Search

In this work, the head-on collisions of the non-stationary dissipative soliton in ultracold neutral plasmas (UNPs) are investigated. The extended Poincare-Lighthill-Kuo (PLK) approach is adopted for reducing the fluid equations of the UNPs to two-counterpropagating damped Korteweg-de Vries (dKdV) equations. The dKdV equation is not an integrable Hamiltonian system, i.e., does not have an exact solution. Thus, one of the main goal of this paper is to find a new general approximate analytical solution to the dKdV equation for investigating the mechanism of the propagation and interaction of the non-stationary dissipative solitons. The residual error is estimated for checking the accuracy of the new obtained solution. The approximate analytical soliton solutions are adopted for deriving the temporal phase shifts after the collision. The impact of physical parameters on the nonstationary dissipative soliton profile and the temporal phase shifts is discussed. The obtained results will contribute to understand the mechanism of propagation and interaction of many nonlinear phenomena in different nonlinear mediums such as ocean, sea, optical fiber, plasma physics, etc.

In this study, the approximate and exact solutions for the stationary-state of the solids model with neglecting reactant consumption for both non-uniform and uniform temperature systems were applied on gas ignition under a constant pressure condition. The criticality conditions for a slab, an infinite cylinder, and a sphere are determined and discussed using dimensionless temperatures under constant ambient and surface temperatures for a non-uniform temperature system. Exact solution for a Semenov model with convection heat loss was also presented. The solution of the Semenov problem for constant volume or density as a solid and constant pressure were compared. The critical parameter δ is calculated and compared with those of Frank-Kamenetskii solution values. The validation of the calculated ignition temperatures with other exact solution and experimental results were offered. The relation between critical parameters form Semenov and F.K. models solution was introduced.

This work focuses on the numerical solution of hyperbolic conservations laws (possibly endowed with a source term) using the Active Flux method. This method is an extension of the finite volume method. Instead of solving a Riemann Problem, the Active Flux method uses actively evolved point values along the cell boundary in order to compute the numerical flux. Early applications of the method were linear equations with an available exact solution operator, and Active Flux was shown to be structure preserving in such cases. For nonlinear PDEs or balance laws, exact evolution operators generally are unavailable. Here, strategies are shown how sufficiently accurate approximate evolution operators can be designed which allow to make Active Flux structure preserving / well-balanced for nonlinear problems.

Introduction The Gross-Pitaevskii equation is a class of the nonlinear Schrodinger equation, whose exact solution, especially soliton solution, is proposed for understanding and studying Bose-Einstein condensate and some nonlinear phenomena occurring in the intersection field of Bose-Einstein condensate with some other fields. It is an important subject to investigate their exact solutions. Objectives We give multi-soliton of a two-dimensional Gross-Pitaevskii system which contains the time-varying trapping potential with a few interactions of multi-soliton. Through analytical and graphical analysis, we obtain one-, two- and three-soliton which are affected by the strength of atomic interaction. The asymptotic expression of two-soliton embodies the properties of solitons. We can give some interactions of solitons of different structures including parabolic soliton, line-soliton and dromion-like structure. Methods By constructing an appropriate Hirota bilinear form, the multi-soliton solution of the system is obtained. The soliton elastic interaction is analyzed via asymptotic analysis. Results The results in this paper theoretically provide the analytical bright soliton solution in the two-dimensional Bose-Einstein condensation model and their interesting interaction. To our best knowledge, the discussion and results in this work are new and important in different fields. The results enrich the existing nonlinear phenomena of the Gross-Pitaevskii model in Bose-Einstein condensation, and prove that the Hirota bilinear method and asymptotic analysis method are powerful and effective techniques in physical sciences and engineering for analyzing nonlinear mathematical-physical equations and their solutions. These provide a valuable basis and reference for the controllability of bright soliton phenomenon in experiments for high-dimensional Bose-Einstein condensation.

The flow through saturated porous soil is described by Darcy equation that generally has no analytical solution. Therefore, we have to use numerical methods to solve it. There exist several numerical methods and the well-known most used one is the finite elements method. In this work, we use a Diamond finite volumes scheme characterized with its conservation property to simulate Darcy equation in the linear and the non linear case in two dimensional case, based on permeability tensor. Different test cases are studied to validate the accuracy and the efficiency of the numerical model. The numerical scheme proves, through the obtained results, high level of accuracy comparing with the exact solution. The diamond finite volumes scheme for its ability to ensure the mass conservation of our model, knowing that the finite elements or the finite differences methods are less efficient on this side, our finite volumes method is the most suited to the Darcy flow and more faithful to physical problems comparing to other methods. We focused in our study on different aspects of the finite volumes scheme, the mass conservation, the non-linearity of the hydraulic conductivity and their interaction with the finite volumes method.

