Representation Of Solution Research Articles

A novelty Multi-Step Associated with Laplace Transform Semi Analytic Technique for Solving Generalized Non-linear Differential Equations

In this work, a novel technique to obtain an accurate solutions to nonlinear form by multi-step combination with Laplace-variational approach (MSLVIM) is introduced. Compared with the traditional approach for variational it overcome all difficulties and enable to provide us more an accurate solutions with extended of the convergence region as well as covering to larger intervals which providing us a continuous representation of approximate analytic solution and it give more better information of the solution over the whole time interval. This technique is more easier for obtaining the general Lagrange multiplier with reduces the time and calculations. It converges rapidly to exact formula with simply computable terms with few time. To investigate of this technique, selected examples to show the ability, validity, accurately and effectiveness.

Exact Solutions of One Nonlinear Countable-Dimensional System of Integro-Differential Equations

In the present paper, a nonlinear countable-dimensional system of integrodifferential equations is investigated, whose vector of unknowns is a countable set of functions of two variables. These variables are interpreted as spatial coordinate and time. The nonlinearity of this system is constructed from two simultaneous convolutions: first convolution is in the sense of functional analysis and the second one is in the sense of linear space of double-sided sequences. The initial condition for this system is a doublesided sequence of functions of one variable defined on the entire real axis. The system itself can be written as a single abstract equation in the linear space of double-sided sequences. As the system may be resolved with respect to the time derivative, it may be presented as a dynamical system. The solution of this abstract equation can be interpreted as an approximation of the solution of a nonlinear integro-differential equation, whose unknown function depends not only on time, but also on two spatial variables. General representation for exact solution of system under study is obtained in the paper. Also two kinds of particular examples of exact solutions are presented. The first demonstrates oscillatory spatio-temporal behavior, and the second one shows monotone in time behavior. In the paper typical graphs of the first components of these solutions are plotted. Moreover, it is demonstrated that using some procedure one can generate countable set of new exact system’s solutions from previously found solutions. From radio engineering point of view this procedure just coincides with procedure of upsampling in digital signal processing.

Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point

Abstract In this article, we focus on the scattering analysis of Sturm-Liouville type singular operator including an impulsive condition and turning point. In the classical literature, there are plenty of papers considering the positive values of the weight function in both continuous and discontinuous cases. However, this article differs from the others in terms of the impulsive condition appearing at the turning point. We generate the scattering function, resolvent operator, and discrete spectrum of the operator using the hyperbolic representations of the fundamental solutions. Finally, we create an example to show the article’s primary conclusions.

On the Spectrum of the Non-Selfadjoint Differential Operator with an Integral Boundary Condition and Negative Weight Function

In this paper, we shall study the spectral properties of the non-selfadjoint operator in the space $L_{\varrho }^{2}\left(\mathbb{R}_{+}\right) $ generated by the Sturm-Liouville differential equation \begin{equation*} -y^{^{\prime \prime }}+q\left( x\right) y=\omega ^{2}\varrho \left( x\right) y, \quad x \in \mathbb{R}_{+} \end{equation*} with the integral type boundary condition \begin{equation*} \int \limits_{0}^{\infty }G\left( x \right) y\left( x\right) dx+ \gamma y^{\prime }\left( 0\right) -\theta y\left( 0\right) =0 \end{equation*} and the non-standard weight function \begin{equation*} \varrho \left( x\right) =-1 \end{equation*} where $\left \vert \gamma \right \vert +\left \vert \theta \right \vert \neq 0$. There are an enormous number of papers considering the positive values of $ \varrho \left( x\right) $ for both continuous and discontinuous cases. The structure of the weight function affects the analytical properties and representations of the solutions of the equation. Differently from the classical literature, we used the hyperbolic type representations of the fundamental solutions of the equation to obtain the spectrum of the operator. Moreover, the conditions for the finiteness of the eigenvalues and spectral singularities were presented. Hence, besides generalizing the recent results, Naimark's and Pavlov's conditions were adopted for the negative weight function case.

