Convergence Of Series Solution Research Articles

Impacts of variable magnetic field on ternary Casson nanofluid flow through ciliated arterial walls incorporating interfacial nanolayer

ABSTRACT The current investigation explores tri-hybrid mediated blood flow through a ciliary annular model, designed to emulate an endoscopic environment. The human circulatory system, driven by the metachronal ciliary waves, is examined in this study to understand how ternary nanoparticles influence wave-like flow dynamics in the presence of interfacial nanolayers. We also analyze the effect of an induced magnetic field on Ag–Cu– A l 2 O 3 /blood flow within the annulus, focusing on thermal radiation, heat sources, buoyancy forces and ciliary motion. The Casson fluid model characterizes the non-Newtonian viscous properties of the biofluid. To describe the steady fluid flow mathematically, we use coupled partial differential equations and apply the homotopy perturbation method to derive rapidly convergent series solutions for the non-linear flow equations. The obtained hemodynamic consequences are graphically represented with the variations of emerging parameters. These are significantly influenced by the rheological factors of the nanofluid flow, improving flow velocity with changes in shear viscosity, while a decrease in flow is observed for intensified Lorentz forces. Ciliary motion accelerates the expansion of the induced magnetic field on nanolayers, while a higher Magnetic Reynolds number decreases the current density distribution. Increased radiative heat generation lowers the temperature, indicating that thermal radiation enhances heat transfer and improves cooling efficiency. In contrast, an increased ciliary length along the wall raises the temperature due to wave-like motion, which strengthens the thermal boundary layer in the fluid flow. Additionally, a higher nanoparticle concentration increases wall shear stress due to frictional forces, while enhanced magnetic forces decrease the shear stress along the ciliary wall. Furthermore, a higher Strommer’s number may regulate the formation of blood boluses in the wavy flow. The key findings play an important role in the development of analytical benchmarks to validate computational methods, ensuring accuracy in clinical research tools and supporting reliable medical applications.

Neutrosophic Approaches to Soliton Solutions for Nonlinear Time-Fractional Coupled Jaulent–Miodek System Using a Modified Laplace Adomian Dec omposition Method

This paper presents a modified Laplace Adomian decomposition method (MLADM) to solve the nonlinear time-fractional coupled Jaulent–Miodek system. The proposed approach provides convergent series solutions with easily computable components, demonstrating both accuracy and simplicity in its application. By employing the Caputo fractional derivative, this study establishes a robust framework for analyzing nonlinear behavior in fractional differential equations. The effectiveness of the method is validated through comparisons with previous studies, with results illustrated using graphical representations. The solutions proposed herein are significant for modeling complex and dynamic real-world phenomena across various scientific disciplines. All computations and graphical results were carried out using Mathematica, emphasizing the method’s reliability, precision, and ease of application to nonlinear fractional systems. The study of fractional nonlinear systems is crucial for modeling complex, dynamic, and uncertain processes, which are core aspects of neutrosophic science. By addressing the intricate behavior of the nonlinear time-fractional coupled Jaulent–Miodek system, this work advances mathematical models that encapsulate uncertainty, indeterminacy, and complex interactions. Such an alignment with the principles of neutrosophic science underscores the relevance of our approach to the objectives of the International Journal of Neutrosophic Science, highlighting its potential to enhance the understanding and practical applications of complex systems.

Numerical investigation of two-dimensional fractional Helmholtz equation using Aboodh transform scheme

PurposeThis paper aims to present a numerical investigation for two-dimensional fractional Helmholtz equation using the Aboodh integral homotopy perturbation transform scheme (AIHPTS).Design/methodology/approachThe proposed scheme combines the Aboodh integral transform and the homotopy perturbation scheme (HPS). This strategy is based on an updated form of Taylor’s series that yields a convergent series solution. This study analyzes the fractional derivatives in the context of Caputo.FindingsThis study illustrates two numerical examples and calculates their approximate results using AIHPTS. The derived findings are also presented in tabular form and graphical representations.Research limitations/implicationsIn addition, He’s polynomials are calculated using HPS, so the minimal computational outcome is a defining feature of this method and gives a competitive advantage over other series solution techniques.Originality/valueNumerical data and graphical illustrations for different fractional order levels confirm the proposed method’s successful performance. The results show that the proposed approach is speedy and straightforward to execute on fractional-ordered models.

