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Journal articles on the topic 'Convergence des solutions'

Author: Grafiati

Published: 4 June 2021

Last updated: 18 February 2022

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1

Hayase, Toshiyuki. "Monotonic Convergence Property of Turbulent Flow Solution With Central Difference and QUICK Schemes." Journal of Fluids Engineering 121, no. 2 (June 1, 1999): 351–58. http://dx.doi.org/10.1115/1.2822213.

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Monotonic convergence of numerical solutions with the computational grid refinement is an essential requirement in estimating the grid-dependent uncertainty of computational fluid dynamics. If the convergence is not monotonic, the solution could be erroneously regarded as convergent at the local extremum with respect to some measure of the error. On the other hand, if the convergence is exactly monotonic, estimation methods such as Richardson extrapolation properly evaluate the uncertainty of numerical solutions. This paper deals with the characterization of numerical schemes based on the property of the monotonic convergence of numerical solutions. Two typical discretization schemes of convective terms were considered; the second-order central difference scheme and the third-order Leonard’s QUICK scheme. A fully developed turbulent flow through a square duct was calculated via a SIMPLER based finite volume method without a turbulence model. The convergence of the numerical solution with the grid refinement was investigated for the mean flow property as well as fluctuations. The comparison of convergence process between the discretization schemes has revealed that the QUICK scheme results in preferable monotonic convergence, while the second-order central difference scheme undergoes non-monotonic convergence. The latter possibly misleads the determination of convergence with the grid refinement, or causes trouble in applying the Richardson extrapolation procedure to estimate the numerical uncertainty.

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2

Xinfu, Chen, Charlie M. Elliot, Gardiner Andy, and Jennifer Jing Zhao. "Convergence of numerical solutions." Applicable Analysis 69, no. 1-2 (June 1998): 95–108. http://dx.doi.org/10.1080/00036819808840645.

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3

Amleh, A. M., and G. Ladas. "Convergence to periodic solutions." Journal of Difference Equations and Applications 7, no. 4 (January 2001): 621–31. http://dx.doi.org/10.1080/10236190108808292.

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4

Wang, Qi. "Application of Optimal Homotopy Analysis Method for Solitary Wave Solutions of Kuramoto-Sivashinsky Equation." Zeitschrift für Naturforschung A 66, no. 1-2 (February 1, 2011): 117–22. http://dx.doi.org/10.1515/zna-2011-1-216.

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In this paper, the optimal homotopy analysis method is applied to find the solitary wave solutions of the Kuramoto-Sivashinsky equation. With three auxiliary convergence-control parameters, whose possible optimal values can be obtained by minimizing the averaged residual error, the method used here provides us with a simple way to adjust and control the convergence region of the solution. Compared with the usual homotopy analysis method, the optimal method can be used to get much faster convergent series solutions.

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5

Lukas, Mark A. "Convergence rates for regularized solutions." Mathematics of Computation 51, no. 183 (September 1, 1988): 107. http://dx.doi.org/10.1090/s0025-5718-1988-0942146-8.

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6

Kindermann, Stefan. "Projection Methods for Ill-Posed Problems Revisited." Computational Methods in Applied Mathematics 16, no. 2 (April 1, 2016): 257–76. http://dx.doi.org/10.1515/cmam-2015-0036.

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AbstractWe consider the discretization of least-squares problems for linear ill-posed operator equations in Hilbert spaces. The main subject of this article concerns conditions for convergence of the associated discretized minimum-norm least-squares solution to the exact solution using exact attainable data. The two cases of global convergence (convergence for all exact solutions) or local convergence (convergence for a specific exact solution) are investigated. We review the existing results and prove new equivalent conditions when the discretized solution always converges to the exact solution. An important tool is to recognize the discrete solution operator as an oblique projection. Hence, global convergence can be characterized by certain subspaces having uniformly bounded angles. We furthermore derive practically useful conditions when this holds and put them into the context of known results. For local convergence, we generalize results on the characterization of weak or strong convergence and state some new sufficient conditions. We furthermore provide an example of a bounded sequence of discretized solutions which does not converge at all, not even weakly.