We present a progressive approximation algorithm for the exact solution of several classes of interdiction games in which two noncooperative players (namely an attacker and a follower) interact sequentially. The follower must solve an optimization problem that has been previously perturbed by means of a series of attacking actions led by the attacker. These attacking actions aim at augmenting the cost of the decision variables of the follower’s optimization problem. The objective, from the attacker’s viewpoint, is that of choosing an attacking strategy that reduces as much as possible the quality of the optimal solution attainable by the follower. The progressive approximation mechanism consists of the iterative solution of an interdiction problem in which the attacker actions are restricted to a subset of the whole solution space and a pricing subproblem invoked with the objective of proving the optimality of the attacking strategy. This scheme is especially useful when the optimal solutions to the follower’s subproblem intersect with the decision space of the attacker only in a small number of decision variables. In such cases, the progressive approximation method can solve interdiction games otherwise intractable for classical methods. We illustrate the efficiency of our approach on the shortest path, 0-1 knapsack and facility location interdiction games. Summary of Contribution: In this article, we present a progressive approximation algorithm for the exact solution of several classes of interdiction games in which two noncooperative players (namely an attacker and a follower) interact sequentially. We exploit the discrete nature of this interdiction game to design an effective algorithmic framework that improves the performance of general-purpose solvers. Our algorithm combines elements from mathematical programming and computer science, including a metaheuristic algorithm, a binary search procedure, a cutting-planes algorithm, and supervalid inequalities. Although we illustrate our results on three specific problems (shortest path, 0-1 knapsack, and facility location), our algorithmic framework can be extended to a broader class of interdiction problems.

The prize-collecting Steiner tree problem (PCSTP) is a well-known generalization of the classic Steiner tree problem in graphs, with a large number of practical applications. It attracted particular interest during the 11th DIMACS Challenge in 2014, and since then, several PCSTP solvers have been introduced in the literature. Although these new solvers further, and often drastically, improved on the results of the DIMACS Challenge, many PCSTP benchmark instances have remained unsolved. The following article describes further advances in the state of the art in exact PCSTP solving. It introduces new techniques and algorithms for PCSTP, involving various new transformations (or reductions) of PCSTP instances to equivalent problems, for example, to decrease the problem size or to obtain a better integer programming formulation. Several of the new techniques and algorithms provably dominate previous approaches. Further theoretical properties of the new components, such as their complexity, are discussed. Also, new complexity results for the exact solution of PCSTP and related problems are described, which form the base of the algorithm design. Finally, the new developments also translate into a strong computational performance: the resulting exact PCSTP solver outperforms all previous approaches, both in terms of runtime and solvability. In particular, it solves several formerly intractable benchmark instances from the 11th DIMACS Challenge to optimality. Moreover, several recently introduced large-scale instances with up to 10 million edges, previously considered to be too large for any exact approach, can now be solved to optimality in less than two hours. Summary of Contribution: The prize-collecting Steiner tree problem (PCSTP) is a well-known generalization of the classic Steiner tree problem in graphs, with many practical applications. The article introduces and analyses new techniques and algorithms for PCSTP that ultimately aim for improved (practical) exact solution. The algorithmic developments are underpinned by results on theoretical aspects, such as fixed-parameter tractability of PCSTP. Computationally, we considerably push the limits of tractibility, being able to solve PCSTP instances with up to 10 million edges. The new solver, which also considerably outperforms the state of the art on smaller instances, will be made publicly available as part of the SCIP Optimization Suite.

Addressing non-convexity plays a fundamental role in solving the optimal electricity-gas flow models. In this paper, an improved spatial branch-and-bound algorithm is proposed to solve the non-convex problem, which is formulated as a mixed-integer bilinear programming, for its exact solution. The core of the algorithm is to divide the non-convex model into small and convex sub-problems by branching on some specific continuous variables, so that the problem can be equivalent to a rooted tree for exploration. The exactness of the algorithm is guaranteed by the same criterion as the classical branch-and-bound algorithm. To alleviate the computational burden, a novel two-stage spatial branching strategy is developed to improve the effectiveness and efficiency of the branching operations. The performance of the proposed algorithm is verified on two integrated electricity-gas systems with different sizes. Numerical results demonstrate that our method achieves a balance among feasibility, optimality and efficiency. The comparison with another 6 convexification-based methods, 3 state-of-the-art non-convex optimization solvers, and 2 spatial branch-and-bound algorithms with classical branching rules further shows the superiority of the proposed algorithm.

