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Journal articles on the topic 'Parabolic Numerical solutions'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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1

Cannon, John R., and Hong-Ming Yin. "Numerical solutions of some parabolic inverse problems." Numerical Methods for Partial Differential Equations 6, no. 2 (1990): 177–91. http://dx.doi.org/10.1002/num.1690060207.

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2

FLOURI, EVANGELIA T., JOHN A. EKATERINARIS, and NIKOLAOS A. KAMPANIS. "HIGH-ORDER ACCURATE NUMERICAL SCHEMES FOR THE PARABOLIC EQUATION." Journal of Computational Acoustics 13, no. 04 (December 2005): 613–39. http://dx.doi.org/10.1142/s0218396x05002888.

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Efficient, high-order accurate methods for the numerical solution of the standard (narrow-angle) parabolic equation for underwater sound propagation are developed. Explicit and implicit numerical schemes, which are second- or higher-order accurate in time-like marching and fourth-order accurate in the space-like direction are presented. The explicit schemes have severe stability limitations and some of the proposed high-order accurate implicit methods were found conditionally stable. The efficiency and accuracy of various numerical methods are evaluated for Cartesian-type meshes. The standard parabolic equation is transformed to body fitted curvilinear coordinates. An unconditionally stable, implicit finite-difference scheme is used for numerical solutions in complex domains with deformed meshes. Simple boundary conditions are used and the accuracy of the numerical solutions is evaluated by comparing with an exact solution. Numerical solutions in complex domains obtained with a finite element method show excellent agreement with results obtained with the proposed finite difference methods.

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3

Choudhury, A. H. "Wavelet Method for Numerical Solution of Parabolic Equations." Journal of Computational Engineering 2014 (February 27, 2014): 1–12. http://dx.doi.org/10.1155/2014/346731.

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We derive a highly accurate numerical method for the solution of parabolic partial differential equations in one space dimension using semidiscrete approximations. The space direction is discretized by wavelet-Galerkin method using some special types of basis functions obtained by integrating Daubechies functions which are compactly supported and differentiable. The time variable is discretized by using various classical finite difference schemes. Theoretical and numerical results are obtained for problems of diffusion, diffusion-reaction, convection-diffusion, and convection-diffusion-reaction with Dirichlet, mixed, and Neumann boundary conditions. The computed solutions are highly favourable as compared to the exact solutions.

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4

Al-Sultani, Mohamed Saleh Mehdi, and Igor Boglaev. "Block monotone iterations for solving coupled systems of nonlinear parabolic equations." ANZIAM Journal 61 (July 28, 2020): C166—C180. http://dx.doi.org/10.21914/anziamj.v61i0.15144.

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The article deals with numerical methods for solving a coupled system of nonlinear parabolic problems, where reaction functions are quasi-monotone nondecreasing. We employ block monotone iterative methods based on the Jacobi and Gauss–Seidel methods incorporated with the upper and lower solutions method. A convergence analysis and the theorem on uniqueness of a solution are discussed. Numerical experiments are presented. References Al-Sultani, M. and Boglaev, I. ''Numerical solution of nonlinear elliptic systems by block monotone iterations''. ANZIAM J. 60:C79–C94, 2019. doi:10.21914/anziamj.v60i0.13986 Al-Sultani, M. ''Numerical solution of nonlinear parabolic systems by block monotone iterations''. Tech. Report, 2019. https://arxiv.org/abs/1905.03599 Boglaev, I. ''Inexact block monotone methods for solving nonlinear elliptic problems'' J. Comput. Appl. Math. 269:109–117, 2014. doi:10.1016/j.cam.2014.03.029 Lui, S. H. ''On monotone iteration and Schwarz methods for nonlinear parabolic PDEs''. J. Comput. Appl. Math. 161:449–468, 2003. doi:doi.org/10.1016/j.cam.2003.06.001 Pao, C. V. Nonlinear parabolic and elliptic equations. Plenum Press, New York, 1992. doi:10.1007/s002110050168 Pao C. V. ''Numerical analysis of coupled systems of nonlinear parabolic equations''. SIAM J. Numer. Anal. 36:393–416, 1999. doi:10.1137/S0036142996313166 Varga, R. S. Matrix iterative analysis. Springer, Berlin, 2000. 10.1007/978-3-642-05156-2 Zhao, Y. Numerical solutions of nonlinear parabolic problems using combined-block iterative methods. Masters Thesis, University of North Carolina, 2003. http://dl.uncw.edu/Etd/2003/zhaoy/yaxizhao.pdf

