Semi-analytical Solution Research Articles

A novel solution for vertical borehole heat exchangers in multilayered ground with the Robin boundary condition

The ground surface condition is crucial in the thermal analysis of borehole heat exchange (BHE) systems, as it directly influences the thermal regime within the subsurface. However, most existing analytical solutions assume a constant temperature at the ground surface equal to the undisturbed ground temperature, overlooking the impact of various anthropogenic and environmental factors that alter ground surface conditions. This study addresses this gap by proposing a new semi-analytical solution that incorporates the Robin boundary condition for BHEs in multilayered ground. The solution, developed using the unsteady surface element method, balances computational efficiency with analytical precision. The proposed solution was verified with an exact solution in a homogeneous body, showing a maximum error of approximately 0.2 %. Compared with prescribing a constant temperature at the ground surface, using a Robin boundary condition better captures the heat exchange between the ground and the surrounding environment. The results show that variations in the heat transfer coefficient at the ground surface can cause differences in the mean borehole wall temperature change of up to 7.45 % over about 31.7 years (109 s) for a 70-metre borehole. Meanwhile, a unit heat flux at the ground surface can raise the mean borehole wall temperature by 0.56 °C for a heat transfer coefficient of 0.25 Wm−2K−1 over 31.7 years. Additionally, the circulating fluid can transmit the impact of the ground surface condition to greater depths. The results indicate that even at the bottom of a 70-metre borehole, using a fixed temperature can underestimate the temperature change by up to 6.11 % over 31.7 years compared with using the Robin boundary condition. These findings emphasize the indispensability of the proposed solution for accurately predicting the thermal performance of BHEs.

A semi-analytical x-space solution for parton evolution — Application to non-singlet and singlet DGLAP equation

We present a novel semi-analytical method for parton evolution. It is based on constructing a family of analytic functions spanning x-space which is closed under the considered evolution equation. Using these functions as a basis, the original integro-differential evolution equation transforms into a system of coupled ordinary differential equations, which can be solved numerically by restriction to a suitably chosen finite subsystem. The evolved distributions are obtained as analytic functions in x with numerically obtained coefficients, providing insight into the analytic behavior of the evolved parton distributions. As a proof-of-principle, we apply our method to the leading order non-singlet and singlet DGLAP equation. Comparing our results to traditional Mellin-space methods, we find good agreement. The method is implemented in the code POMPOM in Mathematica as well as in Python.

Mathematical analysis of batch reactor performance for the enzymatic synthesis of cephalexin: Laplace Homotopy perturbation method and Adomian decomposition method

In this article, a mathematical model of diffusion reaction in kinetically controlled cephalexin synthesis in the batch reactor with penicillin acylase immobilized in glyoxyl-agarose is analyzed. The kinetic model is a non-linear non-stead-state reaction–diffusion equation with non-linear terms related to the Fick’s law. We have presented the approximate analytical expression for the non-steady-state non-linear reaction–diffusion equation by utilizing the Laplace homotopy perturbation method (LHPM) and for steady-state by using LHPM and Adomian decomposition method (ADM). The approximate analytical solution obtained from these methods proved that they are fit for every values of the reaction–diffusion and kinetic parameters. We also present the numerical solution of the considered reaction–diffusion equation by using pdepe tool in MATLAB software. When comparing the semi-analytical solution with the numerical solution, a satisfactory result is noted for all the possible values of the parameters. The closed-form analytical expressions for the effectiveness factor in non-steady and steady-state conditions are obtained and analyzed using LHPM and ADM.