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7

Diblík, J., M. Růžičková, Z. Šmarda, and Z. Šutá. "Asymptotic Convergence of the Solutions of a Dynamic Equation on Discrete Time Scales." Abstract and Applied Analysis 2012 (2012): 1–20. http://dx.doi.org/10.1155/2012/580750.

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The paper investigates a dynamic equationΔy(tn)=β(tn)[y(tn−j)−y(tn−k)]forn→∞, wherekandjare integers such thatk>j≥0, on an arbitrary discrete time scaleT:={tn}withtn∈ℝ,n∈ℤn0−k∞={n0−k,n0−k+1,…},n0∈ℕ,tn<tn+1,Δy(tn)=y(tn+1)−y(tn), andlimn→∞tn=∞. We assumeβ:T→(0,∞). It is proved that, for the asymptotic convergence of all solutions, the existence of an increasing and asymptotically convergent solution is sufficient. Therefore, the main attention is paid to the criteria for the existence of an increasing solution asymptotically convergent forn→∞. The results are presented as inequalities for the functionβ. Examples demonstrate that the criteria obtained are sharp in a sense.

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8

Feireisl, Eduard. "Time-periodic solutions of a quasilinear beam equation via accelerated convergence methods." Applications of Mathematics 33, no. 5 (1988): 362–73. http://dx.doi.org/10.21136/am.1988.104317.

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9

Mikhailov, Mikhail D., and Renato M. Cotta. "CONVERGENCE ACCELERATION OF INTEGRAL TRANSFORM SOLUTIONS." Hybrid Methods in Engineering 3, no. 1 (2001): 6. http://dx.doi.org/10.1615/hybmetheng.v3.i1.70.

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10

Gheorghe, Gabriel, and Daniel Craciun. "SMART GRID / SMART METERING CONVERGENCE SOLUTIONS." EMERG - Energy. Environment. Efficiency. Resources. Globalization 6, no. 1 (2020): 11–29. http://dx.doi.org/10.37410/emerg.2020.1.01.

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11

Raj, Mayank, Avinash Narayan, Sajib Datta, Sajal Das, and Jogen Pathak. "Fixed mobile convergence: challenges and solutions." IEEE Communications Magazine 48, no. 12 (December 2010): 26–34. http://dx.doi.org/10.1109/mcom.2010.5673069.

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12

Knees, Dorothee. "Convergence analysis of time-discretisation schemes for rate-independent systems." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 65. http://dx.doi.org/10.1051/cocv/2018048.

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It is well known that rate-independent systems involving nonconvex energy functionals in general do not allow for time-continuous solutions even if the given data are smooth. In the last years, several solution concepts were proposed that include discontinuities in the notion of solution, among them the class of global energetic solutions and the class of BV-solutions. In general, these solution concepts are not equivalent and numerical schemes are needed that reliably approximate that type of solutions one is interested in. In this paper, we analyse the convergence of solutions of three time-discretisation schemes, namely an approach based on local minimisation, a relaxed version of it and an alternate minimisation scheme. For all three cases, we show that under suitable conditions on the discretisation parameters discrete solutions converge to limit functions that belong to the class of BV-solutions. The proofs rely on a reparametrisation argument. We illustrate the different schemes with a toy example.

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13

Jenschke, Tristan. "Approximation of minimal surfaces with free boundaries: convergence results." IMA Journal of Numerical Analysis 39, no. 3 (July 17, 2018): 1391–420. http://dx.doi.org/10.1093/imanum/dry043.

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Abstract In a previous paper we developed a penalty method to approximate solutions of the free boundary problem for minimal surfaces by solutions of certain variational problems depending on a parameter $\lambda $. There we showed existence and $C^2$-regularity of these solutions as well as convergence to the solution of the free boundary problem for $\lambda \to \infty $. In this paper we develop a fully discrete finite element procedure for approximating solutions of these variational problems and prove a convergence estimate, which includes an order of convergence with respect to the grid size.