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5

Hashimoto, Kouji, Takehiko Kinoshita, and Mitsuhiro T. Nakao. "Numerical Verification of Solutions for Nonlinear Parabolic Problems." Numerical Functional Analysis and Optimization 41, no. 12 (June 12, 2020): 1495–514. http://dx.doi.org/10.1080/01630563.2020.1777159.

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6

Cho, Chien-Hong, and Ying-Jung Lu. "On the numerical solutions for a parabolic system with blow-up." AIMS Mathematics 6, no. 11 (2021): 11749–77. http://dx.doi.org/10.3934/math.2021683.

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<abstract><p>We study the finite difference approximation for axisymmetric solutions of a parabolic system with blow-up. A scheme with adaptive temporal increments is commonly used to compute an approximate blow-up time. There are, however, some limitations to reproduce the blow-up behaviors for such schemes. We thus use an algorithm, in which uniform temporal grids are used, for the computation of the blow-up time and blow-up behaviors. In addition to the convergence of the numerical blow-up time, we also study various blow-up behaviors numerically, including the blow-up set, blow-up rate and blow-up in $ L^\sigma $-norm. Moreover, the relation between blow-up of the exact solution and that of the numerical solution is also analyzed and discussed.</p></abstract>

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7

Shidfar, A., and Z. Darooghehgimofrad. "Numerical solution of two backward parabolic problems using method of fundamental solutions." Inverse Problems in Science and Engineering 25, no. 2 (January 30, 2016): 155–68. http://dx.doi.org/10.1080/17415977.2016.1138947.

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8

Qi-guang, Wu, and Li Ji-chun. "Numerical solutions for singularly perturbed semi-linear parabolic equation." Applied Mathematics and Mechanics 14, no. 9 (September 1993): 793–801. http://dx.doi.org/10.1007/bf02457474.

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9

Thapa, Dhak Bahadur, and Kedar Nath Uprety. "Analytic and Numerical Solutions of Couette Flow Problem: A Comparative Study." Journal of the Institute of Engineering 12, no. 1 (March 6, 2017): 105–13. http://dx.doi.org/10.3126/jie.v12i1.16731.

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In this work, an incompressible viscous Couette flow is derived by simplifying the Navier-Stokes equations and the resulting one dimensional linear parabolic partial differential equation is solved numerically employing a second order finit difference Crank-Nicolson scheme. The numerical solution and the exact solution are presented graphically.Journal of the Institute of Engineering, 2016, 12(1): 105-113

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10

Prakash, Amit, and Manoj Kumar. "Numerical solution of two dimensional time fractional-order biological population model." Open Physics 14, no. 1 (January 1, 2016): 177–86. http://dx.doi.org/10.1515/phys-2016-0021.

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AbstractIn this work, we provide an approximate solution of a parabolic fractional degenerate problem emerging in a spatial diffusion of biological population model using a fractional variational iteration method (FVIM). Four test illustrations are used to show the proficiency and accuracy of the projected scheme. Comparisons between exact solutions and numerical solutions are presented for different values of fractional orderα.

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11

Kong, Tao, Weidong Zhao, and Tao Zhou. "Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs." Communications in Computational Physics 18, no. 5 (November 2015): 1482–503. http://dx.doi.org/10.4208/cicp.240515.280815a.