Symmetry-Optimized Dynamical Analysis of Optical Soliton Patterns in the Flexibly Supported Euler–Bernoulli Beam Equation: A Semi-Analytical Solution Approach

This study employs spatial optimization principles to investigate the nonlinear vibration of a flexibly supported Euler–Bernoulli beam, a (1 + 1)-dimensional system subjected to axial loads. The modified Khater method, a crucial tool in mechanical engineering, is utilized to analyze analytical solutions, which include a symmetric spatial representation of the waveform as an integral part of each solution. Notably, periodic soliton solutions for the nonlinear model closely align with numerical and approximate analytical solutions, demonstrating the accuracy of our modeling approach. Density diagrams, contour diagrams, and Poincaré maps depicting the obtained analytical solutions are presented to elucidate their accuracy and provide visual confirmation of the optimized engineering model’s physical significance. The planar dynamical system is derived through the Galilean transformation by employing mathematical models and appropriate parameter values, thereby further refining problem understanding. Sensitivity analysis is conducted, and phase portraits with equilibrium points are illustrated by analyzing a special case of the investigated dynamical system, emphasizing its symmetrical properties. Lastly, we perform a global analysis to identify periodic, quasi-periodic, and chaotic behaviors, with an extra weak forcing term confirmed by Poincaré maps and a two-dimensional symmetric basin of the largest Lyapunov exponent.

Electro-Osmotic Flow and Mass Transfer through a Rough Microchannel with a Modulated Charged Surface.

In this paper, we investigate the electro-osmotic flow (EOF) and mass transfer of a Newtonian fluid propelled by a pressure gradient and alternating current (AC) electric field in a parallel microchannel with sinusoidal roughness and modulated charged surfaces. The two-wall roughness is described by in-phase or out-of-phase sine functions with a small amplitude δ. By employing the method of perturbation expansion, the semi-analytical solutions of the Poisson-Boltzmann (P-B) equation based on the Debye-Hückel approximation and the modified Navier-Stokes (N-S) equation are obtained. The numerical solution of the concentration equation is obtained by the finite difference method. The effects of sinusoidal roughness, modulated charged surface, and the AC electric field on the potential field, velocity field, and concentration field are discussed. Under the influence of the modulated charged surface and sinusoidal roughness, vortices are generated. The velocity oscillates due to the effect of the AC electric field. The results indicate that solute diffusion becomes enhanced when the oscillation Reynolds number is below a specific critical value, and it slows down when the oscillation Reynolds number exceeds this critical value.

A semi-analytical solution in time domain for evaluating the nonlinear normal modes of a cantilever beam with a tip nonlinearity

A semi-analytical solution in time domain for evaluating the nonlinear normal modes of a cantilever beam with a tip nonlinearity

Legendre-Galerkin spectral algorithm for fractional-order BVPs: Application to the Bagley-Torvik equation

<p>Herein, we provide an efficient spectral Galerkin algorithm, according to a special type of shifted Legendre basis for finding a semi-analytic solution to the Liouville-Caputo fractional boundary value problem. The algorithm’s main goal is to transform the fractional differential problem into a linear system with efficiently invertible, well-structured matrices. The convergence rates of the algorithm are carefully obtained as well as the error bound.</p>

Hydrodynamic performance of a pile-supported oscillating water column breakwater in front of a partially reflecting seawall

Integrating wave energy devices with breakwaters can offer an innovative and sustainable approach by combining wave power extraction with wave attenuation. The performance of this integrated system in offshore areas is influenced by the unique characteristics of the coastline. In this paper, a semi-analytical solution was developed using the matching eigenfunction method for the oscillating water column device integrated into a pile-supported breakwater in front of a partially reflective seawall. The model was validated through the energy conservation law, the Haskind relationship, and experimental data. Detailed examinations were conducted on the effects of the seawall's reflection coefficients, the distance between the system and the seawall, the wall draft, and the chamber breadth on hydrodynamic performance. Results show that the presence of the seawall significantly influences hydrodynamic coefficients (hydrodynamic efficiency, reflection coefficient, the relative transmitted amplitude, etc.), accompanied by the piston and sloshing mode resonances inside the chamber and the confined area between the system and the seawall. Due to energy dissipated by a partially reflective seawall, the magnitude of those hydrodynamic coefficients is mitigated, together with the piston and sloshing mode resonances inside the air chamber. The cancellation of the sloshing mode resonance inside the confined area is observed for the smaller seawall's reflection coefficient. The maximum and minimum hydrodynamic efficiency occur when the system is arranged at the wave nodes and antinodes of the formed standing wave field. Lower wave reflection and better wave power extraction can be achieved by properly adjusting the chamber drafts and breadths.