The Dirichlet problem for the Beltrami equations with sources

The paper is devoted to the study of the Dirichlet problem ${\rm{Re}}\,\omega(z)\to\varphi(\zeta)$ as $z\to\zeta,$ $z\in D,\zeta\in \partial D,$ with continuous boundary data $\varphi :\partial D\to\mathbb R$ for Beltrami equations $\omega_{\bar{z}}=\mu(z) \omega_z+\sigma (z)$, $|\mu(z)|<1$ a.e., with sources $\sigma :D\to\mathbb C$ in the case of locally uniform ellipticity. In this case, we have established a series of effective integral criteria of the BMO, FMO, Calderon-Zygmund, Lehto, and Orlicz types on the singularities of the equations at the boundary for the existence, representation, and regularity of solutions in arbitrary bounded domains $D$ of the complex plane $\mathbb C$ with no boun\-da\-ry component degenerated to a single point for sources $\sigma$ in $L_p(D)$, $p>2$, with compact support in $D$. Moreover, we have proved the existence, representation, and regularity of weak solutions of the Dirichlet problem in such domains for the Poisson-type equation ${\rm div} [A(z)\nabla\,u(z)] = g(z)$, whose source $g\in L_p(D)$, $p>1$, has compact support in $D$ and whose mat\-rix-valued coefficient $A(z)$ guarantees its locally uniform ellipticity.

Multi-task scheduling in cloud remanufacturing system integrating reuse, reprocessing, and replacement under quality uncertainty

This study focuses on the multi-task scheduling problem for the cloud remanufacturing system incorporating reuse, reprocessing, and replacement operations considering quality uncertainty. A framework of the cloud remanufacturing under quality differences is first constructed, and a mathematical model is then established to characterize the multi-task scheduling problem in the cloud platform. In this framework, five quality grades are considered, and each grade is assigned a specific remanufacturing line. To efficiently address this problem, the nonlinear grey wolf optimization (NLGWO) algorithm with a hybrid sequential solution representation for remanufacturing line selection, task sequencing, and service searching and matching is introduced. Additionally, the nonlinear strategy and the crisscross optimization method are embedded in the NLGWO to balance the exploration and exploitation capabilities while enhancing its population updating mechanisms. To evaluate the proposed method, a real-world case study is designed and implemented under actual remanufacturing conditions. Three different optimization problems, including makespan optimization, cost optimization, and multi-objective optimization problems, are addressed using the NLGWO and its four baseline meta-heuristic methods. The computational results demonstrate that the NLGWO can efficiently solve all the optimization problems in the case study, outperforming its baseline algorithms in terms of solution accuracy, computing speed, and convergence performance.

Optimizing partial component activation policy in multi-attempt missions

Active redundancy is an effective technique for improving the mission success probability. However, more components simultaneously performing the mission implies higher cost and higher risk associated with components’ failures. This paper formulates and solves a new optimization problem for an active redundancy system performing a multi-attempt mission, which finds the set/number of components activated in each attempt, referred to as the component activation policy (CAP) to minimize the expected mission cost (EMC). The EMC encompasses the expected operational cost, the expected cost of component losses, and the expected cost associated with damage caused by the mission failure. An analytical solution is first suggested for evaluating the EMC and determining the optimal CAP for a special case of a two-component two-attempt system. A recursive method is then proposed to derive the EMC for the general case of multi-component multi-attempt systems under a given CAP. Based on the suggested string representation of the CAP solution, the genetic algorithm is implemented to solve the EMC minimization problem. Impacts of several model parameters and their interactions on EMC and optimal CAP are examined through a detailed case study of unmanned aerial vehicles performing a multi-attempt reconnaissance mission.

Vehicle routing with heterogeneous service types: Optimizing post-harvest preprocessing operations for fruits and vegetables in short food supply chains