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14

Shah, Rasool, Hassan Khan, Muhammad Arif, and Poom Kumam. "Application of Laplace–Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations." Entropy 21, no. 4 (March 28, 2019): 335. http://dx.doi.org/10.3390/e21040335.

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In the present article, we related the analytical solution of the fractional-order dispersive partial differential equations, using the Laplace–Adomian decomposition method. The Caputo operator is used to define the derivative of fractional-order. Laplace–Adomian decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. The fractional order solutions that are convergent to integer order solutions are also investigated.

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15

Rzezuchowski, Tadeusz. "Strong convergence of selections implied by weak." Bulletin of the Australian Mathematical Society 39, no. 2 (April 1989): 201–14. http://dx.doi.org/10.1017/s0004972700002677.

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In some situations weak convergence in L1, implies strong convergence. Let P, L: T → C∘(ℝd) be measurable multifunctions (C∘(ℝd) being the set of closed, convex subsets of ℝd) the values L(t) affine sets and W(t) = P(t) ∩ L(t) extremal faces of P(t). Let pk be integrable selections of P, the projection of pk,(t) on L(t) and pk(t) on W(t). We prove that if converges weakly to zero then pk − k converges to zero in measure. We give also some extensions of this theorem. As applications to differential inclusions we investigate convergence of derivatives of convergent sequences of solutions and we describe solutions which are in some sense isolated. Finally we discuss what can be said about control functions u when the corresponding trajectories of ẋ = f(t, x, u) are convergent to some trajectory.

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16

Berezansky, L., J. Diblík, M. Růžičková, and Z. Šutá. "Asymptotic Convergence of the Solutions of a Discrete Equation with Two Delays in the Critical Case." Abstract and Applied Analysis 2011 (2011): 1–15. http://dx.doi.org/10.1155/2011/709427.

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A discrete equationΔy(n)=β(n)[y(n−j)−y(n−k)]with two integer delayskandj, k>j≥0is considered forn→∞. We assumeβ:ℤn0−k∞→(0,∞), whereℤn0∞={n0,n0+1,…}, n0∈ℕandn∈ℤn0∞. Criteria for the existence of strictly monotone and asymptotically convergent solutions forn→∞are presented in terms of inequalities for the functionβ. Results are sharp in the sense that the criteria are valid even for some functionsβwith a behavior near the so-called critical value, defined by the constant(k−j)−1. Among others, it is proved that, for the asymptotic convergence of all solutions, the existence of a strictly monotone and asymptotically convergent solution is sufficient.

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17

Cardone, Giuseppe, Aurelien Fouetio, and Jean Louis Woukeng. "Homogenization of a 2D Tidal Dynamics Equation." Mathematics 8, no. 12 (December 12, 2020): 2209. http://dx.doi.org/10.3390/math8122209.

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This work deals with the homogenization of two dimensions’ tidal equations. We study the asymptotic behavior of the sequence of the solutions using the sigma-convergence method. We establish the convergence of the sequence of solutions towards the solution of an equivalent problem of the same type.

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18

Ruch, David K. "On uniformly contractive systems and quadratic equations in Banach space." Bulletin of the Australian Mathematical Society 52, no. 3 (December 1995): 441–55. http://dx.doi.org/10.1017/s0004972700014921.

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The solution of quadratic equations using the contraction mapping principle is considered. A uniqueness result extending that given by Argyros is proved. Uniformly contractive systems theory is used to find approximate solutions and convergence criteria are given. In particular, only pointwise convergence of approximating operators is required to guarantee convergence of the approximate solutions. A theorem and algorithm for a continuation method are presented, and illustrated on Chandrasekhar's equation.