Thermally developing streaming potential-mediated pressure-driven flow of Phan–Thien–Tanner fluid in a microchannel

In this study, we have conducted a semi-analytical investigation into the streaming potential-mediated pressure-driven flow of hydrodynamically fully developed and thermally developing viscoelastic fluid in a parallel plate microchannel. We have utilized a simplified Phan–Thien–Tanner model to describe the rheology of the viscoelastic fluid. Our approach delved into the full Poisson–Boltzmann equation, deriving exact analytical solutions for the electrostatic potential distribution and velocity profile. Furthermore, we concurrently derived semi-analytical solutions for the temperature distribution and Nusselt number, accounting for the effects of heat generation from viscous dissipation and Joule heating in thermally developing flows. We have demonstrated that an increase in the degree of surface charge triggers the streaming potential field, while the volumetric flow rate escalates with the viscoelastic parameter εWik¯2. Moreover, we have observed that the magnitude of the dimensionless temperature decreases with increasing values of the effective Joule heating parameter Speff. Our analysis reveals that the streaming potential effect hampers fluid flow, resulting in an increase in the bulk fluid temperature and consequently reducing the heat transfer rate. We observe that the magnitude of the Nusselt number decreases with increasing Speff. The entropy generation analysis reveals that increasing the Peclet number amplifies flow and temperature gradients, leading to higher fluid irreversibility in microchannels. The Bejan number experiences a significant decrease across the channel, reaching its minimum at a specific axial location before stabilizing further downstream. We find that heat transfer irreversibility predominantly influences system irreversibility, except for the Brinkmann number Br = 0.01, where convective heat transfer dominates at the entrance region, transitioning to friction losses beyond the thermal entry zone.

الحلول شبه التحليلية لمعادلات فيشر الكسرية للزمن عبر الطريقة التكرارية الجديدة

إحدى الطرق الفعالة لحل المعادلات التفاضلية الجزئية غير الخطية ذات المشتقات الكسرية هي الطريقة التكرارية لتحويل سومودو الجديدة (NSTIM). إنه يبرع في حل الألغاز الرياضية الصعبة ويقدم معلومات ثاقبة حول سلوك معادلات فيشر ذات الكسر الزمني. الطريقة، التي تستخدم مشتقات كابوتو الحسية وWolfram في Mathematica، موثوقة وسهلة الاستخدام وتعطي تصويرًا مرئيًا للحل. أظهرت النتائج التحليلية أن الطريقة المقترحة فعالة وبسيطة في توليد حلول دقيقة لمعادلات فيشر الكسرية للزمن. أصبحت النتائج أكثر موثوقية وقابلة للتطبيق من خلال تضمين مشتقات كابوتو الحسية. تعتمد النمذجة الرياضية على فعالية وبساطة منهج NSTIM لحل معادلات فيشر ذات الكسر الزمني لأنها تتيح حلولاً دقيقة دون استخدام الكثير من قوة المعالجة. يعد نهج NSTIM أداة مفيدة للباحثين في مجموعة متنوعة من المجالات لأنه يوفر أيضًا إطارًا مرنًا يمكن تعديله بسهولة مع المعادلات التفاضلية الكسرية الأخرى. أصبح من الممكن الآن فحص ديناميكيات وسلوك الأنظمة المعقدة التي تحكمها معادلات فيشر الكسرية الزمنية بكفاءة وموثوقية، مما يفتح طرقًا بحثية جديدة. إن القدرة على حل معادلات فيشر ذات الكسور الزمنية بكفاءة وموثوقية باستخدام نهج NSTIM لها آثار مهمة على مجالات مختلفة مثل الديناميات السكانية والبيولوجيا الرياضية وعلم الأوبئة. يمكن للباحثين الآن تحليل انتشار الأمراض أو دراسة الديناميكيات السكانية للأنواع بدقة أعلى وجهد حسابي أقل. يمهد هذا التقدم في حل المعادلات التفاضلية الكسرية الطريق لرؤى أعمق حول سلوك وأنماط الأنظمة المعقدة، مما يؤدي في نهاية المطاف إلى تعزيز الفهم العلمي وتقديم إمكانيات جديدة للتطبيقات العملية.