Auto-Bäcklund transformation and exact solutions for a new integrable (2+1)-dimensional shallow water wave equation

Shallow water waves (SWWs) are often used to describe water flow and wave movement in shallow water areas. The article introduces a novel (2 + 1)-dimensional SWW equation. We prove that the equation is integrable and obtain an auto-Bäcklund transformation by truncating Painlevé expansion. Using the bilinear form of the equation, a new auto-Bäcklund transformation and some exact solutions are obtained. Besides, a convergent power series solution is derived using Lie symmetry method. These exact solutions can enrich mathematical modeling and help us understand nonlinear wave phenomena. Finally, conserved vectors are derived.

Cattaneo-Christov fluxes for nonlinear mixed convective entropy generated Reiner-Rivlin material flow

Thermal and solutal transportation processes involve in tremendous useful applications related to heat exchanger, scientific, micro-electronic device technologies and industry sectors like nuclear reactors cooling, marine engineering, nanotechnology, thin-film technology, biomedical applications and thermal conduction in soft tissue. In view of such useful applications the Cattaneo-Christov analysis for nonlinear mixed convective Reiner-Rivlin material flow in presence of constant applied magnetic field is considered. Heat source, magnetohydrodynamics and radiation are incorporated in energy equation. Cattaneo-Christov flux is employed to discuss the thermal and mass transport characteristics. Entropy calculation in radiating flow is discussed. The proposed non-linear systems are transformed into dimensionless ordinary systems through appropriate transformation. Non-linear ordinary expressions are computed for convergent series solutions through Optimal homotopy analysis technique (OHAM). Graphical interpretations for quantities under interest against pertinent variables are explored. Large magnetic parameter has opposite impact for entropy and fluid flow. An increase of temperature through thermal relaxation time parameter is noticed. Velocity has same trend for both mixed convection and Reiner-Rivlin fluid variables. Reduction in concentration is seen for solutal relaxation time parameter. Temperature enhancement is witnessed in presence of heat generation. Entropy rate enhancement for radiation is observed. Decrease in concentration is noticed for reaction. Entropy rate increases against diffusion variable is enhanced.

Nonlinear oscillators dynamics using optimal and modified homotopy perturbation method

This study investigates the several strongly nonlinear oscillators (Van der Pol, Duffing and Rayleigh) for which we have applied the most powerful and advanced optimal semi-analytical technique: optimal and modified homotopy perturbation method (OM-HPM) for the convergent semi-analytical series solution. The numerical simulation demonstrates the high accuracy of the OM-HPM, which is straightforward, does not require any domain decomposition, special transformation, or pade approximations to get the convergent series solution. The key features for the high accuracy of the OM-HPM lies on the best optimal auxiliary linear operator. Therefore, OM-HPM offering a valuable tool for engineers and researcher to analyze the complex nonlinear oscillator.

Melting heat transfer of a quadratic stratified Jeffrey nanofluid flow with inclined magnetic field and thermophoresis

Recently, researchers are facing great challenges to form the homogeneous solutions of various liquids at biomedical and industrial levels. Such homogeneous solutions can be controlled through stratification phenomenon directly which is ignored by the researchers in the existing literature specifically quadratic stratification and melting heat mechanism which is valid only for higher temperature processes. Thus, to form the homogeneous and stable liquid solutions for various industrial processes, we investigate the melting heat phenomenon in quadratic stratified Jeffery Nano fluid flow deformed due to linearly stretching sheet along with stagnation point. Inclined constant magnetic field is implemented on the fluid through an arbitrary angel. Characteristics of heat and transportations are disclosed through viscous dissipation, Brownian motion and thermphoresis. Temperature and concentration of the surrounding fluids are assumed higher than the stretching surface. The resultant governing equations are reduced to ordinary differential equations through proper transformations. Convergent analytical series solutions are computed via homotopic approach. Impact of various emerging parameters on temperature, velocity and concentration fields are illustrated and discussed comprehensively. Sherwood and Nusselt numbers and drag force are scrutinized mathematically, physically and graphically. It is analyzed from the study that (i) dominant melting reduces the temperature field (ii) higher thermal and solutal stratification decay the temperature and concentration fields respectively (iii) higher thermophoresis enhances the temperature field. Further, it is concluded that stable and homogeneous solutions may be made by increasing melting and reducing stratification phenomena. This study will help us to provide actual amount of heat for heating processes in industries and biomedicine which is the basic requirement for excellent quality of products.