This study focuses on the post-harvest preprocessing of fruits and vegetables, aiming to provide an effective way to conduct preprocessing operations in short food supply chains. We consider both a heterogeneous fleet of mobile preprocessing units and the possibility to pick up products for centralized preprocessing. The resulting problem is a variant of the classic heterogeneous fleet vehicle routing problems with time windows (HFVRPTW), with the additional consideration of multi-depot and heterogeneous service types, which we refer to as HFVRPTW-MDHS. These additional considerations are important to include in the development of more efficient food supply chains, but lead to a challenging routing problem. In this paper, we formulate the HFVRPTW-MDHS using a mixed-integer linear programming model. Due to the complexity of the model, we propose a customized adaptive large neighborhood search (ALNS) metaheuristic. We design a multi-level struct-based solution representation to improve the efficiency of the ALNS and develop customized methods for solution evaluation, feasibility checks, and neighborhood search. Comparing our results with the results of an exact algorithm and solutions in the existing literature, we find that our ALNS algorithm can obtain high-quality solutions quickly when solving HFVRPTW-MDHS and related variants of the VRP. Finally, we study the application of our approach in the case of precooling, which is a commonly used preprocessing operation, to illustrate the effectiveness of our approach in a relevant practical context.

A Full-Body Relative Orbital Motion of Spacecraft Using Dual Tensor Algebra and Dual Quaternions

This paper proposes a new non-linear differential equation for the six degrees of freedom (6-DOF) relative rigid bodies motion. A representation theorem is provided for the 6-DOF differential equation of motion in the arbitrary non-inertial reference frame. The problem of the 6-DOF relative motion of two spacecraft in the specific case of Keplerian confocal orbits is proposed. The result is an analytical method without secular terms and singularities. Tensors dual algebra and dual quaternions play a fundamental role, with the solution representation being the relative problem. Furthermore, the representation theorems for the rotation and translation parts of the 6-DOF relative orbital motion problems are obtained.

A Hybrid Evolutionary Algorithm Using Two Solution Representations for Hybrid Flow-Shop Scheduling Problem.

As an extension of the classical flow-shop scheduling problem, the hybrid flow-shop scheduling problem (HFSP) widely exists in large-scale industrial production systems and has been considered to be challenging for its complexity and flexibility. Evolutionary algorithms based on encoding and heuristic decoding approaches are shown effective in solving the HFSP. However, frequently used encoding and decoding strategies can only search a limited area of the solution space, thus leading to unsatisfactory performance during the later period. In this article, a hybrid evolutionary algorithm (HEA) using two solution representations is proposed to solve the HFSP for makespan minimization. First, the proposed HEA searches the solution space by a permutation-based encoding representation and two heuristic decoding methods to find some promising areas. Afterward, a Tabu search (TS) procedure based on a disjunctive graph representation is introduced to expand the searching space for further optimization. Two classical neighborhood structures focusing on critical paths are extended to the problem-specific backward schedules to generate candidate solutions for the TS. The proposed HEA is tested on three public HFSP benchmark sets from the existing literature, including 567 instances in total, and is compared with some state-of-the-art algorithms. Extensive experimental results indicate that the proposed HEA performs much better than the other algorithms. Moreover, the proposed method finds new best solutions for 285 hard instances.

Elastic-plastic problem for a perforated plate under transversal shear

The problem of solving the problem of the transverse shear of a plate along the edges of the holes and weakened by a two-periodic system of rectilinear through cracks with plastic end zones, collinear to the abscissa and ordinate axes of unequal length, is considered. General representations of solutions are constructed that describe a class of problems with a doubly periodic distribution of moments outside circular holes and straight-line cracks with end zones of plastic deformations. Satisfying the boundary conditions, the solution of the problem of the plate shear theory is reduced to two infinite systems of algebraic equations and two singular integral equations. Then each singular integral equation is reduced to a finite system of linear algebraic equations.

A Flux Reconstruction Stochastic Galerkin Scheme for Hyperbolic Conservation Laws

The study of uncertainty propagation poses a great challenge to design high fidelity numerical methods. Based on the stochastic Galerkin formulation, this paper addresses the idea and implementation of the first flux reconstruction scheme for hyperbolic conservation laws with random inputs. High-order numerical approximation is adopted simultaneously in physical and random space, i.e., the modal representation of solutions is based on an orthogonal polynomial basis and the nodal representation is based on solution collocation points. Therefore, the numerical behaviors of the scheme in the (physical-random) phase space can be designed and understood uniformly. A family of filters is developed in multi-dimensional cases to mitigate the Gibbs phenomenon arising from discontinuities in both physical and random space. The filter function is switched on and off by the dynamic detection of discontinuous solutions, and a slope limiter is employed to preserve the positivity of physically realizable solutions. As a result, the proposed method is able to capture the stochastic flow evolution where resolved and unresolved regions coexist. Numerical experiments including a wave propagation, a Burgers’ shock, a one-dimensional Riemann problem, and a two-dimensional shock-vortex interaction problem are presented to validate the current scheme. The order of convergence of the high-order scheme is identified. The capability of the scheme for simulating smooth and discontinuous stochastic flow dynamics is demonstrated. The open-source codes to reproduce the numerical results are available under the MIT license (Xiao et al. in FRSG: stochastic Galerkin method with flux reconstruction. https://github.com/CSMMLab/FRSG, (2021). https://doi.org/10.5281/zenodo.5588317).

