Approximate Solution Research Articles

Polynomial Solutions of Electric Field Equations in Anisotropic Media

The time-dependent system of partial differential equations of the second order describing the electric wave propagation in electrically and magnetically anisotropic homogeneous media is considered in the paper. A method for the computation of the polynomial solutions of the initial value problem for the considered system is proposed. Symbolic computations are used and these symbolic computations are implemented in Maple. It is proved also that these polynomial solutions are approximate solutions of the considered initial value problem with smooth initial data and the inhomogeneous term. The computational experiments confirm the robustness of the suggested method for the computation of electric fields in general electrically and magnetically anisotropic media.

A modified perturbation method for global dynamic analysis of generalized mixed Rayleigh–Liénard oscillator with cubic and quintic nonlinearities

Deriving analytical relationship between the system parameters and amplitude of the limit cycle is a meaningful and challenging task. Currently, numerous existing analytical approximate methods struggle to achieve this goal when expressions of restoring force or nonlinear damping is complicated. To overcome this shortcoming, this study proposes a modified generalized harmonic function perturbation method. Using the proposed method, a generalized mixed RayleighLiénard oscillator with cubic and quintic nonlinearities was investigated. The analytical relationships between the system parameters and amplitude of the limit cycle, as well as the expression of its characteristic quantity, were derived. By employing these analytical relationships, the existence, stability, number, position, and amplitude of each limit cycle are quantitatively analysed. The homoclinic and heteroclinic bifurcations were also predicted using the above analytical relationships. Additionally, analytical approximate solutions for this oscillator were calculated using the proposed method. All results obtained in this study were subsequently confirmed numerically to demonstrate their feasibility and validity. Consequently, the proposed method can be considered an effective supplement to perturbation-based methods. This also implies that the work presented in this paper has a certain theoretical significance and application value in the research area of quantitative analysis methods for strongly nonlinear oscillators.

Approximate bound state solutions of the Dunkl-Schrödinger equation for a hyperbolic double-well interaction

We construct approximate solutions to the stationary, one-dimensional Schrödinger equation for a hyperbolic double-well potential within the Dunkl formalism. Our approximation is applied to an inverse quadratic term contributed by the Dunkl formalism in the effective potential. The solutions we obtain are given in terms of confluent Heun functions. We establish parity of these solutions, discuss their elementary cases, and present an example of a system admitting bound states.

Honeycomb-configured dissipative nanofluid flow within a squeezed channel with entropy generation: regression and numerical evaluations

Purpose Nanosized honeycomb-configured materials are used in modern technology, thermal science and chemical engineering due to their high ultra thermic relevance. This study aims to scrutinize the heat transmission features of magnetohydrodynamic (MHD) honeycomb-structured graphene nanofluid flow within two squeezed parallel plates under Joule dissipation and solar thermal radiation impacts. Design/methodology/approach Mass, energy and momentum preservation laws are assumed to find the mathematical model. A set of unified ordinary differential equations with nonlinear behavior is used to express the correlated partial differential equations of the established models, adopting a reasonable similarity adjustment. An approximate convergent numerical solution to these equations is evaluated by the shooting scheme with the Runge–Kutta–Fehlberg (RKF45) technique. Findings The impression of pertinent evolving parameters on the temperature, fluid velocity, entropy generation, skin friction coefficients and the heat transference rate is explored. Further, the significance of the irreversibility nature of heat transfer due to evolving flow parameters are evaluated. It is noted that the heat transference rate performance is improved due to the imposition of the allied magnetic field, Joule dissipation, heat absorption, squeezing and thermal buoyancy parameters. The entropy generation upsurges due to rising magnetic field strength while its intensification is declined by enhancing the porosity parameter. Originality/value The uniqueness of this research work is the numerical evaluation of MHD honeycomb-structured graphene nanofluid flow within two squeezed parallel plates under Joule dissipation and solar thermal radiation impacts. Furthermore, regression models are devised to forecast the correlation between the rate of thermal heat transmission and persistent flow parameters.