On the generalized time fractional reaction–diffusion equation: Lie symmetries, exact solutions and conservation laws

In this paper, Lie symmetry analysis method is applied to the generalized time fractional reaction–diffusion equation. We obtain a conditional symmetric group and some conservation laws of the governing equation. The obtained Lie symmetries are used to reduce the studied fractional partial differential equation to some fractional ordinary differential equations with Riemann–Liouville fractional derivative or Erdélyi-Kober fractional derivative. Furthermore, we obtained asymptotic stable solutions and convergent power series solutions for the reduced equations. The dynamic behavior of these exact solutions is presented graphically.

Comparison of Approximate Analytical and Numerical Solutions of the Allen Cahn Equation

Allen Cahn (AC) equation is highly nonlinear due to the presence of cubic term and also very stiff; therefore, it is not easy to find its exact analytical solution in the closed form. In the present work, an approximate analytical solution of the AC equation has been investigated. Here, we used the variational iteration method (VIM) to find approximate analytical solution for AC equation. The obtained results are compared with the hyperbolic function solution and traveling wave solution. Results are also compared with the numerical solution obtained by using the finite difference method (FDM). Absolute error analysis tables are used to validate the series solution. A convergent series solution obtained by VIM is found to be in a good agreement with the analytical and numerical solutions.

A neural network approach to explore bioelectromagnetics aspects of blood circulation conveying tetra-hybrid nanoparticles and microbes in a ciliary artery with an endoscopy span

Bioelectromagnetics is gaining remarkableness due to its wide-ranging applications in medical and biochemical industries. Our proposed model delves into the bioelectromagnetics aspects of blood circulation in a scenario involving tetra-hybrid nanoparticles and gyrotactic microbes within a ciliated arterial annulus, particularly one mimicking an endoscopic environment. Our model takes into account influential factors, including cilia beating, buoyancy force, heat source, viscous and Joule warming, arterial wall properties, chemical reaction, and bioconvection. Different geometric shapes of suspended tetra-hybrid nanoparticles (spheres, bricks, cylinders, and platelets) are incorporated into the model formation. The close-form solution of the Poisson equation is found in terms of Bessel’s functions. The homotopy perturbation methodology (HPM) is implemented to track down the rapidly convergent series solutions of the highly non-linear coupled flow equations arising. The hemodynamical attributes of distinct evolving factors on the different dimensionless hemodynamic profiles and entities of interest are elucidated evocatively via a sort of graphs and charts. Our results prove that the factors related to electroosmosis have a dual effect on the blood flow pattern. The bloodstream rapidity is alleviated or boosted for the assisting or opposing electroosmosis process. Cooling of blood within the annulus is achieved with a larger cilia length. The Peclet number positively correlates with the gyrotactic microbes’ density. Blood infused with tetra-hybrid nanoparticles exhibits a superior heat transmission rate across the arterial wall compared to others. Additionally, an artificial neural network (ANN) model, developed using HPM-derived data, accurately predicts shearing stress (SS), achieving high accuracy rates of 99.98% in testing and 100% in validation for MB flow. The model’s new findings could be noteworthy for engineers and medical device designers to develop innovative medical devices and technologies based on bioelectromagnetic principles.

Exact and explicit analytical solution for the Sakiadis boundary layer

It has recently been established [Naghshineh et al. IMA J. of Appl. Math., 88, 1 (2023)] that a convergent series solution may be obtained for the Sakiadis boundary layer problem once key parameters are determined iteratively using the series itself. Here, we provide exact and explicit analytical expressions for these parameters, including that associated with wall shear, thus completing the exact analytical solution. The resulting solution to the Sakiadis problem is summarized here for direct use.

A New Method for Solving a Neutral Functional-Differential Equation with Proportional Delays

This study presents and implements a new hybrid technique that combines the Sawi transform (ST) and Homotopy perturbation method (HPM) to solve neutral functional-differential equations with proportional delays. Some of the important properties of the method are established and validated. We start the method by first applying ST to obtain the recurrence relation. We, next, implement HPM to find convergent series solutions of the recurrence relation. The series is free of assumptions and restrictions, highlighting its adaptability and robustness. Moreover, the convergence of the method is established through convincing proof. To demonstrate its effectiveness and applicability, we provide five examples. The method yields accurate approximate solutions, or in some cases exact solutions, with a few number of iterations, reinforcing its reliability and validity. Moreover, the performance of the method is compared with some available methods and demonstrates its superiority and efficiency.