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AbstractIn this paper, we are concerned with probabilistic high order numerical schemes for Cauchy problems of fully nonlinear parabolic PDEs. For such parabolic PDEs, it is shown by Cheridito, Soner, Touzi and Victoir [4] that the associated exact solutions admit probabilistic interpretations, i.e., the solution of a fully nonlinear parabolic PDE solves a corresponding second order forward backward stochastic differential equation (2FBSDEs). Our numerical schemes rely on solving those 2FBSDEs, by extending our previous results [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.]. Moreover, in our numerical schemes, one has the flexibility to choose the associated forward SDE, and a suitable choice can significantly reduce the computational complexity. Various numerical examples including the HJB equations are presented to show the effectiveness and accuracy of the proposed numerical schemes.

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12

Cockburn, Bernardo. "Continuous dependence and error estimation for viscosity methods." Acta Numerica 12 (May 2003): 127–80. http://dx.doi.org/10.1017/s0962492902000107.

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In this paper, we review some ideas on continuous dependence results for the entropy solution of hyperbolic scalar conservation laws. They lead to a complete L^\infty(L^1)-approximation theory with which error estimates for numerical methods for this type of equation can be obtained. The approach we consider consists in obtaining continuous dependence results for the solutions of parabolic conservation laws and deducing from them the corresponding results for the entropy solution. This is a natural approach, as the entropy solution is nothing but the limit of solutions of parabolic scalar conservation laws as the viscosity coefficient goes to zero.

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13

FAYYAD, D., and N. NASSIF. "APPROXIMATION OF BLOWING UP SOLUTIONS TO SEMILINEAR PARABOLIC EQUATIONS USING "MASS CONTROLLED" PARABOLIC SYSTEMS." Mathematical Models and Methods in Applied Sciences 09, no. 07 (October 1999): 1077–88. http://dx.doi.org/10.1142/s0218202599000488.

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This paper considers asymptotic approximations to the solutions of the semilinear parabolic equation: [Formula: see text] where the function f(u) is such that the solution to (0.1) blows up in a finite time Tb. In order to control the explosive behavior of this problem, we consider a "perturbation" to (0.1) defined by: [Formula: see text] where ε is a small positive number. The boundary and initial conditions on uε are those of u. For vε, the initial and boundary conditions are chosen to be 1. Note that system (0.2) belongs to a class of coupled semilinear parabolic equations, with positive solutions and "mass control" property, (see Ref. 10). The solution {uε, vε} of such systems is known to be global. As such, (0.2) appears to be a regular perturbation to a singular problem (0.1). In this work, our basic theorem is a convergence proof for uε and [Formula: see text] to u and ut, respectively, in the L∞ norm. These results constitute a framework for designing in subsequent work, numerical algorithms for the computation of blow-up times (see Ref. 6).

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14

Mbehou, M., R. Maritz, and P. M. D. Tchepmo. "Numerical Analysis for a Nonlocal Parabolic Problem." East Asian Journal on Applied Mathematics 6, no. 4 (October 19, 2016): 434–47. http://dx.doi.org/10.4208/eajam.260516.150816a.

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AbstractThis article is devoted to the study of the finite element approximation for a nonlocal nonlinear parabolic problem. Using a linearised Crank-Nicolson Galerkin finite element method for a nonlinear reaction-diffusion equation, we establish the convergence and error bound for the fully discrete scheme. Moreover, important results on exponential decay and vanishing of the solutions in finite time are presented. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

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15

Kravchenko, Vladislav V., Josafath A. Otero, and Sergii M. Torba. "Analytic Approximation of Solutions of Parabolic Partial Differential Equations with Variable Coefficients." Advances in Mathematical Physics 2017 (2017): 1–5. http://dx.doi.org/10.1155/2017/2947275.

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A complete family of solutions for the one-dimensional reaction-diffusion equation, uxx(x,t)-q(x)u(x,t)=ut(x,t), with a coefficient q depending on x is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristic Cauchy problem with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper, the Dirichlet boundary conditions are considered; however, the proposed method can be easily extended onto other standard boundary conditions. The proposed numerical method is shown to reveal good accuracy.