FORECASTING THE BEHAVIOR OF FRACTIONAL MODEL OF EMDEN–FOWLER EQUATION WITH CAPUTO–KATUGAMPOLA MEMORY

The main aim of this paper is to analyze the behavior of time-fractional Emden–Fowler (EF) equation associated with Caputo–Katugampola fractional derivative occurring in mathematical physics and astrophysics. A powerful analytical approach, which is an amalgamation of q-homotopy analysis approach and generalized Laplace transform with homotopy polynomials, is implemented to obtain approximate analytical solution of the time-fractional EF equation. Main advantage of this research work is that the implemented technique contains an auxiliary parameter to control the convergence region of obtained series solution. Some examples are considered to illustrate the accuracy and efficiency of the applied technique. Numerical results are demonstrated in the form of tabular and graphical representations.

Novel Numerical Investigations of Some Problems Based on the Darcy–Forchheimer Model and Heat Transfer

In this study, we introduced a new type of basis function and subsequently a Chebyshev delta shaped collocation method (CDSC). We then use this method to numerically investigate both the natural convective flow and heat transfer of nanofluids in a vertical rectangular duct on the basis of a Darcy–Brinkman–Forchheimer model and the electroosmosis-modulated Darcy–Forchheimer flow of Casson nanofluid over stretching sheets with Newtonian heating problems. The approximate solution is represented in terms of Chebyshev delta shaped basis functions. Novel error estimates for interpolating polynomials are derived. Computational experiments were carried out to corroborate the theoretical results and to compare the present method with the existing Chebyshev pseudospectral method. To demonstrate our proposed approach, we also compared the numerical solutions with analytic solutions of the Poisson equation. Computer simulations show that the proposed method is computationally cheap, fast, and spectrally accurate and more importantly the obtained approximate solution can easily be used by researchers in this field.

The Hydraulic Evolution of Groundwater-Fed Pit Lakes After Mine Closure.

Open pit mining frequently requires regional water tables to be lowered to access ore deposits. When mines close, dewatering ceases allowing the water table to recover. In arid and semi-arid mining regions, the developing pit lakes are predominantly fed by groundwater during this recovery phase and pit lakes develop first into "terminal sinks" for the surrounding groundwater system. With time, the re-establishment of regional hydraulic gradients can cause pit lakes to develop into throughflow systems, in which pit lake water outflows into adjacent aquifers. In this study, we use numerical groundwater modeling to aid process understanding of how regional hydraulic gradients, aquifer properties, net evaporation rates, and pit geometry determine the hydraulic evolution of groundwater-fed pit lakes. We find that before the recovery of the regional water table to its new equilibrium, pit lakes frequently transition to throughflow systems. Throughflow from pit lakes to downstream aquifers can develop within two decades following cessation of dewatering even under low hydraulic gradients (e.g., 5 × 10-4) or high net evaporation rates (e.g., 2.5 m/year). Pit lakes remain terminal sinks only under suitable combinations of high evaporation rates, low hydraulic gradients, and low hydraulic conductivities. In addition, we develop an approximate analytical solution for a rapid assessment of the hydraulic status of pit lakes under steady-state conditions. Understanding whether pit lakes remain terminal sinks or transition into throughflow systems largely determines the long-term water quality of pit lakes and downstream aquifers. This knowledge is fundamental for mine closure and planning post-mining land use.

Numerical Solution of Generalized Delay Integro-Differential Equations via Galerkin-Vieta-Lucas Polynomials

In this article, the Galerkin-Vieta-Lucas scheme is presented to find an approximate solution to the generalised delay integro-differential equation using the Vieta-Lucas polynomial as an approximation. The Galerkin approach transforms the delay integro-differential equation into a set of n × (n + m) algebraic equations, which, together with the attached conditions, give (n + m) × (n + m) equations. The effectiveness and accuracy of the proposed technique were tested on some existing examples in the literature, and obviously, the results obtained justify the accuracy of the proposed scheme.