In this research, the Shehu transform is coupled with the Adomian decomposition method for obtaining the exact-approximate solution of the plasma fluid physical model, known as the Zakharov-Kuznetsov equation (briefly, ZKE) having a fractional order in the Caputo sense. The Laplace and Sumudu transforms have been refined into the Shehu transform. The action of weakly nonlinear ion acoustic waves in a plasma carrying cold ions and hot isothermal electrons is investigated in this study. Important fractional derivative notions are discussed in the context of Caputo. The Shehu decomposition method (SDM), a robust research methodology, is effectively implemented to generate the solution for the ZKEs. A series of Adomian components converge to the exact solution of the assigned task, demonstrating the solution of the suggested technique. Furthermore, the outcomes of this technique have generated important associations with the precise solutions to the problems being researched. Illustrative examples highlight the validity of the current process. The usefulness of the technique is reinforced via graphical and tabular illustrations as well as statistics theory.

In this paper, we propose Push-SAGA, a decentralized stochastic first-order method for finite-sum minimization over a directed network of nodes. Push-SAGA combines node-level variance reduction to remove the uncertainty caused by stochastic gradients, network-level gradient tracking to address the distributed nature of the data, and push-sum consensus to tackle the challenge of directed communication links. We show that Push-SAGA achieves linear convergence to the exact solution for smooth and strongly convex problems and is thus the first linearly-convergent stochastic algorithm over arbitrary strongly connected directed graphs. We also characterize the regimes in which Push-SAGA achieves a linear speed-up compared to its centralized counterpart and achieves a network-independent convergence rate. We illustrate the behavior and convergence properties of Push-SAGA with the help of numerical experiments on strongly convex and non-convex problems.

Nonlinear contaminant transports through porous media are important in many scientific and engineering applications. In this paper, we develop and analyze fourth-order block-centered compact difference scheme (BCCDS) for the nonlinear contaminant transport equations with adsorption process in porous media. Based on block-centered mesh, a fourth order compact difference scheme of solution and its flux is derived. We prove the mass conservation of the proposed scheme and its unconditional stability. We analyze the convergence and obtain the fourth-order error estimate under the smooth regularity of exact solution. Numerical experiments are presented to show the performance of the schemes.

In this paper, we investigate the strong convergence of the exponential Euler method to stochastic delay differential equations with fractional Brownian motion (FSDDEs) of Hurst parameter H ∈ ( 1 2 , 1 ) . We establish the strong convergence rate H of the method for FSDDEs to the exact solution. Also we justify our theoretical results with some numerical examples of these equations alongside insignificant step size.

Aggregate nearest neighbor ( Ann ) query in both the euclidean space and road networks has been extensively studied, and the flexible aggregate nearest neighbor ( Fann ) problem further generalizes Ann by introducing an extra flexibility parameter $\phi$ ϕ that ranges in $(0, 1]$ ( 0 , 1 ] . In this article, we focus on Fann on road networks, denoted as Fann $_\mathcal {R}$ R , and its keyword-aware variant, denoted as KFann $_\mathcal {R}$ R . To solve these problems, we propose a series of universal (i.e., suitable for both max and sum ) algorithms, including a Dijkstra-based algorithm that enumerates $P$ P instead of $\phi |Q|$ ϕ | Q | -combinations of $Q$ Q , a queue-based approach that processes data points from-near-to-far, and a framework that combines incremental euclidean restriction (IER) and $k$ k NN. We also propose a specific exact solution to max - Fann $_\mathcal {R}$ R and a constant-factor ratio approximate solution to sum - Fann $_\mathcal {R}$ R . These specific algorithms are easy to implement and can achieve excellent performance in some scenarios. Besides, we further extend this problem to top- $k$ k and multiple Fann $_\mathcal {R}$ R (resp., KFann $_\mathcal {R}$ R ) queries. We conduct a comprehensive experimental evaluation for the proposed algorithms on real datasets to demonstrate their superior efficiency and high quality.

We revisit the classical mechanics problem of a particle moving under the influence of a force that depends on its velocity. Using the properties of the rotation matrix and associated operators, we show that it is possible to find an exact analytical solution to a number of problems where the differential equation of motion depends on the velocity. First, we apply our method to the well-known cases of a particle under the influence of the Lorentz force and Coriolis force, providing the complete analytical solution in each case for the motion of the particle in three dimensions. We also show that the complete solution can be obtained when the centrifugal force is included, showing the applicability to cases where there is simultaneous dependence on the position and velocity. This method, which is not currently discussed in a typical course in elementary mechanics, provides an alternative approach to the traditional methods that are used to solve these types of problems.