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16

Markov and Rodionov. "Numerical Simulation Using Finite-Difference Schemes with Continuous Symmetries for Processes of Gas Flow in Porous Media." Computation 7, no. 3 (August 24, 2019): 45. http://dx.doi.org/10.3390/computation7030045.

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This article presents the applications of continuous symmetry groups to the computational fluid dynamics simulation of gas flow in porous media. The family of equations for one-phase flow in porous media, such as equations of gas flow with the Klinkenberg effect, is considered. This consideration has been made in terms of difference scheme constructions with the preservation of continuous symmetries, which are presented in original parabolic differential equations. A new method of numerical solution generation using continuous symmetry groups has been developed for the equation of gas flow in porous media. Four classes of invariant difference schemes have been found by using known group classifications of parabolic differential equations with partial derivatives. Invariance of necessary conditions for stability has been shown for the difference schemes from the presented classes. Comparison with the classical approach for seeking numerical solutions for a particular case from the presented classes has shown that the calculation speed is greater by several orders than for the classical approach. Analysis of the accuracy for the presented method of numerical solution generation on the basis of continuous symmetries shows that the accuracy of generated numerical solutions depends on the accuracy of initial solutions for generations.

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17

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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18

Mungkasi, Sudi, and Friska Dwi Mesra Mangadil. "Numerical solutions to a parabolic model of two-layer fluids." Journal of Physics: Conference Series 1090 (September 2018): 012050. http://dx.doi.org/10.1088/1742-6596/1090/1/012050.

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19

Aloy, R., M. C. Casabán, and L. Jódar. "Constructing unconditionally time-stable numerical solutions for mixed parabolic problems." Computers & Mathematics with Applications 53, no. 11 (June 2007): 1773–83. http://dx.doi.org/10.1016/j.camwa.2006.06.015.

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20

Geng, Jun, and Zhongwei Shen. "Asymptotic expansions of fundamental solutions in parabolic homogenization." Analysis & PDE 13, no. 1 (January 6, 2020): 147–70. http://dx.doi.org/10.2140/apde.2020.13.147.

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21

Naeem, Muhammad, Omar Fouad Azhar, Ahmed M. Zidan, Kamsing Nonlaopon, and Rasool Shah. "Numerical Analysis of Fractional-Order Parabolic Equations via Elzaki Transform." Journal of Function Spaces 2021 (August 31, 2021): 1–10. http://dx.doi.org/10.1155/2021/3484482.

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This research article is dedicated to solving fractional-order parabolic equations, using an innovative analytical technique. The Adomian decomposition method is well supported by Elzaki transformation to establish closed-form solutions for targeted problems. The procedure is simple, attractive, and preferred over other methods because it provides a closed-form solution for the given problems. The solution graphs are plotted for both integer and fractional-order, which shows that the obtained results are in good contact with problems’ exact solution. It is also observed that the solution of fractional-order problems is convergent to the integer-order problem. Moreover, the validity of the proposed method is analyzed by considering some numerical examples. The theory of the suggested approach is fully supported by the obtained results for the given problems. In conclusion, the present method is a straightforward and accurate analytical technique that can solve other fractional-order partial differential equations.

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22

Tadi, M., and Miloje Radenkovic. "A Numerical Method for 1-D Parabolic Equation with Nonlocal Boundary Conditions." International Journal of Computational Mathematics 2014 (November 20, 2014): 1–9. http://dx.doi.org/10.1155/2014/923693.

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This paper is concerned with a local method for the solution of one-dimensional parabolic equation with nonlocal boundary conditions. The method uses a coordinate transformation. After the coordinate transformation, it is then possible to obtain exact solutions for the resulting equations in terms of the local variables. These exact solutions are in terms of constants of integration that are unknown. By imposing the given boundary conditions and smoothness requirements for the solution, it is possible to furnish a set of linearly independent conditions that can be used to solve for the constants of integration. A number of examples are used to study the applicability of the method. In particular, three nonlinear problems are used to show the novelty of the method.