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19

Che, Hai-tao, Xin-tian Pan, Lu-ming Zhang, and Yi-ju Wang. "Numerical Analysis of a Linear-Implicit Average Scheme for Generalized Benjamin-Bona-Mahony-Burgers Equation." Journal of Applied Mathematics 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/308410.

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A linear-implicit finite difference scheme is given for the initial-boundary problem of GBBM-Burgers equation, which is convergent and unconditionally stable. The unique solvability of numerical solutions is shown. A priori estimate and second-order convergence of the finite difference approximate solution are discussed using energy method. Numerical results demonstrate that the scheme is efficient and accurate.

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20

Li, Sheng-Kun. "Iterative Hermitian R-conjugate solutions to general coupled sylvester matrix equations." Filomat 31, no. 7 (2017): 2061–72. http://dx.doi.org/10.2298/fil1707061l.

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For a given symmetric orthogonal matrix R, i.e., RT = R, R2 = I, a matrix A ? Cnxn is termed Hermitian R-conjugate matrix if A = AH, RAR = ?. In this paper, an iterative method is constructed for finding the Hermitian R-conjugate solutions of general coupled Sylvester matrix equations. Convergence analysis shows that when the considered matrix equations have a unique solution group then the proposed method is always convergent for any initial Hermitian R-conjugate matrix group under a loose restriction on the convergent factor. Furthermore, the optimal convergent factor is derived. Finally, two numerical examples are given to demonstrate the theoretical results and effectiveness.

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21

Sukhinov, A. I., V. V. Sidoryakina, and Andrey A. Sukhinov. "Sufficient convergence conditions for positive solutions of linearized two-dimensional sediment transport problem." COMPUTATIONAL MATHEMATICS AND INFORMATION TECHNOLOGIES 1, no. 1 (2017): 21–35. http://dx.doi.org/10.23947/2587-8999-2017-1-1-21-35.

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22

DESPRÉS, BRUNO. "DISCRETE COMPRESSIVE SOLUTIONS OF SCALAR CONSERVATION LAWS." Journal of Hyperbolic Differential Equations 01, no. 03 (September 2004): 493–520. http://dx.doi.org/10.1142/s0219891604000226.

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We prove the convergence of numerical approximations of compressive solutions for scalar conservation laws with convex flux. This new proof of convergence is fully discrete and does not use Kuznetsov's approach. We recover the well-known rate of convergence in O(Δx½). With the same fully discrete approach, we also prove a rate of convergence in O(Δx) uniformly in time, if the initial data is a shock, or asymptotically after the compression of the initial profile. Numerical experiments confirm the theoretical analysis.

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23

Awati, Vishwanath B., and Manjunath Jyoti. "Homotopy analysis method for the solution of lubrication of a long porous slider." Applied Mathematics and Nonlinear Sciences 1, no. 2 (November 1, 2016): 507–16. http://dx.doi.org/10.21042/amns.2016.2.00040.

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AbstractIn this article, the lubrication of a long porous slider in which the fluid is injected into the porous bottom is considered. The similarity transformations reduce the governing problem of Navier-Stokes equations to coupled nonlinear ordinary differential equations which are solved by HAM. Solutions are obtained for much larger values of Reynolds number compared to analytical and numerical methods. The results comprise good agreement between approximate and numerical solutions. HAM gives rapid convergent series solutions which show that this method is efficient, accurate and has advantages over other methods. Further, homotopy-pade’ technique is used to accelerate the convergence of series solution.

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24

Lin, Y. Y., and A. P. Pisano. "Three-Dimensional Dynamic Simulation of Helical Compression Springs." Journal of Mechanical Design 112, no. 4 (December 1, 1990): 529–37. http://dx.doi.org/10.1115/1.2912642.

