Asymptotic Approximation Of Solution Research Articles

Approximate Analytical Solutions for Strongly Coupled Systems of Singularly Perturbed Convection–Diffusion Problems

This work presents a reliable algorithm to obtain approximate analytical solutions for a strongly coupled system of singularly perturbed convection–diffusion problems, which exhibit a boundary layer at one end. The proposed method involves constructing a zero-order asymptotic approximate solution for the original system. This approximation results in the formation of two systems: a boundary layer system with a known analytical solution and a reduced terminal value system, which is solved analytically using an improved residual power series approach. This approach combines the residual power series method with Padé approximation and Laplace transformation, resulting in an approximate analytical solution with higher accuracy compared to the conventional residual power series method. In addition, error estimates are extracted, and illustrative examples are provided to demonstrate the accuracy and effectiveness of the method.

Piecewise approximate analytical solutions of high-order reaction-diffusion singular perturbation problems with boundary and interior layers

<abstract> <p>This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor.</p> </abstract>

Analytical solutions of fractal and anomalous diffusion models for pressure and rate transient analysis

In this study, we examine the modeling of the naturally fractured reservoirs based on the fractal, Metzler, and Raghavan anomalous diffusion models which are different from Darcy flux law. The governing equations of these models are presented for a vertical well producing at either a constant rate (CR) or constant bottomhole pressure (CBHP) production in an infinite-acting reservoir. New analytical and early- and late-time asymptotic approximate solutions were derived for the finite-wellbore problem for the Raghavan anomalous model. The approximate solutions identify the fractal and anomalous diffusion parameters that influence behaviors of the early- and late-time pressure and rate transient data. The method of Laplace transformation is used to find the solutions in dimensionless forms which are then inverted to the real-time domain using a numerical inversion algorithm. In addition, we compare the solutions for the fractal, Metzler anomalous, and Raghavan anomalous models and delineate the similarities and differences among these solutions. We show the effect of fractal parameters on diffusion models, the relationship between the fractal model, and Raghavan and Metzler anomalous models, in terms of dimensionless pressure and dimensionless pressure derivative responses in the reservoir. We show that the “dimensionless” pressure defined by Raghavan is not a dimensionless quantity unless the Euclidean dimension d=2 (or for a fractal structure with the mass fractal dimension df=2). To the best of our knowledge, in the literature, there exists no study providing a detailed review and comparing the applicability of each diffusion model on real field well-test data as well as providing new analytical solutions and early- and late-time asymptotic approximate solutions for the anomalous diffusion for both the CR and CBHP cases for the finite-wellbore problem. Hence, this study fills this gap. In addition, we apply the three diffusion models to an interference well test data from the Kizildere geothermal reservoir in Turkey by using the ensemble smoother with multiple data assimilation (ES-MDA) method. The results show that the fractal model in terms of the goodness of fit (root-mean-square error – RMSE) best matches the field test data.

ПРИБЛИЖЕННЫЕ АСИМПТОТИЧЕСКИЕ ВЫРАЖЕНИЯ ЭЛЕКТРОМАГНИТНОГО ПОЛЯ ИСТОЧНИКОВ ВБЛИЗИ ГЛАДКОЙ ВЫПУКЛОЙ ПРОВОДЯЩЕЙ ПОВЕРХНОСТИ ВРАЩЕНИЯ

Approximate asymptotic solution is developed for the electromagnetic field due of a surface magnetic and electric currents located near an arbitrary smooth convex conducting surface of rotation of large size. The expressions of field are valid in a near-surface layer with a thickness of the order of the wavelength and have the form of a sum of a series of azimuthal harmonics and an integral over a continuous spectrum of eigenfunctions of the field. The coefficients of the series are expressed by integrals from the product of a given distribution of sources with eigenfunctions. For eigenfunctions, both general integral representations and closed expressions in terms of Airy functions are obtained, which are uniformly valid along surface of a rotation for the case of one pole by the parabolic equation method. The field expressions take into account the inseparability of the field by the sum in term of E- and H-type fields, are uniformly valid in the near-surface layer, excluding the vicinity of the poles of the rotation surface, and have no discontinuities on the caustics of surface rays. The expressions obtained were used to calculate the mutual admittances between two annular slots on a semi-infinite and smoothly truncated conical surface. Numerical results obtained by the proposed and rigorous methods for a semi-infinite cone were compared. It is shown that in the case of in-phase annular slots in a semi-infinite cone, the obtained asymptotic expression for mutual admittance coincides with the main term of the asymptotic of a strict solution based on the method of eigenfunctions.