A Radial Hybrid Estimation of Distribution Algorithm for the Truck and Trailer Routing Problem

The truck and trailer routing problem (TTRP) has been widely studied under different approaches. This is due to its practical characteristic that makes its research interesting. The TTRP continues to be attractive to developing new evolutionary algorithms. This research details a new estimation of the distribution algorithm coupled with a radial probability function from hydrogen. Continuous values are used in the solution representation, and every value indicates, in a hydrogen atom, the distance between the electron and the core. The key point is to exploit the radial probability distribution to construct offspring and to tackle the drawbacks of the estimation of distribution algorithms. Various instances and numerical experiments are presented to illustrate and validate this novel research. Based on the performance of the proposed scheme, we can make the conclusion that incorporating radial probability distributions helps to improve the estimation of distribution algorithms.

Bifurcations and the Exact Solutions of the Time-Space Fractional Complex Ginzburg-Landau Equation with Parabolic Law Nonlinearity

This paper studies the bifurcations of the exact solutions for the time–space fractional complex Ginzburg–Landau equation with parabolic law nonlinearity. Interestingly, for different parameters, there are different kinds of first integrals for the corresponding traveling wave systems. Using the method of dynamical systems, which is different from the previous works, we obtain the phase portraits of the the corresponding traveling wave systems. In addition, we derive the exact parametric representations of solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions, peakon solutions, periodic peakon solutions and compacton solutions under different parameter conditions.

Exact integrability conditions for cotangent vector fields

In Quantum Hydro-Dynamics the following problem is relevant: let (sqrt{rho },Lambda ) in {W^{1,2}}({mathbb {R}}^d,{mathcal {L}}^d,{mathbb {R}}^+) times L^2({mathbb {R}}^d,mathcal L^d,{mathbb {R}}^d) be a finite energy hydrodynamics state, i.e. Lambda = 0 when rho = 0 and E=∫Rd12|∇ρ|2+12Λ2Ld<∞.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} E = \\int _{{\\mathbb {R}}^d} \\frac{1}{2} \\big | \ abla \\sqrt{\\rho } \\big |^2 + \\frac{1}{2} \\Lambda ^2 {\\mathcal {L}}^d < \\infty . \\end{aligned}$$\\end{document}The question is under which conditions there exists a wave function psi in {W^{1,2}}({mathbb {R}}^d,{mathcal {L}}^d,{mathbb {C}}) such that ρ=|ψ|,J=ρΛ=ℑ(ψ¯∇ψ).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sqrt{\\rho } = |\\psi |, \\quad J = \\sqrt{\\rho } \\Lambda = \\Im \\big ( {\\bar{\\psi }} \ abla \\psi ). \\end{aligned}$$\\end{document}The second equation gives for psi = sqrt{rho } w smooth, |w| = 1, that i Lambda = sqrt{rho } {bar{w}} nabla w. Interpreting rho {mathcal {L}}^d as a measure in the metric space {mathbb {R}}^d, this question can be stated in generality as follows: given metric measure space (X,d,mu ) and a cotangent vector field v in L^2(T^* X), is there a function w in {W^{1,2}}(X,mu ,{mathbb {S}}^1) such that dw=iwv.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} dw = i w v. \\end{aligned}$$\\end{document}Under some assumptions on the metric measure space (X,d,mu ) (conditions which are verified on Riemann manifolds with the measure mu = rho textrm{Vol} or more generally on non-branching textrm{MCP}(K,N)), we show that the necessary and sufficient conditions for the existence of w is that (in the case of differentiable manifold) ∫v(γ(t))·γ˙(t)dt∈2πZ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int v(\\gamma (t)) \\cdot {\\dot{\\gamma }} (t) dt \\in 2\\pi {\\mathbb {Z}}\\end{aligned}$$\\end{document}for pi -a.e. gamma , where pi is a test plan supported on closed curves. This condition generalizes the condition that the vorticity is quantized. We also give a representation of every possible solution. In particular, we deduce that the wave function psi = sqrt{rho } w is in W^{1,2}(X,mu ,{mathbb {C}}) whenever sqrt{rho } in W^{1,2}(X,mu ,{mathbb {R}}^+).