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The dynamic equations for general helical springs are solved and classified according to the number of energy terms used to formulate them. Solutions of several sets of dynamic equations, each with a different number of energy terms, are compared with experimental data. It is found that at higher compression speeds the numerical solution with a traditional, fixed boundary represents a physically impossible situation. A moving boundary technique is applied to improve the numerical solution and bring it into agreement with physical reality. Since a convergence proof for a numerical algorithm for nonlinear partial differential equations with a moving boundary is not available, a grid study has been performed to demonstrate convergence. The agreement between the solutions of different grid sizes and the experimental data is taken to show that the numerical algorithm was convergent. This three dimensional spring simulation model can be used in the simulation of high-speed mechanical machinery utilizing helical springs, and in particular, for design optimization of automotive valve springs.

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25

SVANSTEDT, NILS. "CONVERGENCE OF QUASI-LINEAR HYPERBOLIC EQUATIONS." Journal of Hyperbolic Differential Equations 04, no. 04 (December 2007): 655–77. http://dx.doi.org/10.1142/s0219891607001306.

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Multiscale stochastic homogenization is studied for quasilinear monotone hyperbolic problems with a linear damping term. It is shown by classical G-convergence methods that the sequence of solutions to a class of multi-scale highly oscillatory (possibly random) hyperbolic problems converges in the appropriate Sobolev space to the solution to a homogenized quasilinear hyperbolic problem.

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26

Bär, Christian. "Some Properties of Solutions to Weakly Hypoelliptic Equations." International Journal of Differential Equations 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/526390.

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A linear different operatorLis called weakly hypoelliptic if any local solutionuofLu=0is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which coverall elliptic, overdetermined elliptic, subelliptic, and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients, we show that Liouville's theorem holds, any bounded solution must be constant, and anyLp-solution must vanish.

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27

Lu, Peng. "Convergence of fundamental solutions of linear parabolic equations under Cheeger–Gromov convergence." Mathematische Annalen 353, no. 1 (May 26, 2011): 193–217. http://dx.doi.org/10.1007/s00208-011-0679-7.

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28

Aşik, M. Zülfü, and C. V. G. Vallabhan. "On the convergence of nonlinear plate solutions." Computers & Structures 65, no. 2 (October 1997): 225–29. http://dx.doi.org/10.1016/s0045-7949(96)00283-0.

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29

Kent, C. M. "Convergence of solutions in a nonhyperbolic case." Nonlinear Analysis: Theory, Methods & Applications 47, no. 7 (August 2001): 4651–65. http://dx.doi.org/10.1016/s0362-546x(01)00578-8.

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30

PEIRONE, ROBERTO. "Convergence of solutions of linear transport equations." Ergodic Theory and Dynamical Systems 23, no. 3 (June 2003): 919–33. http://dx.doi.org/10.1017/s014338570200144x.

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31

Gulledge, Thomas, Rainer Sommer, and Georg Simon. "Analyzing convergence alternatives across existing SAP solutions." Industrial Management & Data Systems 104, no. 9 (December 2004): 722–34. http://dx.doi.org/10.1108/02635570410567711.

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32

Rybka, Piotr, and Karl-Heinz Hoffnlann. "Convergence of solutions to cahn-hilliard equation." Communications in Partial Differential Equations 24, no. 5-6 (January 1999): 1055–77. http://dx.doi.org/10.1080/03605309908821458.

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33

Flåm, Sjur D., and Alain Fougères. "Infinite horizon programs; convergence of approximate solutions." Annals of Operations Research 29, no. 1 (December 1991): 333–50. http://dx.doi.org/10.1007/bf02283604.

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34

Erisova, I. A. "Convergence of solutions of backward stochastic equations." Ukrainian Mathematical Journal 61, no. 7 (July 2009): 1093–112. http://dx.doi.org/10.1007/s11253-009-0261-6.

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35

Bondarev, Andrei Sergeevich. "THE STRONG-NORM CONVERGENCE OF A PROJECTION-DIFFERENCE METHOD OF SOLUTION OF A PARABOLIC EQUATION WITH THE PERIODIC CONDITION ON THE SOLUTION." Tambov University Reports. Series: Natural and Technical Sciences, no. 124 (2018): 617–23. http://dx.doi.org/10.20310/1810-0198-2018-23-124-617-623.