Temperature-reducing shocks in optically thin radiative MHD—Analytical and numerical results

Shocks are often invoked as heating mechanisms in astrophysical systems, with both adiabatic compression and dissipative heating that leading to the increase in temperature. While shocks are reasonably well understood for ideal magnetohydrodynamic (MHD) systems, in many astrophysical plasmas, radiation is an important phenomena, which can allow energy to leave the system. As such, energy becomes non-conservative, which can fundamentally change the behavior of shocks. The energy emitted through optically thin radiation post-shock can exceed the thermal energy increase, resulting in shocks that reduce the temperature of the medium, i.e., cooling shocks that have a net decrease in temperature across the interface. In this paper, semi-analytical solutions for radiative shocks are derived to demonstrate that both cooling (temperature decreasing) and heating (temperature increasing) shock solutions are possible across the whole temperature range in optically thin radiative MHD. Numerical simulations of magnetic reconnection for solar-like temperatures and plasma-β with optically thin radiative losses also yield both heating and cooling shocks in roughly equal abundances. The detected cooling shocks feature a significantly lower pressure jump across the shock than their heating counterparts. The compression at the shock front leads to locally enhanced radiative losses, resulting in significant cooling within a few grid cells in the upstream and downstream directions. The presence of temperature-reducing (cooling) shocks is critical in determining the thermal evolution, and heating or cooling, across a wealth of radiative astrophysical plasmas including magnetic reconnection in the solar corona.

A reliable semi-analytic solution for the Caputo fractional human liver model

A reliable semi-analytic solution for the Caputo fractional human liver model

A hybrid modeling approach: dynamic simulation of two-phase fluid flow in compacting gas condensate reservoirs using potential flow theory

Simulating the flow processes of complex hydrocarbon mixtures, such as gas condensate mixtures and volatile oils, in deep reservoirs is one of the most challenging problems in reservoir hydrodynamics. The challenge lies in meeting three requirements simultaneously when creating a simulator: high accuracy, low computational cost, and high reliability. These metrics determine the performance of the simulation. Meeting all these requirements at the same time necessitates a hybrid approach to solving the problem and optimizing the computational algorithm in terms of CPU time. A new technique has been developed for hybrid modeling of the development of deposits of complex hydrocarbon mixtures. This technique integrates the theory of potential flow, the Binary Flow Model (which accounts for rock compaction, PVT properties of reservoir fluids, phase transformation, and mass transfer between phases), and the material balance equations of the hydrocarbon system using time discretization. In this case, to linearize nonlinear differential equations for the flow of a gascondensate mixture, the method of averaging reservoir pressure along the radial coordinate was used. By introducing the Khrestianovich function, an algorithm was obtained to determine the rate of inflow of the gas-condensate mixture to the well. Using material balance equations for gas and condensate, it was possible to obtain differential equations that describe changes in reservoir pressure and condensate saturation over time. Based on this technique, a simulator for a gas condensate reservoir was created, enabling computer studies to be conducted. The results demonstrated the proposed technique's good accuracy compared with the results of the semi-analytical solution. Keywords: dynamic modeling; simulation; monitoring; streamline; hybrid modeling; binary model.

Hydrodynamic performance of multi-chamber oscillating water columns in a caisson array

A theoretical model based on the linear potential flow theory and eigenfunction expansion-matching method is developed to analyze the interaction between water waves and multi-chamber oscillating water columns (OWCs) embedded in a caisson array. The semi-analytical solution consists of the diffraction and radiation components of periodic OWCs. The velocity singularity at the tip of the OWC chamber walls is resolved by implementing a Chebyshev polynomial expansion. The unknown expansion coefficients are determined by the continuity equations of velocity and pressure at the interface of subdomains. The semi-analytical solution is verified using the Haskind relation and the conservation law of wave energy flux. It is found that multi-chamber OWCs have higher hydrodynamic efficiency over a broader frequency bandwidth than traditional single- and dual-chamber OWCs. The maximum hydrodynamic efficiency η increases with increasing of chamber number (J), from 0.5 (J = 1) to nearly 1.0 (J = 8 and 12). It is also found that a larger incident wave angle leads to a narrower frequency bandwidth for energy capturing. Increasing the incident wave angle from π/3 to 5π/12 results in a decrease of the first cutoff frequency kc from 2.41 to 2.02, and therefore results in the energy capture bandwidth narrowing accordingly. Both constructive and destructive hydrodynamic interactions between caissons array and OWCs are observed over the tested frequency range. Interestingly, when configuration of the OWC chambers is reversed, the transmission coefficient remains unchanged.