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23

Secer, Aydin. "Numerical Solution and Simulation of Second-Order Parabolic PDEs with Sinc-Galerkin Method Using Maple." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/686483.

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An efficient solution algorithm for sinc-Galerkin method has been presented for obtaining numerical solution of PDEs with Dirichlet-type boundary conditions by using Maple Computer Algebra System. The method is based on Whittaker cardinal function and uses approximating basis functions and their appropriate derivatives. In this work, PDEs have been converted to algebraic equation systems with new accurate explicit approximations of inner products without the need to calculate any numeric integrals. The solution of this system of algebraic equations has been reduced to the solution of a matrix equation system via Maple. The accuracy of the solutions has been compared with the exact solutions of the test problem. Computational results indicate that the technique presented in this study is valid for linear partial differential equations with various types of boundary conditions.

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24

Andreianov, Boris, Mostafa Bendahmane, Alfio Quarteroni, and Ricardo Ruiz-Baier. "Solvability analysis and numerical approximation of linearized cardiac electromechanics." Mathematical Models and Methods in Applied Sciences 25, no. 05 (March 8, 2015): 959–93. http://dx.doi.org/10.1142/s0218202515500244.

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This paper is concerned with the mathematical analysis of a coupled elliptic–parabolic system modeling the interaction between the propagation of electric potential and subsequent deformation of the cardiac tissue. The problem consists in a reaction–diffusion system governing the dynamics of ionic quantities, intra- and extra-cellular potentials, and the linearized elasticity equations are adopted to describe the motion of an incompressible material. The coupling between muscle contraction, biochemical reactions and electric activity is introduced with a so-called active strain decomposition framework, where the material gradient of deformation is split into an active (electrophysiology-dependent) part and an elastic (passive) one. Under the assumption of linearized elastic behavior and a truncation of the updated nonlinear diffusivities, we prove existence of weak solutions to the underlying coupled reaction–diffusion system and uniqueness of regular solutions. The proof of existence is based on a combination of parabolic regularization, the Faedo–Galerkin method, and the monotonicity-compactness method of Lions. A finite element formulation is also introduced, for which we establish existence of discrete solutions and show convergence to a weak solution of the original problem. We close with a numerical example illustrating the convergence of the method and some features of the model.

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25

Yin, Hong-Ming. "The classical solutions for nonlinear parabolic integro- differential equations." Journal of Integral Equations and Applications 1, no. 2 (June 1988): 249–64. http://dx.doi.org/10.1216/jie-1988-1-2-249.

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26

Li, Buyang, and Weiwei Sun. "Maximal Regularity of Fully Discrete Finite Element Solutions of Parabolic Equations." SIAM Journal on Numerical Analysis 55, no. 2 (January 2017): 521–42. http://dx.doi.org/10.1137/16m1071912.

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27

Shishkin, G., L. Shishkina, and K. Cronin. "A NUMERICAL METHOD FOR A STEFAN-TYPE PROBLEM." Mathematical Modelling and Analysis 16, no. 1 (April 8, 2011): 119–42. http://dx.doi.org/10.3846/13926292.2011.562930.

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A Stefan-type problem is considered. This is an initial-boundary value problem on a composite domain for a parabolic reaction-diffusion equation with a moving interface boundary. At the moving boundary between the two subdomains, an interface condition is prescribed for the solution of the problem and its derivatives. A finite difference scheme is constructed that approximates the initial-boundary value problem. An iterative Newton-type method for the solution of the difference scheme and a numerical method for the analysis of the errors of the computed discrete solutions are both developed.

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28

González Parra, Gilberto, Abraham J. Arenas, and Miladys Cogollo. "Analytical-Numerical Solution of a Parabolic Diffusion Equation Under Uncertainty Conditions Using DTM with Monte Carlo Simulations." Ingeniería y Ciencia 11, no. 22 (July 31, 2015): 49–72. http://dx.doi.org/10.17230/ingciencia.11.22.3.