General approach on the best fitted linear operator and basis function for homotopy methods and application to strongly nonlinear oscillators

In this study, we propose a universal mathematical guideline for the selection of auxiliary linear operator L(u) and the basis function u0(x), which are the principal parts of Homotopy methods: Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM). We assume the coefficients of derivatives involved in L(u) as a function of auxiliary roots of L(u)=0. Based on the residual error minimization we compute unknown roots and thereby obtain the optimal linear operator at an order of approximation. Also, an effort is made to explore the role of the number of unknowns embedded into L(u) (for different choices of real roots of L(u)=0) on the rate of converges of the solution and CPU uses. Hence the best fitted linear-operator and corresponding basis function are proposed for a particular problem. Also, we propose to discretize the exact square residual using Simpson’s 1/3 algorithms, and compare its accessibility in comparison to other existing residual calculation methods. Initially, we applied our algorithms to three IVPs (Logistic, Stiff system and Riccati system) and three singular BVPs, and then finally to a strongly nonlinear oscillator (cubic-quintic Duffing equation). We compare our technique’s accuracy, efficiency and simplicity with other contemporary analytical and numerical methods. It is demonstrated that our optimal linear operator, which is updated at every order of approximation, is much more efficient, important (than the artificial parameters and functions of developed HAM), and self-sufficient for the convergence of series solutions. Our approach is more effective, straightforward and easy to use when applied to many nonlinear problems arising in mechanics and engineering. It is seen for the initial value problems that our technique provides the globally convergent series solution, while the solutions by HPM and optimal HAM are locally convergent. This novel development will make homotopy methods more powerful.

Convergent piecewise series solutions of ordinary differential equations

Convergent piecewise series solutions of ordinary differential equations

On the steady-state interfacial waves with two-dimensional type-A double exact resonance

Steady-state interfacial waves under two-dimensional (2D) type-A exact triad resonance and other related resonances are researched in a two-layer liquid model with a free surface in contact with air. Five groups (groups 1–5) of convergent series solutions are achieved via the homotopy analysis method. It is found that the phenomenon of double exact resonance could exist in periodic interfacial waves if physical parameters correspond to the intersection of two exact resonance curves. The double exact resonance considered here contains a 2D type-A triad resonance and an other resonance. Under the 2D type-A exact triad resonance, the other resonant triad could obviously enlarge or reduce the wave amplitudes and energy proportions of primary and resonant components. Nevertheless, other resonant quartet, quintet, sextet, and septet all produce no influence on interfacial waves when the 2D type-A exact triad resonance occurs. The above-mentioned results indicate that in the neighborhood of the double exact triad resonance, small perturbations of wave vector of a primary component can cause huge changes on wave profiles of free surface and interface, wave amplitude spectrum, and energy distribution of internal waves in real ocean. In addition, the closer the interfacial waves are to the double exact triad resonance, the more possible energy combinations exist in the wave system, and the greater the number of steady-state interfacial wave solutions. All of this should deepen our understanding of nonlinear resonance interactions in short-crested internal waves.

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>

The shape of an axisymmetric meniscus in a static liquid pool: effective implementation of the Euler transformation

Abstract We examine the classical problem of the height of a static liquid interface that forms on the outside of a solid vertical cylinder in an unbounded stagnant pool exposed to air. Gravitational and surface tension forces compete to affect the interface shape as characterized by the Bond number. Here, we provide a convergent power series solution for interface shapes that rise above or fall below the horizontal pool as a function of contact angle and Bond number. We find that the power series solution expressed in terms of the radial distance from the wall is divergent, and thus rewrite the divergent series as a new power series expressed as powers of an Euler transformed variable; this series is modified to match the large distance asymptotic behaviour of the meniscus. The Euler transformation maps non-physical singularities to locations that do not restrict series convergence in the physical domain, while the asymptotic modification increases the rate of convergence of the series overall. We demonstrate that when the divergent series coefficients are used to implement the Euler transformation, finite precision errors are incurred, even for a relatively small number of terms. To avoid such errors, the independent variable in the governing differential equation is changed to that of the Euler transform, and the power series is developed directly without using the divergent series. The resulting power series solution is validated by comparison with a numerical solution of the interface shape and the small Bond number matched asymptotic solution for the height of the interface along the cylinder developed by Lo (1983, J. Fluid Mech., 132, p.65-78). The convergent power series expansion has the ability to exceed the accuracy of the matched asymptotic solution for any Bond number given enough terms, and the recursive nature of the solution makes it straightforward to implement.