Vertical matrix advection dominates transient anomalous diffusion within fracture-matrix systems using a modified diffusion model

Pollutant transport in fractured media has been well analyzed using process-based diffusion models and concise, time-nonlocal models like the tempered fractional derivative (TFD) model. While traditional diffusion models are limited to specific transport processes, TFD model parameters lack hydrogeological meaning. In addressing these knowledge gaps, a modified diffusion model is proposed to augment the traditional diffusion model and offer hydrogeological interpretations for TFD parameters. This modified diffusion model incorporates transverse advection in low-permeability matrices, allowing for the quantification of transient anomalous diffusion in a single fracture-matrix system. This model offers a semi-analytical solution, systematically validated for its applicability. Memory function analysis establishes a quantitative connection between the modified diffusion model and the TFD model, uncovering a pivotal discovery in this study. Parameter sensitivity analysis further shows that the matrix characteristic rate defined as a function of (Vm)2/(4Dm), where Vm is the transverse matrix velocity and Dm is the transverse matrix diffusion coefficient, controls the temporal evolution of anomalous sub-diffusion in the fracture. Applications are performed to check the feasibility of the modified diffusion model. The developed modified diffusion model sheds light on quantifying complex transport in fracture-matrix systems and enhances the predictability of nonlocal transport models.

Semi-Analytical Solution for a Single-Phase Thermal Conduction Problem of a Hollow Cylinder with a Growing or Receding Inner Radius

Abstract A semi-analytical solution for the thermal conduction of a single phase, homogeneous, circular, hollow-cylinder with a growing or receding inner radius at a constant rate under unit-loading was derived using conformal mapping, Laplace transformation, and a Zakian series representation of the inverse Laplace transform. All solutions allow for convection on the fixed outer radius. Predictions were compared to finite-element simulations with excellent agreement observed. Given the changing thickness, thermal transients could not reach true steady-state equilibrium, especially for faster growth or recession rates. Indeed, the temperature states become somewhat linear with respect to time, reflecting the constant velocity of growth or recession. In practice, the resulting solutions can be used to determine temperatures during machining, wear, erosion, corrosion, and/or additive manufacturing, especially for lower temperature solid-state methods such as cold-spray.

Semi-analytical solution of MHD free convective flow of a nanofluid over an exponentially stretching porous sheet in the presence of heat source/sink

In this study, the magnetohydrodynamic flow and heat transfer of electrically conducting nanofluid over an exponentially stretching porous sheet is investigated. By introducing similarity variables, the non-linear set of partial differential governing equations, are transformed into a system of ordinary nonlinear differential equations. The equations are solved semi-analytically by the Optimal Collocation Method (OCM) which is a powerful method to solve equations containing infinite boundary conditions. The accuracy of the OCM results is examined by the Richardson method. The effects of the different physical parameters such as magnetic field strength, nanofluid volume fraction, suction strength, and the heat sink/source value are discussed on the flow and heat transfer characteristics of the problem. The obtained results show that the velocity and temperature of nanofluid increase with the increase in volume fraction. Also, when suction increases, the thickness of the boundary layers decreases. As the strength of the magnetic field (Hartmann) increases, the temperature slightly increases but the flow rate of the nanofluid decreases. The skin friction coefficient and Nusselt number increase when the values of suction parameters, volume fraction, magnetic field, and the field angle increase.

