Concentric Cylinders

A Hybrid Analytical Approximate Technique for Solving Two-dimensional Incompressible Flow in Lid-driven Square Cavity Problem

This paper suggests a new technique for finding the analytical approximate solutions to two-dimensional kinetically reduced local Navier-Stokes equations. This new scheme depends combines the q-Homotopy analysis method (q-HAM) , Laplace transform, and Padé approximant method. The power of the new methodology is confirmed by applying it to the flow problem of the lid-driven square cavity. The numerical results obtained by using the proposed method showed that the new technique has good convergence, high accuracy, and efficiency compared with the earlier studies. Moreover, the graphs and tables demonstrate the new approach’s validity.

Theoretical analysis of amperometric response towards PPO-based rotating disk bioelectrodes: Taylors series and hyperbolic function method

A mathematical model for the electroenzymatic process of a rotating disk electrode in which polyphenol oxidase occurs is discussed. The model is related to a system of reaction-diffusion equations that includes a nonlinear factor relating to the enzymatic reaction's Michaelis-Menten kinetics. The approximate analytical techniques (Taylor's series and Hyperbolic function) are used to solve the nonlinear differential equations describing the diffusion-coupled enzymatic reactions. For all parameter values γE(=γL2) and α1(αS∞). Closed analytical equations for the concentration of the substrate, the concentration of the product, and the related current response have been obtained. These analytical results are in good agreement when compared to simulation results ( Matlab program). The findings are reliable over the whole solution domain.

Illustration of reacting species on the magnetohydrodynamic flow of micropolar fluid through an expanding surface with ohmic heating

The existing investigation intended to expose the control of dissipative heat on the flow of micropolar fluid past a moving surface embedding with permeable medium. The flow profiles are considerably boosted by the Darcy-Forchheimer inertial drag associated with Joule and Darcy dissipation. Because of the magnetic interaction as well as the permeability of the medium it is not wise to neglect these effects. The application of the current study is vital in the production process of certain materials in industries, applied engineering vis-a-vis biomedical areas such as drug delivery system, peristaltic pumping, etc. The use of standard transformation rules for the stream function the profiles are get rid into ordinary nonlinear from the system of nonlinear partial. Further, the system of equations is solved by using approximate analytical technique generally known as ‘Adomian Decomposition Method’. Graphs are used to analyse many constraints affecting the flow phenomena, and tabular data is used to investigate on the simulated findings for the rate coefficients. Further, the important outcomes are; the non-Newtonian parameter favours in enhancing the fluid velocity in comparison to the Newtonian fluid whereas the inertial drag retards it significantly.

Illustration of free convection tangent hyperbolic fluid in conjunction with magnetic field within a microchannel

This article investigates the physical interpretation of the magnetic field on the free convection of hyperbolic tangent fluid within a microchannel. The radiating heat through the permeable surface, dissipative heat and the additional heat source experience an important role. The interaction of magnetic dissipation and Darcy-dissipation can be neglected as the flow is supposed to be affected by including a magnetic field and the medium’s permeability. The dimensional form of the designed model is converted to its corresponding non-dimensional form for the use of the similarity variables. Further, these are handled using an approximate analytical technique known as the Adomian decomposition method. The physical behaviour of certain characterizing parameters on the non-Newtonian flow phenomena is exhibited and displayed graphically, and the tabular form is used for the numerical simulation of the rate coefficients.

Survival probability determination of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

An approximate analytical technique based on a combination of statistical linearization and stochastic averaging is developed for determining the survival probability of stochastically excited nonlinear/hysteretic oscillators with fractional derivative elements. Specifically, approximate closed form expressions are derived for the oscillator non-stationary marginal, transition, and joint response amplitude probability density functions (PDF) and, ultimately, for the time-dependent oscillator survival probability. Notably, the technique can treat a wide range of nonlinear/hysteretic response behaviors and can account even for evolutionary excitation power spectra with time-dependent frequency content. Further, the corresponding computational cost is kept at a minimum level since it relates, in essence, only to the numerical integration of a deterministic nonlinear differential equation governing approximately the evolution in time of the oscillator response variance. Overall, the developed technique can be construed as an extension of earlier efforts in the literature to account for fractional derivative terms in the equation of motion. The numerical examples include a hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative terms. The accuracy degree of the technique is assessed by comparisons with pertinent Monte Carlo simulation data.

Response of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitations: A Path Integral approach based on Laplace’s method of integration

In this paper, an approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional element and subjected to evolutionary excitations. This is achieved by a novel formulation of the Path Integral (PI) approach. Specifically, a stochastic averaging/linearization treatment of the original fractional order governing equation of motion yields a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is the Chapman–Kolmogorov (CK) equation governing the evolution in time of the non-stationary response amplitude PDF. Next, the PI technique is employed, which is based on a discretized version of the CK equation solved in short time steps. This is done relying on the Laplace’s method of integration which yields an approximate analytical solution of the integral involved in the CK equation. In this manner, the repetitive integrations generally required in the classical numerical implementation of the procedure are avoided. Thus, the non-stationary response amplitude PDF is approximately determined in closed-form in a computationally efficient manner. Notably, the technique can also account for arbitrary excitation evolutionary power spectrum forms, even of the non-separable kind. Applications to oscillators with Van der Pol and Duffing type nonlinear restoring force models, and Preisach hysteretic models, are presented. Appropriate comparisons with Monte Carlo simulation data are shown, demonstrating the efficiency and accuracy of the proposed approach.