The present article discusses a charged spherical solution in modified f ( G , T ) gravity under the embedding class I condition, where f ( G , T ) = G 2 + χ T , where G is a Gauss–Bonnet term and T denotes a trace of energy momentum tensor, while χ is called coupling constant. The Exact solution of the Einstein–Maxwell’s field equations in f ( G , T ) gravity is obtained by taking a particular ansatz for radial component of the line element. The validity of the charged spherical solution has been tested via different physical analysis. The physical analysis shows that the present solution is well behaved. Moreover, we plotted the graphs for a particular star Vela X-1 for different χ , which show the trends of the thermodynamical observables like pressure, density, electric charge, energy conditions as well as impact of χ parameter on the above thermodynamical observables. On the other hand, we predicted the radii for 12 stars with different χ . The mass–radius ratio and surface red-shift are also calculated. Also, we have discussed the stability and equilibrium condition of the solution. Overall, the above analysis shows that obtained solution describes a physically viable and stable compact star. The prediction of radii from GW 170817 is also in agreement with our M − R curve.

In this work, we construct two numerical approximations methods to deal with approximations of a linear Fredholm integro-differential equation. In order to show the existence and uniqueness of the solution we start by the reformulation of the integro-differential equation to a system of integral equations. We explain the general framework of the projection method which helps to clarify the basic ideas of the Collocation and Kantorovich methods. We introduce a new iterative method to avoid inversing the block operator matrix. Next, we apply the iterative projection methods and we present theorems to show the convergence of the constructed solutions to the exact solution. Finally, to observe the error behavior of the two methods, we give two numerical examples.

In this paper, the series solutions of a non-linear delay integral equations are considered by a modified approach of homotopy analysis method (MAHAM). We split the function into infinite sums. The outcomes of the illustrated examples are included to confirm the accuracy and efficiency of the MAHAM. The exact solution can be obtained using special values of the convergence parameter.

In this article, a singular perturbation delay problem of convection–diffusion (C–D) type having an integral boundary condition is considered. The analytical solution of the considered problem has a weak interior layer in addition to the boundary layer at the right end of the domain. Some a priori estimates are given on the exact solution which are useful for the error analysis. The numerical approximation is composed of a hybrid finite difference scheme on a generalized Shishkin mesh. For the proposed scheme, almost second order $$\varepsilon $$ -uniform convergence is established. Numerical experiments are conducted to corroborate the theoretical results. A comparison with the existing scheme (J Appl Math Comput 63:813–828) is also performed.

Accurate prediction of temperature distribution during thermotherapy is a significant factor in the thermotherapy process. Thermotherapy equipment produces a different distribution spatial and time-dependent heat fluxes in the thermotherapy processes. This paper presents an exact analytical solution of steady and unsteady Pennes and porous bioheat equations in a cylindrical coordinate system for multi-layer skin with different spatial and time-dependent heat fluxes on the surface. The proposed analytical solutions are useful to make accurate temperature distribution in multi-layer skin tissue with various properties. The results show that the unsteady temperature distributions in both Pennes and porous models are the same in the initial times of process. When the temperature rises, the cooling effect of blood perfusion in the Pennes model increases the difference in temperature distribution for these two models. The skin surface temperature is linear versus time in constant and linear fluxes, and skin temperature increment has a second-degree trend versus time in the quadratic flux. The magnitude of the flux coefficient and the time of applying flux to the tissue are effective in increasing the temperature of the tissue and the skin surface.

Using a suitable gauge transformation matrix, we present a N -fold Darboux transformation for a Vakhnenko lattice system. This transformation preserves the form of Lax pair of the Vakhnenko lattice system. Applying the obtained Darboux transformation, we arrive at an exact solution of the Vakhnenko lattice system.

В настоящей работе представлен новый метод решения уравнений движения заряженных частиц в электромагнитных полях и проведено его сравнение с различными известными модификациями метода Бориса. Созданные двумерный и трехмерный алгоритмы основаны на использовании точного решения дифференциального уравнения для скорости заряженной частицы на шаге по времени. Сравнительный анализ метода Бориса и его модификаций проводился как по точности методов, так и по времени их работы. Новая модификация метода Бориса позволяет точнее вычислять траекторию и скорость заряженной частицы без значительного увеличения сложности расчетов. Показано, что при выборе модификации метода Бориса для решения задачи в первую очередь следует обращать внимание на точность решения, так как более простая и быстрая схема может не дать выигрыша по времени. The article proposes a new method for solving the equations of motion of charged particles in electromagnetic fields and compares this method with various known modifications of the Boris method. The created two-dimensional and three-dimensional algorithms are based on the use of an exact solution of the differential equation for the velocity of a charged particle at a time step. A comparative analysis of the Boris method and its modifications was carried out both in terms of the accuracy of the methods and the time of their operation. A new modification of the Boris method allows more accurate calculations of the trajectory and velocity of a charged particle without a significant increase in the complexity of calculations. It is shown that, when choosing a modification of the Boris method to solve a problem, one should pay attention first of all to the accuracy of the solution, since a simpler and faster scheme may not give a gain in time.