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A smooth soluble abstract linear parabolic equation with the periodic condition on the solution is treated in a separable Hilbert space. This problem is solved approximately by a projection-difference method using the Galerkin method in space and the implicit Euler scheme in time. Effective both in time and in space strong-norm error estimates for approximate solutions, which imply convergence of approximate solutions to the exact solution and order of convergence rate depending of the smoothness of the exact solution, are obtained.

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36

Kendall, S. R., and H. V. Rao. "Detection of multiple solutions using a mid-cell back substitution technique applied to computational fluid dynamics." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 214, no. 11 (November 1, 2000): 1401–7. http://dx.doi.org/10.1243/0954406001523371.

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Computational models for fluid flow based on the Navier-Stokes equations for compressible fluids led to numerical procedures requiring the solution of simultaneous non-linear algebraic equations. These give rise to the possibility of multiple solutions, and hence there is a need to monitor convergence towards a physically meaningful flow field. The number of possible solutions that may arise is examined, and a mid-cell back substitution technique (MCBST) is developed to detect and avoid convergence towards apparently spurious solutions. The MCBST was used successfully for flow modelling in micron-sized flow passages, and was found to be particularly useful in the early stages of computation, optimizing the speed of convergence.

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37

Grunert, Katrin, Anders Nordli, and Susanne Solem. "Numerical conservative solutions of the Hunter–Saxton equation." BIT Numerical Mathematics 61, no. 2 (January 21, 2021): 441–71. http://dx.doi.org/10.1007/s10543-020-00835-y.

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AbstractIn the article a convergent numerical method for conservative solutions of the Hunter–Saxton equation is derived. The method is based on piecewise linear projections, followed by evolution along characteristics where the time step is chosen in order to prevent wave breaking. Convergence is obtained when the time step is proportional to the square root of the spatial step size, which is a milder restriction than the common CFL condition for conservation laws.

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38

Janavičius, A. J., P. Norgela, and D. Jurgaitis. "UNIQUENESS AND CONVERGENCE OF THE ANALYTICAL SOLUTION OF NONLINEAR DIFFUSION EQUATION." Mathematical Modelling and Analysis 6, no. 1 (June 30, 2001): 77–84. http://dx.doi.org/10.3846/13926292.2001.9637147.

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We have discussed the problems of uniqueness of the physical solution of the nonlinear diffusion equation. Here are considered two different ways to express the solutions in the power series. In the first case we will use the power‐series expansion about the zero point. The accuracy of the obtained physical solution is evaluated. However, in this case we get an infinity of different solutions and the problem of the choice of the unique physical solution is considered using the expansion about the point of maximum penetration of the impurities. Then we get only two solutions which differ one from other only in the directions of the diffusion.

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39

Morgado, M. Luísa, Magda Rebelo, and Luís L. Ferrás. "Stable and Convergent Finite Difference Schemes on NonuniformTime Meshes for Distributed-Order Diffusion Equations." Mathematics 9, no. 16 (August 18, 2021): 1975. http://dx.doi.org/10.3390/math9161975.

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In this work, stable and convergent numerical schemes on nonuniform time meshes are proposed, for the solution of distributed-order diffusion equations. The stability and convergence of the numerical methods are proven, and a set of numerical results illustrate that the use of particular nonuniform time meshes provides more accurate results than the use of a uniform mesh, in the case of nonsmooth solutions.

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40

Xu, Wenfei, and Shanshan Chen. "The convergence between approximate solutions and the unique solutions of SDEs." Pure Mathematical Sciences 3 (2014): 121–27. http://dx.doi.org/10.12988/pms.2014.4614.

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41

Joldes, Grand Roman, Peter Teakle, Adam Wittek, and Karol Miller. "Computation of accurate solutions when using element-free Galerkin methods for solving structural problems." Engineering Computations 34, no. 3 (May 2, 2017): 902–20. http://dx.doi.org/10.1108/ec-01-2016-0017.