A new analytical tool for the elastic/plastic buckling design of pressure vessels subjected to external pressure

This paper deals with the elastic/plastic buckling design of pressure vessels under external pressure loading. The objective is to build a simple analytical tool devoted to the elastic and elastoplastic buckling analysis of pressure equipments. This tool is aimed to provide the critical external pressure of cylindrical pressure vessels with ellipsoidal (or possibly hemispherical) heads. Depending on the geometry of the equipment and material parameters, the first buckling pressure may involve either the cylindrical shell or the closure ends, in the elastic or plastic range. The calculation tool will be thus based on a series of exact or approximate analytical solutions of the critical pressure of each part of the considered equipment. In this paper, the elementary shell structures involved in such pressure equipments are first individually examined. The fundamental critical pressures for each of them are recalled or introduced, both in elasticity and plasticity. The new solutions are validated against reference numerical results obtained through finite element computations. Then, approximate solutions of the critical pressure are derived for the same shells, taking account of specific boundary conditions (and not idealized ones, as before) due to the presence of the remaining part of the equipment. In the end, a VBA Excel software is developed. It is able to provide the mappings of the buckling mode type and the corresponding critical pressures for large ranges of geometric and material parameters in a very efficient way. These analytical solutions are finally confronted to numerical calculations performed on complete pressure equipments and they are shown to give quite reliable results. A discussion is also initiated on the use of such a program to update the current design rules in this scope of application.

An Approximate Analytical Solution of Obesity Epidemic Model by Homotopy Analysis Method

An Approximate Analytical Solution of Obesity Epidemic Model by Homotopy Analysis Method

Inverse localization and global approximation for some Schrödinger operators on hyperbolic spaces

We consider the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the d-dimensional hyperbolic space of constant curvature −κ2 are flexible enough to approximate an arbitrary solution of the Helmholtz equation Δh + h = 0 on Rd, over the natural length scale O(λ−1/2) determined by the eigenvalue λ ≫ 1. This problem is motivated by the fact that, by the asymptotics of the local Weyl law, approximate Laplace eigenfunctions do have this approximation property on any compact Riemannian manifold. In this paper we are specifically interested in the Coulomb and harmonic oscillator operators on the hyperbolic spaces Hd(κ). As the dimension of the space of bound states of these operators tends to infinity as κ ↘ 0, one can hope to approximate solutions to the Helmholtz equation by eigenfunctions for some κ > 0 that is not fixed a priori. Our main result shows that this is indeed the case, under suitable hypotheses. We also prove a global approximation theorem with decay for the Helmholtz equation on manifolds that are isometric to the hyperbolic space outside a compact set, and consider an application to the study of the heat equation on Hd(κ). Although global approximation and inverse approximation results are heuristically related in that both theorems explore flexibility properties of solutions to elliptic equations on hyperbolic spaces, we will see that the underlying ideas behind these theorems are very different.

Polynomial solution of Cauchy-type singular integro-differential equations with bivariate kernels

In this paper, we present Chebyshev and ultraspherical polynomials method to obtain numerical solutions of Cauchy-type singular integro-differential equations. We prove that the singular integro-differential equations with smooth/non-smooth bivariate kernels and general boundary conditions can be transformed into a constrained least square problem and solved via generalized QR factorization. By using open-source Chebfun code, low-rank approximate solutions of singular integro-differential equations can be obtained efficiently in the spectral accuracy range. The method can be easily extended to solve those singular integro-differential equations with smooth bivariate kernels, or convolution kernels with absolute value symbols (non-smooth cases), or when the solutions have (inverse) square root singularities.

Efficient Parallel Algorithm for Minimum Cost Submodular Cover Problem with Lower Adaptive Complexity

In this paper, we study the Minimum Cost Submodular Cover (MCSC) problem over the ground set of size [Formula: see text], which aims at finding a subset with the minimal cost required so that the utility submodular function exceeds a given threshold. The problem has recently attracted a lot of attention due to its applications in various domains of operations research and artificial intelligence. However, the existing algorithms for this problem may not be effectively parallelized because of their costly adaptive complexity. This paper proposes an efficient parallel algorithm that returns a [Formula: see text]-bicriteria approximation solution within [Formula: see text] adaptive complexity, where [Formula: see text] are fixed parameters. Our algorithm requires [Formula: see text] query complexity, however, it can reduce to [Formula: see text] instead while retaining a low adaptive complexity of [Formula: see text]. Therefore, our algorithm not only achieves the same approximation guarantees as the state of the art but also significantly improves the best-known low adaptive complexity algorithm for the above problem.