Multi-manned assembly lines are generally used to produce large-sized volume products such as cars and trucks. This article addresses the multi-manned two sided assembly line balancing problem with the objectives sharing tool between workstations. This paper presents a mathematical model and a Lingo -16 solvers based exact algorithm for multi-manned two-sided assembly line system configuration with tool sharing between adjacent workstations for companies that need intelligent solutions to satisfy customized demands on time with existing resources. The results obtained indicate that tool shared between parallel stations of two or more parallel lines beneficial for assembly line to minimize workstations, idle time and reduce tool cost.

A condensed review is presented for two basic topics in the theory of pattern formation in nonlinear dissipative media: (i) domain walls (DWs, alias grain boundaries), which appear as transient layers between different states occupying semi-infinite regions, and (ii) two- and three-dimensional (2D and 3D) quasiperiodic (QP) patterns, which are built as a superposition of plane–wave modes with incommensurate spatial periodicities. These topics are selected for the present review, dedicated to the 70th birthday of Professor Michael I. Tribelsky, due to the impact made on them by papers of Prof. Tribelsky and his coauthors. Although some findings revealed in those works may now seem “old”, they keep their significance as fundamentally important results in the theory of nonlinear DW and QP patterns. Adding to the findings revealed in the original papers by M.I. Tribelsky et al., the present review also reports several new analytical results, obtained as exact solutions to systems of coupled real Ginzburg–Landau (GL) equations. These are a new solution for symmetric DWs in the bimodal system including linear mixing between its components; a solution for a strongly asymmetric DWs in the case when the diffusion (second-derivative) term is present only in one GL equation; a solution for a system of three real GL equations, for the symmetric DW with a trapped bright soliton in the third component; and an exact solution for DWs between counter-propagating waves governed by the GL equations with group-velocity terms. The significance of the “old” and new results, collected in this review, is enhanced by the fact that the systems of coupled equations for two- and multicomponent order parameters, addressed in this review, apply equally well to modeling thermal convection, multimode light propagation in nonlinear optics, and binary Bose–Einstein condensates.

This paper presents a novel Petrov-Galerkin method for a class of linear systems of fractional differential equations. New fractional-order generalized Jacobi functions are introduced, and their approximation properties are investigated. We show that these functions satisfy the given supplementary conditions and have the same asymptotic behavior as the exact solution, which are essential properties to design a high-order spectral Petrov-Galerkin method. For implementing our scheme, we represent the approximate solution by a linear combination of fractional-order generalized Jacobi functions and minimize the residual using shifted fractional Jacobi functions. The numerical solvability of the resultant algebraic system is justified as well as convergence and stability properties of the proposed scheme are explored. Finally, the reliability of the method is evaluated using various analytical and realistic problems.

This paper is devoted to introducing a novel methodology to prove the convergence and stability of a Crank–Nicolson difference approximation for a class of multi-term time-fractional diffusion equations with nonlinear delay and space fractional derivatives in case of sufficient smooth solutions. The temporal fractional derivatives are approximated by a specific form of L1 scheme at t k + 1 / 2 . A fourth-order difference approximation for the spatial fractional derivatives is employed by using the weighted average of the shifted Grunwald formulae. This methodology is based on a class of discrete fractional Gronwall inequalities convenient with the quadrature formula used to approximate the Caputo derivative at t k + 1 / 2 . In the present work, the method of energy inequalities is utilized to show that the used difference scheme is stable and converges to the exact solution with order O ( τ 2 − α J + h 4 ) , in the case that 0 α J 1 satisfies 3 α J ≥ 3 2 , which means that 0.369 ≤ α J 1 , such that α J is the maximum α-th order in the multi-order fractional operators.

This work constructs and analyzes a nonlocal evolution equation with a weakly singular kernel in three-dimensional space. In the temporal direction, the Crank-Nicolson (CN) method and product-integration (PI) rule are employed, from which the non-uniform meshes are used to eliminate the singular behaviour of the exact solution at t = 0 . Then, a fully discrete scheme is obtained by the spatial discretization based on the finite difference method. Simultaneously, an alternating direction implicit (ADI) algorithm is designed to reduce the computational cost. The stability in L 2 norm and convergence are derived via the energy method, in which the convergence orders are O ( k 2 + h 2 ) with certain suitable assumptions, where k and h are corresponding space-time step sizes, respectively. Numerical results confirm the theoretical analysis.

This paper is concerned with the numerical approximation of the time fractional Burgers equation with nonsmooth solutions. A nonlinear fully discrete scheme is presented based on the nonuniform Alikhanov formula of the Caputo time fractional derivative and Fourier spectral approximation in space. The solvability of the scheme is proved by the fixed point theorem and a priori estimate. Using the error estimation method, the proposed scheme is stable and convergent with order O ( τ min ⁡ { γ σ , 2 } + N 1 − m ) , where τ , N , γ , m and σ are the maximum time step size, polynomial degree, grading parameter, spatial regularity and temporal regularity parameter of the exact solution, respectively. A numerical example is performed to support the theoretical results.