IСНУВАННЯ У ПРОСТОРI ШВАРЦА ТА ВЛАСТИВОСТI РОЗВ’ЯЗКIВ РIВНЯНЬ ТИПУ РIВНЯННЯ ХОПФА ЗI ЗМIННИМИ КОЕФIЦIЄНТАМИ

The issue of the existence of solutions of the Cauchy problem for a first-order quasi-linear differential equation with partial derivatives and variable coefficients is considered. The studied equation is a generalization of the classic Hopf equation, which is used in the study of many mathematical models of hydrodynamics. This equation arises when constructing approximate (asymptotic) solutions of the Korteweg–de Vries equation and other equations with variable coefficients and a singular perturbation, in particular, when finding their asymptotic step-type soliton-like solutions. For the mentioned differential equation of the first order, the solution of the Cauchy problem is obtained in analytical form, and the statement about sufficient conditions for the existence of solutions of the initial problem in the space of rapidly decreasing functions is proved. A similar problem for a first-order differential equation with partial derivatives with variable coefficients and quadratic nonlinearity is considered.

RENORMALIZATION GROUP METHOD FOR A CLASS OF LAGRANGE MECHANICAL SYSTEMS

Considering the important role of small parameter perturbation term in mechanical systems, the perturbed dynamic differential equations of Lagrange systems are established. The basic idea and method of solving ordinary differential equations by normal renormalization group method are transplanted into a kind of Lagrange mechanical systems, the renormalization group equations of Euler-Lagrange equations are obtained, and the first-order uniformly valid asymptotic approximate solution of Lagrange systems with a single-degree-of-freedom is given. Two examples are used to show the calculation steps of renormalization group method in detail as well as to verify the correctness of the method. The innovative finding of this paper is that for integrable Lagrange systems, its renormalization group equations are also integrable and satisfy the Hamilton system's structure.

Generalisation of the Frobenius Formula in the Theory of Block Operators on Normed Spaces

The efficient construction and employment of block operators are vital for contemporary computing, playing an essential role in various applications. In this paper, we prove a generalisation of the Frobenius formula in the setting of the theory of block operators on normed spaces. A system of linear equations with the block operator acting in Banach spaces is considered. Existence theorems are proved, and asymptotic approximations of solutions in regular and irregular cases are constructed. In the latter case, the solution is constructed in the form of a Laurent series. The theoretical approach is illustrated with an example, the construction of solutions for a block equation leading to a method of solving some linear integrodifferential system.

Hamilton-Green solver for the forward and adjoint problems in photoacoustic tomography

The majority of the solvers for the acoustic problem in Photoacoustic Tomography (PAT) rely on full solution of the wave equation, which makes them less suitable for real-time and dynamic applications where only partial data is available. This is in contrast to other tomographic modalities, e.g. X-ray tomography, where partial data implies partial cost for the application of the forward and adjoint operators. In this work we present a novel solver for the forward and adjoint wave equations for the acoustic problem in PAT. We term the proposed solver Hamilton-Green as it approximates the fundamental solution to the respective wave equation along the trajectories of the Hamiltonian system resulting from the high frequency asymptotic approximate solution for the wave equation. This approach is flexible and scalable in the sense that it allows computing the solution for each sensor independently at a fraction of the cost of the full wave solution. The theoretical foundations of our approach are rooted in results available in seismics and ocean acoustics. To demonstrate the feasibility of our approach we present results for 2D domains with homogeneous and heterogeneous sound speeds and evaluate them against a full wave solution obtained with a pseudospectral finite difference method implemented in the k-Wave toolbox [1].

ANALYTICAL CALCULATION OF THE DEFLECTIONOF A SPATIAL HINGE-ROD FRAME WITH AN ARBITRARY NUMBER OF PANELS

Statement of the problem. The task is to obtain in symbolic form the dependence of the deflection of the proposed scheme of a statically definable spatial truss of a regular type on the number of panels under various loads, including the load from the truss plane. A truss has two independent parameters that define its proportions.Results. For several types of loading according to the Maxwell - Mohr formula, analytical dependences of the deflections of the structure on the number of panels, load, and dimensions are derived. When generalizing a series of partial solutions with a given number of panels to an arbitrary number of panels, together with operators of the Maple computer mathematics system, the induction method is used. Asymptotic approximations of solutions are obtained.Conclusions. The proposed model of a spatial frame with two independent numbers of panels that define the proportions of the structure allows an analytical solution of the problem of deflection under different types of loading. The derived formulas can be used as test formulas for evaluating approximate numerical solutions and for optimization problems.

Extremal Solutions for a Class of Tempered Fractional Turbulent Flow Equations in a Porous Medium

In this paper, we are concerned with the existence of the maximum and minimum iterative solutions for a tempered fractional turbulent flow model in a porous medium with nonlocal boundary conditions. By introducing a new growth condition and developing an iterative technique, we establish new results on the existence of the maximum and minimum solutions for the considered equation; at the same time, the iterative sequences for approximating the extremal solutions are performed, and the asymptotic estimates of solutions are also derived.

