10 March 2021
Evaporation of chemicals is an important source of inhalative exposure. We analyzed here the ConsExpo evaporation model, which is characterized by a set of nonlinear differential equations only solvable by numerical means. It shows qualitatively diff...
This paper considers a theoretical substantiation of the influence of a perturbation of a moving singular point on the analytical approximate solution to the Van der Pol equation obtained earlier by the author. A priori estimates of the error of the...
In the present work, a new fractional analytical scheme (NFAS) is developed to obtain the approximate results of fourth-order parabolic fractional partial differential equations (FPDEs). The fractional derivatives are considered in the Caputo sense....
During circumferential tensioning of prestressing strands in U-shaped aqueducts, longitudinal tensile stresses may develop and impair crack resistance. Most existing studies rely on three-dimensional finite element (FE) simulations. Although accurate...
The spontaneous scalarization of Schwarzscild-AdS is investigated in the Einstein-scalar-Gauss–Bonnet (ESGB) theory. Firstly, we construct scalarized AdS black holes numerically. Secondly, making use of the homotopy analysis method (HAM), we ob...
The homotopy analysis method (HAM) is a useful method to derive analytical approximate solutions of black holes in modified gravity theories. In this paper, we study the Einstein–Weyl gravity coupled with Maxwell field and obtain analytical app...
In the article, the existence of solutions for the Van der Pol differential equation is proved, and the approximate structure of such solutions in the analyticity domain is obtained. In the proof, the majorant method was applied not to the right side...
In this paper, we substantiate the analytical approximate method for Cauchy problem of the Van der Pol equation in the complex domain. These approximate solutions allow analytical continuation for both real and complex cases. We follow the influence...
We employ the Polynomial Least Squares Method as a relatively new and very straightforward and efficient method to find accurate approximate analytical solutions for a class of systems of fractional nonlinear integro-differential equations. A compari...
We present newly derived approximate analytical solutions (AAS) of the motion, free and forced modes of dynamics of two viscoelastic rheological Maxwell–Faraday discrete dynamic systems (RMFDDSFTPEP), creeper type, and fractional type, with pie...
This paper presents the final stage in the study of the analytical approximate solution to a class of nonlinear differential equations unsolvable in quadrature in the general case in the neighborhood of a perturbed value of a moving singular point. A...
In this paper, the fuzzy Volterra integral equations’ solutions are calculated using a hybrid methodology. The combination of the Elzaki transform and Adomian decomposition method results in the development of a novel regime. The precise fuzzy...
The classical Black–Scholes model assumes Markovian dynamics and cannot capture the long-range dependence and gradual memory decay observed in real markets. We formulate the two-dimensional time-fractional Black–Scholes equation for a Eur...
An energetic Lagrangian approach is applied to formulate coupled dynamic equations for unknown time-dependent displacements with respect to girder mid-span and vehicle suspension interaction, along with their rates and accelerations. The paper provid...
We construct approximate analytical solutions of the Boussinesq equation for horizontal unconfined aquifers in the buildup phase under constant recharge and zero-inflow conditions. We employ a variety of methods, which include wave solutions, self-si...
In this paper, the Aboodh transform is utilized to construct an approximate analytical solution for the time-fractional Zakharov–Kuznetsov equation (ZKE) via the Adomian decomposition method. In the context of a uniform magnetic flux, this framework...
The response laws of groundwater dynamics on the riverbank to river level variations are highly dependent on the river level fluctuation process. Analytical solutions are widely used to infer the groundwater flow behavior. In analytical calculations,...
The paper considers the specifics of nonlinear differential equations that have applications in different areas. Earlier, the authors proved the existence and uniqueness theorem for a solution to a class of non-linear differential equations in a neig...
Partial differential equations are known to be increasingly important in today’s research, and their solutions are paramount for tackling numerous real-life applications. This article extended the analytical scheme of the homotopy analysis meth...
We study a particular case of a convergence model of interest rates. The bond prices are given as solutions of a parabolic partial differential equation and we consider different possibilities of approximating them, using approximate analytical solut...
The modified differential transform method (MDTM), Laplace transform and Padé approximants are used to investigate a semi-analytic form of solutions of nonlinear oscillators in a large time domain. Forced Duffing and forced van der Pol oscillators un...
Blasingame production decline is an effective method to obtain permeability and single-well controlled reserves. The accurate Blasingame production decline curve needs an accurate wellbore pressure approximate solution of the real-time domain. Theref...
A variety of methods that provide approximate piecewise- analytical solutions to initial-value problems governed by scalar, nonlinear, first-order, ordinary differential equations is presented. The methods are based on fixing the independent variable...