We propose a high-order numerical scheme for nonlinear time fractional reaction-diffusion equations with initial singularity, where L2- 1 σ scheme on graded mesh is used to approximate Caputo fractional derivative and Legendre spectral method is applied to discrete spatial variable. We give the priori estimate, existence and uniqueness of numerical solution. Then the unconditional stability and convergence are proved. The rate of convergence is O ( M − min ⁡ { r α , 2 } + N − m ) , which is obtained without extra regularity assumption on the exact solution. Numerical results are given to confirm the sharpness of error analysis.

We connected the two ends of a finite spin-1/2 antiferromagnetic Ising chain with a magnetic impurity at one end to form a closed ring, and studied its magnetic susceptibility exactly by using the transfer matrix method. We calculated the magnetic susceptibility in the whole temperature range and gave the phase diagram at ground state of the system about the anisotropy of the impurity and strength of the connection exchange interaction for spin-1 and -3/2 impurities. We also gave the ground state entropy of the system and derived the asymptotic expression of the magnetic susceptibility multiplied by temperature at the zero-temperature limit and the high-temperature limit. It was found that a degenerate phase may exist in some parameter region at zero temperature for the odd spin number system, and the ground state entropy is ln(2) in the nondegenerate phase and is dependent on the spin number in the degenerate phase. The magnetic susceptibility of the system at low temperature exhibits ferromagnetic behavior, and the Curie constant is related to the spin configuration at ground state. When the ground state is nondegenerate, the Curie constant is equal to the square of the net spin, regardless of the parity of the number of the spin. When the spin number is odd and the ground state is degenerate, the Curie constant may be related to the total spin number. At the high-temperature limit, the magnetic susceptibility multiplied by temperature is related to the spin quantum number of the impurity and the spin number in the ring.

The idea of the paper is to consolidate Mahgoub transform and variational iteration method (MTVIM) to solve fractional delay differential equations (FDDEs). The fractional derivative was in Caputo sense. The convergences of approximate solutions to exact solution were quick. The MTVIM is characterized by ease of application in various problems and is capable of simplifying the size of computational operations. Several non-linear (FDDEs) were analytically solved as illustrative examples and the results were compared numerically. The results for accentuating the efficiency, performance, and activity of suggested method were shown by comparisons with Adomian Decomposition Method (ADM), Laplace Adomian Decomposition Method (LADM), Modified Adomian Decomposition Method (MADM) and Homotopy Analysis Method (HAM).

In this paper, extended H $$\ddot{o}$$ lder class $$H_\alpha ^{(w)}[0,\mu )$$ is considered. This class is the generalization of H $$\ddot{o}$$ lder class $$H_\alpha [0,\mu )$$ . Three new estimators $$E_N^{(1)}(f), E_N^{(2)}(f)$$ and $$E_N^{(3)}(f)$$ of functions of classes $$H_\alpha [0,\mu )$$ and $$H_\alpha ^{(w)}[0,\mu )$$ have been obtained. These estimators are best in approximation of functions by wavelet methods. The estimators obtained in this paper and the solution of ordinary and partial differential equations by extended Haar wavelet operational matrix method in the interval $$[0,\mu )$$ and its comparison with exact solution for different values of $$\mu$$ are the significant achievement of this research paper in approximation theory as well as Wavelet Analysis.

This paper considers the numerical treatment of the time-dependent Gross–Pitaevskii equation. In order to conserve the time invariants of the equation as accurately as possible, we propose a Crank–Nicolson-type time discretization that is combined with a suitable generalized finite element discretization in space. The space discretization is based on the technique of Localized Orthogonal Decompositions and allows to capture the time invariants with an accuracy of order O ( H 6 ) \mathcal {O}(H^6) with respect to the chosen mesh size H H . This accuracy is preserved due to the conservation properties of the time stepping method. Furthermore, we prove that the resulting scheme approximates the exact solution in the L ∞ ( L 2 ) L^{\infty }(L^2) -norm with order O ( τ 2 + H 4 ) \mathcal {O}(\tau ^2 + H^4) , where τ \tau denotes the step size. The computational efficiency of the method is demonstrated in numerical experiments for a benchmark problem with known exact solution.