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Purpose This paper aims to investigate the application of adaptive integration in element-free Galerkin methods for solving problems in structural and solid mechanics to obtain accurate reference solutions. Design/methodology/approach An adaptive quadrature algorithm which allows user control over integration accuracy, previously developed for integrating boundary value problems, is adapted to elasticity problems. The algorithm allows the development of a convergence study procedure that takes into account both integration and discretisation errors. The convergence procedure is demonstrated using an elasticity problem which has an analytical solution and is then applied to accurately solve a soft-tissue extension problem involving large deformations. Findings The developed convergence procedure, based on the presented adaptive integration scheme, allows the computation of accurate reference solutions for challenging problems which do not have an analytical or finite element solution. Originality/value This paper investigates the application of adaptive quadrature to solid mechanics problems in engineering analysis using the element-free Galerkin method to obtain accurate reference solutions. The proposed convergence procedure allows the user to independently examine and control the contribution of integration and discretisation errors to the overall solution error. This allows the computation of reference solutions for very challenging problems which do not have an analytical or even a finite element solution (such as very large deformation problems).

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42

Wang, Peiguang, Zhifang Li, and Yonghong Wu. "Rapid Convergence of Solution for Hybrid System with Causal Operators." Mathematical Problems in Engineering 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/849731.

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We investigated the convergence of iterative sequences of approximate solutions to a class of periodic boundary value problem of hybrid system with causal operators and established two sequences of approximate solutions that converge to the solution of the problem with rate of orderk≥2.

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43

Yu, Gan, Hongzhi Zhou, and Hui Wang. "Improving Artificial Bee Colony Algorithm Using a Dynamic Reduction Strategy for Dimension Perturbation." Mathematical Problems in Engineering 2019 (July 14, 2019): 1–11. http://dx.doi.org/10.1155/2019/3419410.

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To accelerate the convergence speed of Artificial Bee Colony (ABC) algorithm, this paper proposes a Dynamic Reduction (DR) strategy for dimension perturbation. In the standard ABC, a new solution (food source) is obtained by modifying one dimension of its parent solution. Based on one-dimensional perturbation, both new solutions and their parent solutions have high similarities. This will easily cause slow convergence speed. In our DR strategy, the number of dimension perturbations is assigned a large value at the initial search stage. More dimension perturbations can result in larger differences between offspring and their parent solutions. With the growth of iterations, the number of dimension perturbations dynamically decreases. Less dimension perturbations can reduce the dissimilarities between offspring and their parent solutions. Based on the DR, it can achieve a balance between exploration and exploitation by dynamically changing the number of dimension perturbations. To validate the proposed DR strategy, we embed it into the standard ABC and three well-known ABC variants. Experimental study shows that the proposed DR strategy can efficiently accelerate the convergence and improve the accuracy of solutions.

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44

Jeong, Darae, Yibao Li, Chaeyoung Lee, Junxiang Yang, Yongho Choi, and Junseok Kim. "Verification of Convergence Rates of Numerical Solutions for Parabolic Equations." Mathematical Problems in Engineering 2019 (June 23, 2019): 1–10. http://dx.doi.org/10.1155/2019/8152136.

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In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

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45

Cranny, T. R. "Convergence of approximate solutions of a quasilinear partial differential equation." Bulletin of the Australian Mathematical Society 50, no. 3 (December 1994): 425–33. http://dx.doi.org/10.1017/s0004972700013538.

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This article is a sequel to a paper in which a quasilinear partial differential equation with nonlinear boundary condition was approximated using mollifiers, and the existence of solutions to the approximating problem shown under quite general conditions. In this paper we show that standard a priori Hölder estimates ensure the convergence of these solutions to a classical solution of the original problem. Some partial results giving such estimates for special cases are described.

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Al Khawaja, U., and Qasem M. Al-Mdallal. "Convergent Power Series of sech⁡(x) and Solutions to Nonlinear Differential Equations." International Journal of Differential Equations 2018 (2018): 1–10. http://dx.doi.org/10.1155/2018/6043936.