Approximate solution of time-fractional non-linear parabolic equations arising in Mathematical Physics

Approximate solution of time-fractional non-linear parabolic equations arising in Mathematical Physics

Approximate solution of the problem of redundancy optimization of a complex technological system of ore grinding

Approximate solution of the problem of redundancy optimization of a complex technological system of ore grinding

New Algorithms for Steiner Tree Reoptimization

AbstractReoptimization is a setting in which we are given a good approximate solution of an optimization problem instance and a local modification that slightly changes the instance. The main goal is that of finding a good approximate solution of the modified instance. We investigate one of the most studied scenarios in reoptimization known as Steiner tree reoptimization. Steiner tree reoptimization is a collection of strongly $$\textsf {NP}$$ NP -hard optimization problems that are defined on top of the classical Steiner tree problem and for which several constant-factor approximation algorithms have been designed in the last decades. In this paper we improve upon all these results by developing a novel technique that allows us to design polynomial-time approximation schemes. Remarkably, prior to this paper, no approximation algorithm better than recomputing a solution from scratch was known for the elusive scenario in which the cost of a single edge decreases. Our results are best possible since none of the problems addressed in this paper admits a fully polynomial-time approximation scheme, unless $$\textsf {P}=\textsf {NP}$$ P = NP

An efficient and robust varying-parameter projection neural network for sparse signal reconstruction

Sparse signal reconstruction plays an important role in many practical applications, and therefore, many scholars strive to study its various effective (approximate) solution approaches, including some projection neural network models with finite-time or fixed-time convergence due to their parallel nature and convenience for hardware implementation. However, we experimentally find that noise has a significant influence on the convergence performance of these models. To overcome this problem, in this paper, we propose a new efficient varying-parameter projection neural network model (VPPNN) using the sliding mode control technique and a novel time-varying parameter for sparse signal reconstruction. Theoretical analysis shows that this new model not only has shorter fixed convergence time under certain conditions but also has noise-tolerance when compared to the existing projection neural network models. Numerical simulation experiments (via sparse signal/image reconstruction) are performed to illustrate the superiority of the new proposed model over the existing ones in terms of the convergence rate and robustness.

Numerical solution of the multi-order fractional differential equation using Legendre wavelets and eigenfunction approach

An eigenfunction approach is implemented in this article to solve the multi-order fractional differential equations (FDEs) with boundary conditions. The approximate unknown solution is expressed as a linear combination of eigenfunctions in the present paper. The proposed approach solved the problem via complex and real eigenfunction. By using the eigenvalue problem Dαϕ(x)=λϕ(x) subjected to u(0)=u(1)=0, first we calculate the eigenvalues and their corresponding eigenfunction and then use the output of this problem in the present scheme to simplify. Exact fractional integration has been utilized to solve the eigenvalue problem. Collocation points have also been used to reduce the problem to a system of algebraic equations. The obtained solutions are compared with the exact solution. In addition, the numerical convergence has also been studied of the proposed technique through three examples. These examples also demonstrate the applicatory and simplicity of the proposed scheme.

Exploring fractional-order new coupled Korteweg-de Vries system via improved Adomian decomposition method.

This paper aims to extend the applications of the projected fractional improved Adomian Decomposition method (fIADM) to the fractional order new coupled Korteweg-de Vries (cKdV) system. This technique is significantly recognized for its effectiveness in addressing nonlinearities and iteratively handling fractional derivatives. The approximate solutions of the fractional-order new cKdV system are obtained by employing the improved ADM in fractional form. These solutions play a crucial role in designing and optimizing systems in engineering applications where accurate modeling of wave phenomena is essential, including fluid dynamics, plasma physics, nonlinear optics, and other mathematical physics domains. The fractional order new cKdV system, integrating fractional calculus, enhances accuracy in modeling wave interactions compared to the classical cKdV system. Comparison with exact solutions demonstrates the high accuracy and ease of application of the projected method. This proposed technique proves influential in resolving fractional coupled systems encountered in various fields, including engineering and physics. Numerical results obtained using Mathematica software further verify and demonstrate its efficacy.

Approximate Solutions for Non-Linear Evolution Stochastic Equations with Variations of Drift Parameters

This paper considered system of stochastic differential equations with emphasis on variations of drift parameters as it affects financial markets. The Ito’s method was applied in solving the problems analytically which resulted to three different investment solutions accordingly. The appropriate conditions were accomplished which controlled various drift parameters in the assessment financial markets. Hence, the impressions on each solution of investors in financial markets were analyzed graphically. Finally, the influences of the relevant parameters of stochastic variables were effectively discussed all in this paper.