Asymptotic Solutions of Fifth Order Overdamped-Oscillatory Nonlinear Systems

Combo overdamp-oscillatory system plays an important role in natural phenomena in many engineering problems. In this paper, fifth order nonlinear damped-oscillatory differential system is studied to investigate an asymptotic analytical approximate solution in the fashion of overdamp-oscillations via an extension of the Krylov-Bogoliubov-Mitropolskii (KBM) method. The proposed method is demonstrated by its applications on a Duffing oscillators in the combined form of overdamp and oscillatory effects. The result obtained by the presented extended technique good agreement with the numerical solutions of the fourth order Runge-Kutta method.

Asymptotic Approximations for Solutions of Sturm–Liouville Differential Equations with a Large Complex Parameter and Nonsmooth Coefficients

This article deals with fundamental solutions of the Sturm–Liouville differential equations $$\left( P\left( x\right) y^{\prime }\right) ^{\prime }+\left( R\left( x\right) -\xi ^{2}Q\left( x\right) \right) y=0$$ with a complex parameter $$\xi $$ , $$\left\vert \xi \right\vert \gg 1$$ , whose coefficients $$P\left( x\right) $$ and $$Q\left( x\right) $$ are positive piecewise continuous functions while $$\left( P\left( x\right) Q\left( x\right) \right) ^{\prime }$$ and $$R\left( x\right) $$ belong to $$L^{p}$$ , $$p>1$$ . Two methods are suggested to reduce the problem to special Volterra or Volterra–Hammerstein integral equations with “small” integral operators, which can be treated by iterations that constitute asymptotic scales or by direct numerical methods. Asymptotic analysis of the iterations permits constructing uniform asymptotic approximations for fundamental solutions and their derivatives. A uniform with respect to integration limits analog of Watson’s lemma for finite range Laplace integrals is proved in this connection. When the coefficients belong to $$C^{N}$$ on intervals of continuity, a new simple direct method is suggested in order to derive explicit uniform asymptotic approximations of fundamental solutions as $$\operatorname{Re} \left( \xi \right) \gg 1$$ by a certain recurrent procedure. This method is expected to be more efficient than the WKB method. Theorems estimating errors of such approximations are given. Efficacy of the suggested uniform approximations is demonstrated by computing approximations for a combination of Bessel functions of large indices and arguments.

The existence, uniqueness and asymptotic estimates of solutions for third-order full nonlinear singularly perturbed vector boundary value problems

In this paper, we discuss third-order full nonlinear singularly perturbed vector boundary value problems. We first present the existence of solutions for the nonlinear vector boundary value problems without perturbation by using the upper and lower solutions method and topological degree theory. Then the existence, uniqueness and asymptotic estimates of solutions for the singularly perturbed vector boundary value problems are established by constructing appropriate a lower solution–upper solution pair, as well as analysis technique. Some known results are extended.

Об исследовании спектральных свойств дифференциальных операторов с гладкой весовой функцией

In this paper we study the spectral properties of a third-order differential operator with a summable potential with a smooth weight function. The boundary conditions are separated. The method of studying differential operators with summable potential is a development of the method of studying operators with piecewise smooth coefficients. Boundary value problems of this kind arise in the study of vibrations of rods, beams and bridges composed of materials of different densities. The differential equation defining the differential operator is reduced to the solution of the Volterra integral equation by means of the method of variation of constants. The solution of the integral equation is found by the method of successive Picard approximations. Using the study of an integral equation, we obtained asymptotic formulas and estimates for the solutions of a differential equation defining a differential operator. For large values of the spectral parameter, the asymptotics of solutions of the differential equation that defines the differential operator is derived. Asymptotic estimates of solutions of a differential equation are obtained in the same way as asymptotic estimates of solutions of a differential operator with smooth coefficients. The study of boundary conditions leads to the study of the roots of the function, presented in the form of a third-order determinant. To get the roots of this function, the indicator diagram wasstudied. The roots of this equation are in three sectors of an infinitely small size, given by the indicator diagram. The article studies the behavior of the roots of this equation in each of the sectors of the indicator diagram. The asymptotics of the eigenvalues of the differential operator under study is calculated. The formulas found for the asymptotics of eigenvalues allow us to study the spectral properties of the eigenfunctions of the differential operator under study.

Modeling and Simulation in Engineering Modeling of the Electrocoagulation Processes in Nonisothermal Conditions

The paper suggests an approach to modeling the electrocoagulation process that is based on the generalization of the equations of incompressible fluid flow in nonisothermal conditions. In the model was taking into account the ratio between the values of the parameters which characterize the domination of convective and mass-exchange components of the process over diffusion. An asymptotic approximation of solutions of corresponding boundary value problems is constructed. Based on the received solutions, we conducted a computer simulation of the process of distribution of iron concentration inside the reactor that allows predicting various hydrodynamic phenomena such as internal recirculation and dead zones that affect the formation of a coagulant. The influence of current strength on the concentration of the target component at the exit from the reactor was investigated using the developed mathematical model. In addition, our findings also show the effect of the rate of heat formation from the electrodes on the efficiency of obtaining of coagulant.