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It is known that power series expansion of certain functions such as sech⁡(x) diverges beyond a finite radius of convergence. We present here an iterative power series expansion (IPS) to obtain a power series representation of sech⁡(x) that is convergent for all x. The convergent series is a sum of the Taylor series of sech⁡(x) and a complementary series that cancels the divergence of the Taylor series for x≥π/2. The method is general and can be applied to other functions known to have finite radius of convergence, such as 1/(1+x2). A straightforward application of this method is to solve analytically nonlinear differential equations, which we also illustrate here. The method provides also a robust and very efficient numerical algorithm for solving nonlinear differential equations numerically. A detailed comparison with the fourth-order Runge-Kutta method and extensive analysis of the behavior of the error and CPU time are performed.

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Ghiasi, Emran Khoshrouye, and Reza Saleh. "Constructing analytic solutions on the Tricomi equation." Open Physics 16, no. 1 (April 18, 2018): 143–48. http://dx.doi.org/10.1515/phys-2018-0022.

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AbstractIn this paper, homotopy analysis method (HAM) and variational iteration method (VIM) are utilized to derive the approximate solutions of the Tricomi equation. Afterwards, the HAM is optimized to accelerate the convergence of the series solution by minimizing its square residual error at any order of the approximation. It is found that effect of the optimal values of auxiliary parameter on the convergence of the series solution is not negligible. Furthermore, the present results are found to agree well with those obtained through a closed-form equation available in the literature. To conclude, it is seen that the two are effective to achieve the solution of the partial differential equations.

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TSAI, CHIA-CHENG, and PO-HO LIN. "ON THE EXPONENTIAL CONVERGENCE OF THE METHOD OF FUNDAMENTAL SOLUTIONS." International Journal of Computational Methods 10, no. 02 (March 2013): 1341007. http://dx.doi.org/10.1142/s0219876213410077.

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It is well known that the method of fundamental solutions (MFS) is a numerical method of exponential convergence. In other words, the logarithmic error is proportional to the node number of spatial discretization. In this study, the exponential convergence of the MFS is demonstrated by solving the Laplace equation in domains of rectangles, ellipses, amoeba-like shapes, and rectangular cuboids. In the solution procedure, the sources of the MFS are located as far as possible and the instability resulted from the ill-conditioning of system matrix is avoided by using the multiple precision floating-point reliable (MPFR) library. The results converge faster for the cases of smoother boundary conditions and larger area/perimeter ratios. For problems with discontinuous boundary data, the exponential convergence is also accomplished using the enriched method of fundamental solutions (EMFS), which is constructed by the fundamental solutions and the local singular solutions. The computation is scalable in the sense that the required time increases only algebraically.

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Bahuguna, D., and S. K. Srivastava. "Approximation of solutions to evolution integrodifferential equations." Journal of Applied Mathematics and Stochastic Analysis 9, no. 3 (January 1, 1996): 315–22. http://dx.doi.org/10.1155/s1048953396000299.

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In this paper we study a class of evolution integrodifferential equations. We first prove the existence and uniqueness of solutions and then establish the convergence of Galerkin approximations to the solution.

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Yuan, Haiyan. "Some Properties of Numerical Solutions for Semilinear Stochastic Delay Differential Equations Driven by G-Brownian Motion." Mathematical Problems in Engineering 2021 (July 3, 2021): 1–26. http://dx.doi.org/10.1155/2021/1835490.

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This paper is concerned with the numerical solutions of semilinear stochastic delay differential equations driven by G-Brownian motion (G-SLSDDEs). The existence and uniqueness of exact solutions of G-SLSDDEs are studied by using some inequalities and the Picard iteration scheme first. Then the numerical approximation of exponential Euler method for G-SLSDDEs is constructed, and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent, and it can reproduce the stability of the analytical solution under some restrictions. Numerical experiments are presented to confirm the theoretical results.

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