In this paper, meant as a companion to arXiv:2006.04458, we consider a class of non-integrable $2D$ Ising models in cylindrical domains, and we discuss two key aspects of the multiscale construction of their scaling limit. In particular, we provide a detailed derivation of the Grassmann representation of the model, including a self-contained presentation of the exact solution of the nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic estimates of the fermionic Green's function in the cylinder, required for the multiscale analysis of the model. We also review the multiscale construction of the effective potentials in the infinite volume limit, in a form suitable for the generalization to finite cylinders. Compared to previous works, we introduce a few important simplifications in the localization procedure and in the iterative bounds on the kernels of the effective potentials, which are crucial for the adaptation of the construction to domains with boundaries., 69 pages. Minor revisions. Version accepted by Annales Henri Poincar\'e. arXiv admin note: substantial text overlap with arXiv:2006.04458

This article presents triple Laplace transform coupled with iterative method to obtain the exact solution of two-dimensional nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions. The noise term in this equation is vanished by successive iterative method. The proposed technique has the advantage of producing exact solution, and it is easily applied to the given problems analytically. Four test problems from mathematical physics are taken to show the accuracy, convergence, and the efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.

In this paper, the problem of finding the source function for the Rayleigh–Stokes equation is considered. According to Hadamard’s definition, the sought solution of this problem is both unstable and independent of continuous data. By using the fractional Tikhonov method, we give the regularized solutions and then deal with a priori error estimate between the exact solution and its regularized solutions. Finally, the proposed regularized methods have been verified by simple numerical experiments to check error estimate between the sought solution and the regularized solution.

In this paper, an effective numerical technique based on Lucas and Fibonacci polynomials coupled with finite differences is developed for the solution of nonlinear reaction–diffusion Brusselator system. The system arises in modeling of chemical processes such as enzymatic reactions, plasma and laser physics in multiple coupling between modes and in the formation of ozone by atomic oxygen via a triple collision. The proposed scheme first converts the problem to discrete form and then with a collocation approach to a system of linear equations which is easily solvable. Performance of the method is checked by solving one- and two-dimensional test problems. Validation of the results is examined in terms of L ∞ , L 2 and relative error L R norms. In case of no exact solution, quality of computed solution is examined for small value of diffusion coefficient η . It is observed that when 1 - ξ + β > 0 , the solutions converges to equilibrium points ( β , ξ / β ) .

This work adopts to the time-fractional Klein–Gordon equation (FKGE) in the Caputo sense. We present a new technique using the clique polynomial as basis function for the operational matrices to obtain solution of time-FKGE. The key advantage of this technique is converting the time-FKGE to algebraic equations, which can be simply solved the problem under study. For the approximation of a bivariate function using the clique polynomial, an error bound is given. Numerical results derived using the proposed technique are compared with the exact solution. The results show that the proposed technique is very user friendly for solving the time-FKGE and accurate.

The leading objective of this article is to investigate the analytical and numerical solutions of the Generalized Benjamin-Bona-Mahony (GBBM) equation. We also aim to compare the performance of the considered methods for solving this equation. The exact solution is obtained analytically while the numerical solutions are demonstrated using some techniques, namely, the adaptive moving mesh and uniform mesh methods. The exact solution is presented in a form of convergent power series. The finite differences are also applied to discretise the BBM equation. Under a suitable selection of parameters, some 2D and 3D surfaces for the obtained theoretical and numerical results are shown to compare the exact and numerical solutions.

On the basis of the general relativistic statistical and kinetic theory, a consistent closed cosmological model is formulated. It is based on a statistical system of scalar charged fermions interacting by means of classical and phantom scalar fields. Based on the study of the microscopic dynamics of scalar charged particles, within the framework of the Lagrangian as well as Hamiltonian formalism, a function of the dynamic mass of scalar charged particles was constructed and it was shown that for the consistency of the theory, it is necessary to remove the nonnegativity condition for this function. On the basis of the Lagrangian formalism, equations of gravitational and scalar fields with singular sources are formulated and microscopic conservation laws are obtained. Within the framework of the general relativistic kinetic theory, macroscopic equations of gravitational and scalar fields are formulated and macroscopic conservation laws are obtained. These equations' full correspondence to microscopic equations with singular sources is shown. Further, on the basis of the obtained equations, a cosmological model for a degenerate system of scalarly charged fermions is formulated. An exact solution of the constitutive equations for a degenerate scalar-charged plasma in the cosmological model is obtained, which made it possible to significantly simplify the original system of equations. On the basis of the obtained solution of the constitutive equations, two fundamentally different cosmological models are formulated, one of which has two types of singly scalarly charged fermions, while the second has one kind of fermions charged with two charges of various nature. A qualitative analysis of the obtained 6-dimensional dynamic system for a two-component model is carried out., 31 pages, 70 references

The purpose of this article is to clarify certain ambiguities associated with commonly adopted solution strategies of the problem of thin rectangular plates subjected to equal and opposite colinear point loads. The problem is treated as a plane-stress problem where the boundary conditions are prescribed exclusively in terms of tractions. Both an elaborate exact solution and a simplified approximate Single Fourier Series (SFS) solution are employed, shedding light on the accuracy of the latter. This is accomplished, first, by showing that the error implicit in the sine single-Fourier series solution becomes significant for column plates, and, secondly, by re-evaluating the stress distribution at various cross-sections away from the compressed sides, through the application of a truly exact solution to the problem at hand, in which the free-edge conditions are precisely fulfilled. It is shown that the restrictions usually associated with the SFS technique may not be severe and that a wider range of plate shapes with free vertical sides could be amenable to analysis utilizing this simple approximate scheme.