We apply the polynomial least squares method to obtain approximate analytical solutions for a very general class of nonlinear Fredholm and Volterra integro-differential equations. The method is a relatively simple and straightforward one, but its pre...
This paper applies the Polynomial Least Squares Method (PLSM) to the case of fractional Lane-Emden differential equations. PLSM offers an analytical approximate polynomial solution in a straightforward way. A comparison with previously obtained resul...
This paper introduces a new analytical technique (NAT) for solving a system of nonlinear fractional partial differential equations (NFPDEs) in full general set. Moreover, the convergence and error analysis of the proposed technique is shown. The appr...
In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations wit...
We present a relatively new and very efficient method to find approximate analytical solutions for a very general class of nonlinear fractional Volterra and Fredholm integro-differential equations. The test problems included and the comparison with p...
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 mechani...
The paper studies a nonlinear equation including both fractional and ordinary derivatives. The singular Cauchy problem is considered. The theorem of the existence of uniqueness of a solution in the neighborhood of a fixed singular point of algebraic...
The article considers the Van der Pol equation nonlinearity aspect related to a moving singular point. The fact of the existence of moving singular points and the uniqueness of their solution for complex domains have been proved. An answer to the que...
This paper studies the uniqueness of the bounded solution to a new Cauchy problem of the fractional nonlinear partial integro-differential equation based on the multivariate Mittag–Leffler function as well as Banach’s contractive principl...
In this paper, the temperature shock phenomenon is considered. This phenomenon occurs during the operation of engineering structures on Earth and in outer space. A rectangular plate has been selected as a structural element exposed to temperature sho...
The paper deals with the results of a study of a third-order nonlinear differential equation with moving singular points and critical poles. So far, this type of equation cannot be solved in quadratures. The development of the author’s approach...
This paper is proposed for solving a partial differential equation of second order with a fractional derivative with respect to time (the vibration string equation), where the fractional derivative order is in the range from zero to two. We propose a...
In this work, conceptual approximations of the Boussinesq equation were introduced and analyzed, resulting into a very accurate and well-applicable model for horizontal unconfined aquifers during the pure drainage phase, without any recharge and zero...
In this paper, least squares homotopy perturbation is presented as a straightforward and accurate method to compute approximate analytical solutions for systems of ordinary differential equations. The method is employed to solve a problem related to...
In this work, we present a modified generalized Mittag–Leffler function method (MGMLFM) and Laplace Adomian decomposition method (LADM) to get an analytic-approximate solution for nonlinear systems of partial differential equations (PDEs) of fr...
This study focuses on deriving and presenting an infinite series as the analytical solution for transient electroosmotic and pressure-driven flows in microtubes. Such a mathematical presentation of fluid dynamics under simultaneous electric field and...
The correct evaluation of the curvatures of ball screw grooves allows the accurate design of the constructive parameters of this mechanism and enhancing its performance. The formulation commonly used in the literature, however, refers to ball bearing...
As one of the most representative passive control devices, tuned mass dampers (TMDs) are widely used in civil engineering, aerospace, machinery, and other fields, after years of research and improvement. However, due to their large displacement and t...
In this article, we use the homotopy perturbation method and the Adomian decomposition method with the Yang transformation to discover analytical solution to the time-fractional coupled Schrödinger–KdV equation. In the Caputo sense, fracti...
To improve the pull-out bearing capacity of the anchor plate support system, a support system with series-connected anchor plates was proposed. Based on the Winkler foundation model, a theoretical analysis method for the pull-out resistance–dis...
Analytical solutions of the one-dimensional (1D) advection–dispersion equations, describing the substance transport in streams, are often used because of their simplicity and computational speed. Practical computations, however, clearly show th...
The analytical solutions of the radiative transfer equation are needed for the solution of various applied atmospheric and snow optics problems. In this paper, we propose a simple analytical equation for the top-of-atmosphere (TOA) spectral reflectan...
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 cons...
The Parameter Expansion Method (PEM) is employed to study nonlinear Jerk equations, which are often difficult to solve because of their strong nonlinearity. This method provides higher accuracy and broader applicability, enabling analytical insights...
The paper oresents the analytical construction of approximate solutions to the generalized Fisher–Kolmogorov equation in the complex domain. The existence and uniqueness of such solutions are established within an analytic domanin of the comple...
The analysis of the time-fractional nonlinear Sharma–Tasso–Olver (STO) equation with various initial conditions has been shown in this work. Finding the appropriate approximate solution of the problems under consideration is carried out b...
The paper considers a class of nonlinear differential equations which are not solvable in quadratures in general case. The author’s technology for solving such equations contains six problems. In this article, the solution to one of these probl...
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