Response of an Oscillating Water Column spanning a circle sector and embedded in a circular platform

This paper develops a semi-analytical solution of a water waves problem concerning the interaction between linear waves and an Oscillating Water Column (OWC) embedded into a circular platform. Unlike similar studies available in the literature, the proposed investigation concerns a realistic situation involving the use of an OWC having limited angular width. Such a configuration is likely to be employed in the arrangement of platforms in which the use of a large OWC is neither efficient nor feasible.The proposed solution is developed in the framework of the linear potential flow theory. The solution relies on the use of a domain decomposition approach involving eigen-function expansions of velocity potentials with unknown coefficients, which are calculated via a matching technique. The solution is utilized for conducting relevant parameter studies concerning the effects of the main geometrical parameters on the overall system performance and on the system hydrodynamic parameters.

Study on the initiation criterion for double symmetric radial perforations emanating from circular hole under high water pressure considering T stress

Hydraulic fracturing is common technique to break the rock mass down under high water pressure. In this paper, the tensile failure of double symmetric radial perforations emanating from circular hole under far-field compression and high water pressure is studied. And the existing theoretical model does not consider the unequal water pressure applied on the surfaces of perforation and circular hole, it also neglects the effect of T stress, so the maximum tangential stress (MTS) criterion could be revised with considering T stress. Firstly, considering the unequal water pressure on the perforation surfaces and circular hole, the stress intensity factors (SIFs) are derived with conformal mapping method and compared with the semi-analytical solution. Secondly, the terms of T stress are derived by analogy to the open-crack under high water pressure. On this basis, the revised MTS criterion is obtained, and its validity is verified with model tests. Finally, the effects of the lateral pressure coefficient k, minimum horizontal principal stress σh(σh/σH remains constant), radius of circular hole a, hydraulic coefficient λ and radius of process zone rc on the initiation angle −θc, critical water pressure Pc and SIFs are analyzed respectively. It is found that −θc always increases firstly and then decreases with increasing the inclination angle α, whereas Pc firstly decreases and then increases as a/(c-a) varies from 0 to 1 and increases as α increases in the range of a/(c-a) = 1∼+∞. Then −θc increases with increasing σh and k, whereas Pc can intersect each other with increasing k and increases with increasing σh. As c-a remains constant, −θc increases firstly and then decreases with increasing the ratio a/c, whereas Pc keeps decreasing. Also −θc and Pc increase with decreasing λ. As rc increases, −θc decreases and Pc increases. As for SIFs, when α increases, KI decreases and KII increases firstly, then decreases. As α remains constant, KI increases firstly, then decreases with increasing k and KII keeps increasing. In the range of a/(c-a) = 0 ∼ 1, KI decreases with increasing a, and KII keeps increasing. Within the range of a/(c-a) = 1∼+∞, KI increases firstly, then decreases with increasing a, the increment of KII is not obvious. And KI decreases with decreasing λ.

Computational approach and convergence analysis for interval-based solution of the Benjamin–Bona–Mahony equation with imprecise parameters

PurposeTo provide a new semi-analytical solution for the nonlinear Benjamin–Bona–Mahony (BBM) equation in the form of a convergent series. The results obtained through HPTM for BBM are compared with those obtained using the Sine-Gordon Expansion Method (SGEM) and the exact solution. We consider the initial condition as uncertain, represented in terms of an interval then investigate the solution of the interval Benjamin–Bona–Mahony (iBBM).Design/methodology/approachWe employ the Homotopy Perturbation Transform Method (HPTM) to derive the series solution for the BBM equation. Furthermore, the iBBM equation is solved using HPTM to the initial condition has been considered as an interval number as the coefficient of it depends on several parameters and provides lower and upper interval solutions for iBBM.FindingsThe obtained numerical results provide accurate solutions, as demonstrated in the figures. The numerical results are evaluated to the precise solutions and found to be in good agreement. Further, the initial condition has been considered as an interval number as the coefficient of it depends on several parameters. To enhance the clarity, we depict our solutions using 3D graphics and interval solution plots generated using MATLAB.Originality/valueA new semi-analytical convergent series-type solution has been found for nonlinear BBM and interval BBM equations with the help of the semi-analytical technique HPTM.