In this article, the mixed Fourier Legendre spectral Galerkin (MFLSG) methods are considered to solve the two-dimensional Fredholm integral equations ( fie s) on the Banach spaces with smooth kernel. The same methods are also considered to find the eigenvalues of the eigenvalue problems ( evp s) associated with the two-dimensional fie s. Making use of these methods, we establish the error between the approximated solution as well as iterated approximate solution versus exact solution for two-dimensional fie s in both L 2 and L ∞ norms. We also establish the error between approximated eigen-values, eigen-vectors and iterated eigen-vectors and exact eigen-elements by MFLSG methods in L 2 and L ∞ norms. The numerical illustrations are introduced for the error of these methods.

Interior-penalized weak Galerkin (IPWG) finite element methods are proposed and analyzed for solving second order elliptic equations. The new methods employ the element $(\mathbb{P}_{k}, \mathbb{P}_{k}, \mathcal{RT}_{k})$, with dimensions of space $d = 2, 3$, and the optimal a priori error estimates in discrete $H^1$-norm and $L^2$-norm are established. Moreover, provided enough smoothness of the exact solution, superconvergence in $H^1$ and $L^2$ norms can be derived. Some numerical experiments are presented to demonstrate flexibility, effectiveness and reliability of the IPWG methods. In the experiments, the convergence rates of the IPWG methods are optimal in $L^2$-norm, while they are suboptimal for NIPG and IIPG if the polynomial degree is even.

In this paper, we consider an inverse problem for time-fractional advection-dispersion equation in a multi-layer composite medium. The main goal of our paper is to approximate the initial information, which is inaccessible for measurement, from the observation data at a certain point in second layer by constructing a regularized solution using a filter regularization method. Under appropriate regularity assumptions of the exact solution, we give convergence rate of the error between the reconstructed solution and the exact one. A numerical example is provided to illustrate the main results.

In this paper, an exact solution for nonlinear static behaviors of functionally graded (FG) beams with porosities resting on the elastic foundation is presented. The FG material properties with porosities are assumed to vary along the thickness of the beam, and two types of porosity distributions are considered. Actually, the geometrical middle surface of the FG beam selected in computations is very popular in the literature. By contrast, in this study, the physical neutral surface of the beam is utilized. Based on the Timoshenko beam theory, von Kármán nonlinear assumption, together with neutral surface concept, the nonlinear governing equations of the FG beam resting on the elastic foundation are derived. By using the physical neutral surface, the nonlinear governing equations have simple forms and can be solved directly. The exact solution for the problem with all immovable and moveable boundary conditions is conducted in detail. Some numerical investigations to show the effects of boundary conditions, material properties, length-to-thickness ratio, elastic foundation coefficients and several types of applied load on nonlinear static bending behaviors of the beam are given.

This paper presents an analysis solution for systems of partial differential equations using a new modification of the decomposition method to overcome the computational difficulties. Convergence of series solution was discussed with two illustrated examples, and the method showed a high-precision, being a fast approach to solve the non-linear system of PDEs with initial conditions. There is no need to convert the nonlinear terms into the linear ones due to the Adomian polynomials. The method does not require any discretization or assumption for a small parameter to be present in the problem. The steps of the suggested method are easily implemented, with high accuracy and rapid convergence to the exact solution, compared with other methods that can be used to solve systems of PDEs.

Lower and upper semicontinuity results for the approximate solution mapping of the parametric bifunction-set optimization problem are established under new assumptions that are quite different from the ones used previously in the case of exact solution mapping. Applications to the stability study of the approximate solution mapping of a parametric Kuroiwa set optimization problem and a parametric vector Ky Fan inequality problem are given. To our knowledge, our stability results are original. Several examples are provided.

The main purpose of this paper is to introduce a new formulation of the Tau-collocation method for solving a class of nonlinear weakly singular delay Volterra integro-differential equations based on fractional Muntz Jacobi polynomials. So the paper consists of two main parts: studying the regularity of solution and introducing a simple structure and efficient numerical method. The method generates approximate solution with fractional power terms having the same behavior of the exact solution of the given problem. The convergence analysis of the method is also investigated in $$L^{2}$$ -norm. Some illustrative examples are given to confirm the theoretical results and the accuracy of the approximate solution. The paper is closed by providing real application of the method to some physical problems.