EDL Induced Electro-magnetized Modified Hybrid Nano-blood Circulation in an Endoscopic Fatty Charged Arterial Indented Tract.

The electrokinetic process for streaming fluids in magnetic environments is emerging due to its immense applications in medical and biochemical industrial domains. In this context, our proposed model seeks to inquire into the hemodynamic characteristics of electro-magnetized blood blended with trihybrid nanoparticles circulation induced by electro-osmotic forces in an endoscopic charged arterial annular indented tract. This steaming model also invokes the consequences of variable Lorentz attractive force, buoyancy force, heat source, viscous and Joule warming, arterial wall properties, and sliding phenomena for featuring more realistic problems in blood flows. Different shapes of suspended trihybrid nanoparticles, such as spheres, bricks, cylinders, and platelets, are included in the model formation. Electro-magnetized modified hybrid nano-blood is an electro-conductive solution comprising blood as base fluid and magnetized trihybrid nanoparticles (copper, gold, and alumina). Closed-form solution in terms of Bessel's functions is gotten for electro-osmotic potential due to the electric double layer (EDL). The homotopy perturbation methodology is implemented in order to track down the convergent series solutions of non-linear coupled flow equations being elicited. The physical attributes of distinct evolving parameters on the different dimensionless hemodynamic profiles and quantities of interest are elucidated evocatively via a sort of graphs and charts. The ancillary outcomes proved that the Debye-Hückel parameter and Helmholtz-Smoluchowski velocity have a dual impact on the ionized bloodstream. The bloodstream rapidity is alleviated/boosted for the assisting/opposing electroosmosis process. Cooling of ionized blood in the endoscopic arterial conduit is achieved with lower Hartmann numbers. Copper-gold-alumina/blood exhibits a superior heat transmission rate across the arterial wall than copper-gold-blood, copper-blood, and pure blood. Additionally, the contour topology for the bloodstream in the flow domain is briefly elaborated. The contour distribution is significantly amended due to the variant of the Debye-Hückel parameter. The model's new findings may be invaluable in electro-magneto-endoscopic operation, electro-magneto-treatment for cancer, surgical process, etc.

Numerical simulation and analysis of Airy's-type equation

Abstract In this article, we propose a novel new iteration method and homotopy perturbation method (HPM) along with the Elzaki transform to compute the analytical and semi-analytical approximations of fractional Airy’s-type partial differential equations (FAPDEs) subjected to specific initial conditions. A convergent series solution form with easily commutable coefficients is used to examine and compare the performance of the suggested methods. Using Maple graphical method analysis, the behavior of the estimated series results at various fractional orders ς \varsigma and its modeling in two-dimensional (2D) and three-dimensional (3D) spaces are compared with actual results. Also, detailed descriptions of the physical and geometric implications of the calculated graphs in 2D and 3D spaces are provided. As a result, the obtained solutions of FAPDEs that are subject to particular initial values quite closely match the exact solutions. In this way, to solve FAPDEs quickly, the proposed approaches are considered to be more accurate and efficient.

Computational Analysis of Fractional-Order KdV Systems in the Sense of the Caputo Operator via a Novel Transform

The main features of scientific efforts in physics and engineering are the development of models for various physical issues and the development of solutions. In order to solve the time-fractional coupled Korteweg–De Vries (KdV) equation, we combine the novel Yang transform, the homotopy perturbation approach, and the Adomian decomposition method in the present investigation. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. The fractional derivative is regarded in the Caputo meaning. These approaches apply straightforward steps through symbolic computation to provide a convergent series solution. Different nonlinear time-fractional KdV systems are used to test the effectiveness of the suggested techniques. The symmetry pattern is a fundamental feature of the KdV equations and the symmetrical aspect of the solution can be seen from the graphical representations. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Additionally, the system’s approximative solution is illustrated graphically. The results show that these techniques are extremely effective, practically applicable for usage in such issues, and adaptable to other nonlinear issues.