The analytic extension of solutions to initial-boundary value problems outside their domain of definition

AbstractWe examine the analytic extension of solutions of linear, constant-coefficient initial-boundary value problems outside their spatial domain of definition. We use the Unified Transform Method or Method of Fokas, which gives a representation for solutions to half-line and finite-interval initial-boundary value problems as integrals of kernels with explicit spatial and temporal dependence. These solution representations are defined within the spatial domain of the problem. We obtain the extension of these representation formulae via Taylor series outside these spatial domains and find the extension of the initial condition that gives rise to a whole-line initial-value problem solved by the extended solution. In general, the extended initial condition is not differentiable or continuous unless the boundary and initial conditions satisfy compatibility conditions. We analyse dissipative and dispersive problems, and problems with continuous and discrete spatial variables.

Green location routing problem with flexible multi-compartment for source-separated waste: A Q-learning and multi-strategy-based hyper-heuristic algorithm

In this paper, we extend a novel model for source-separated waste collection and transportation, the green location routing problem with multi-compartment (GLRPFMC), for which we design a Q-learning and multi-strategy-based hyper-heuristic algorithm (QLMSHH). The remarkable merits of this paper can be highlighted as the following threefold: (1) The GLRPFMC is novel in that it constructs a variant of the location routing problem with carbon emissions and flexible multi-compartment sizes that occurs in a source-separated waste transportation context. (2) For the methodological contribution, the QLMSHH is presented to design a hyper-heuristic model by intelligently selecting appropriate high-level heuristic components during different stages of the optimization process. (3) The proposed method incorporates the design of solution representations, evolution-acceptance pairs for high-level heuristic construction, the repairing solution scheme, and the local search strategy. Finally, sufficient experiments are conducted on the benchmark, new instances, and simulation data of GLRPFMC and draw some managerial insights. The satisfactory results highlight the efficiency and universality of the proposed model and method.

Novel solutions to the coupled KdV equations and the coupled system of variant Boussinesq equations

This article examines the coupled KdV equations and the coupled system of variant Boussinesq equations with beta time derivative and explores their travelling wave solutions. This work explains the evolution of waves with fractional parameter. The simple ansatz approach produces a variety of novel solutions in terms of hyperbolic and periodic functions. Graphical representation of solutions are also presented.

Real representations of powers of real matrices and its applications

We give real representations of (or ) based on for a real square matrix A, where is the projection to the generalized eigenspace associated with an imaginary eigenvalue μ of A. Our method is based on the spectral decomposition theorem. As applications, we can easily obtain realifications of representations of solutions of inhomogeneous linear difference equations with constant coefficients.

Study of the Hypergeometric Equation via Data Driven Koopman-EDMD Theory

We consider a data-driven method, which combines Koopman operator theory with Extended Dynamic Mode Decomposition. We apply this method to the hypergeometric equation which is the Fuchsian equation with three regular singular points. The space of solutions at any of its singular points is a two-dimensional linear vector space on the field of reals when the independent variable is restricted to take values in the real axis and the unknown function is restricted to be a real-valued function of a real variable. A basis of the linear vector space of solutions is spanned by the hypergeometric function and its products with appropriate powers of the independent variable or the logarithmic function depending on the roots of the indicial equation of the hypergeometric equation. With our work, we obtain a new representation of the fundamental solutions of the hypergeometric equation and relate them to the spectral analysis of the finite approximation of the Koopman operator associated with the hypergeometric equation. We expect that the usefulness of our results will come more to the fore when we extend our study into the complex domain.