Asymptotic solutions of fractional interval differential equations with nonsingular kernel derivative.

Realizing the behavior of the solution in the asymptotic situations is essential for repetitive applications in the control theory and modeling of the real-world systems. This study discusses a robust and definitive attitude to find the interval approximate asymptotic solutions of fractional differential equations (FDEs) with the Atangana-Baleanu (A-B) derivative. In fact, such critical tasks require to observe precisely the behavior of the noninterval case at first. In this regard, we initially shed light on the noninterval cases and analyze the behavior of the approximate asymptotic solutions, and then, we introduce the A-B derivative for FDEs under interval arithmetic and develop a new and reliable approximation approach for fractional interval differential equations with the interval A-B derivative to get the interval approximate asymptotic solutions. We exploit Laplace transforms to get the asymptotic approximate solution based on the interval asymptotic A-B fractional derivatives under interval arithmetic. The techniques developed here provide essential tools for finding interval approximation asymptotic solutions under interval fractional derivatives with nonsingular Mittag-Leffler kernels. Two cases arising in the real-world systems are modeled under interval notion and given to interpret the behavior of the interval approximate asymptotic solutions under different conditions as well as to validate this new approach. This study highlights the importance of the asymptotic solutions for FDEs regardless of interval or noninterval parameters.

Analysis and Optimization of Random Caching in $K$ -Tier Multi-Antenna Multi-User HetNets

Combining cache and heterogeneous cellular networks (HetNets) together is proposed to satisfy the increasing capacity requirements in 5G networks. In this paper, we consider the analysis and optimization of random caching in the $K$ -tier multi-antenna multi-user HetNets. We derive the expressions of the cache hit probability and the potential throughput, using tools from stochastic geometry. Limiting to the fully loaded interference-limited scenario, we obtain the closed-form exact expressions and the upper bounds of the metrics. The linear asymptotic expression of the cache hit probability in the high signal to interference and noise ratio (SINR) threshold region is also obtained. Based on analytical results, we consider the potential throughput maximization problem via optimizing the caching distribution. For a general scenario, we obtain a local optimal solution and a simple approximate asymptotic optimal solution in the high SINR threshold region. For a special case where each BS serves the largest number of users per resource block, we numerically solve a non-convex problem and propose a sub-optimal caching strategy by approximation, where a closed-form result is obtained in each iteration. Numerical simulations show that our proposed caching schemes outperform existing caching schemes. Analysis and optimization results provide insightful design guidelines for the cache-enabled multi-antenna HetNets.

Scalar Singularly Perturbed Cauchy Problem for a Differential Equation of Fractional Order

In this paper we consider initial problem for an ordinary differential equation of fractional order with a small parameter for the derivative. S.A. Lomov regularization method is used to construct an asymptotic approximate solution of the problem with accuracy up to any power of a small parameter. Using the computer mathematics system (CMS) Maple, a symbolic solution of the original problem is obtained, and solution schedules are constructed, depending on the initial data and various values of the small parameter. It is shown that the asymptotic solution presented in the form of a specific convergent series and the solution represented by the CMS Maple coincides with the exact solution of the original problem.Â

A new complete Calabi\u2013Yau metric on {mathbb {C}}^3

Motivated by the study of collapsing Calabi–Yau 3-folds with a Lefschetz K3 fibration, we construct a complete Calabi–Yau metric on {mathbb {C}}^3 with maximal volume growth, which in the appropriate scale is expected to model the collapsing metric near the nodal point. This new Calabi–Yau metric has singular tangent cone at infinity {mathbb {C}}^2/{mathbb {Z}}_2 times {mathbb {C}}, and its Riemannian geometry has certain non-standard features near the singularity of the tangent cone, which are more typical of adiabatic limit problems. The proof uses an existence result in H-J. Hein’s Ph.D. thesis to perturb an asymptotic approximate solution into an actual solution, and the main difficulty lies in correcting the slowly decaying error terms.

INDUCTIVE ANALYSIS OF THE DEFORMATION OF THE ARCHED TRUSS

An analytical solution to the problem of the deflection of a flat arched statically determinate truss un-der the action of a uniformly distributed and concentrated load is obtained and analyzed. To obtain the depend-ence of the deflection on the number of panels, an induction method was used in two parameters — the number of panels in the crossbar and the number of panels in the supporting side trusses. All transformations and analysis are performed in the system of computer mathematics Maple. Expressions for the forces in individual rods are found. The asymptotic approximation of solutions is obtained.