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A numerical method to solve a general random linear parabolic equationwhere the diffusion coefficient, source term, boundary and initial condi-tions include uncertainty, is developed. Diffusion equations arise in manyfields of science and engineering, and, in many cases, there are uncertaintiesdue to data that cannot be known, or due to errors in measurements andintrinsic variability. In order to model these uncertainties the correspon-ding parameters, diffusion coefficient, source term, boundary and initialconditions, are assumed to be random variables with certainprobabilitydistributions functions. The proposed method includes finite differenceschemes on the space variable and the differential transformation methodfor the time. In addition, the Monte Carlo method is used to deal withthe random variables. The accuracy of the hybrid method is investigatednumerically using the closed form solution of the deterministic associated equation. Based on the numerical results, confidence intervals and ex-pected mean values for the solution are obtained. Furthermore, with theproposed hybrid method numerical-analytical solutions are obtained.

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29

MIKHIN, DMITRY. "ENERGY-CONSERVING AND RECIPROCAL SOLUTIONS FOR HIGHER-ORDER PARABOLIC EQUATIONS." Journal of Computational Acoustics 09, no. 01 (March 2001): 183–203. http://dx.doi.org/10.1142/s0218396x01000450.

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The energy conservation law and the flow reversal theorem are valid for underwater acoustic fields. In media at rest the theorem transforms into well-known reciprocity principle. The presented parabolic equation (PE) model strictly preserves these important physical properties in the numerical solution. The new PE is obtained from the one-way wave equation by Godin12 via Padé approximation of the square root operator and generalized to the case of moving media. The PE is range-dependent and explicitly includes range derivatives of the medium parameters. Implicit finite difference scheme solves the PE written in terms of energy flux. Such formalism inherently provides simple and exact energy-conserving boundary condition at vertical interfaces. The finite-difference operators, the discreet boundary conditions, and the self-starter are derived by discretization of the differential PE. Discreet energy conservation and flow reversal theorem are rigorously proved as mathematical properties of the finite-difference scheme and confirmed by numerical modeling. Numerical solution is shown to be reciprocal with accuracy of 10–12 decimal digits, which is the accuracy of round-off errors. Energy conservation and wide-angle capabilities of the model are illustrated by comparison with two-way normal mode solutions including the ASA benchmark wedge.

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Alvino, Angelo, Roberta Volpicelli, and Bruno Volzone. "Comparison results for solutions of nonlinear parabolic equations." Complex Variables and Elliptic Equations 55, no. 5-6 (April 20, 2010): 431–43. http://dx.doi.org/10.1080/17476930903276191.

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Lord, G. J., and J. Rougemont. "Numerical computation of ε-entropy for parabolic equations with analytic solutions." Physica D: Nonlinear Phenomena 194, no. 1-2 (July 2004): 65–74. http://dx.doi.org/10.1016/j.physd.2004.01.040.

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32

Hagstrom, Thomas, and H. B. Keller. "The Numerical Calculation of Traveling Wave Solutions of Nonlinear Parabolic Equations." SIAM Journal on Scientific and Statistical Computing 7, no. 3 (July 1986): 978–88. http://dx.doi.org/10.1137/0907065.

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Khater, A. H., A. B. Shamardan, M. H. Farag, and A. H. Abel-Hamid. "Analytical and numerical solutions of a quasilinear parabolic optimal control problem." Journal of Computational and Applied Mathematics 95, no. 1-2 (August 1998): 29–43. http://dx.doi.org/10.1016/s0377-0427(98)00066-1.

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34

Minamoto, T. "Numerical existence and uniqueness proof for solutions of semilinear parabolic equations." Applied Mathematics Letters 14, no. 6 (August 2001): 707–14. http://dx.doi.org/10.1016/s0893-9659(01)80031-8.

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35

Wei, Dongming. "Existence, Uniqueness, and Numerical Analysis of Solutions of a Quasilinear Parabolic Problem." SIAM Journal on Numerical Analysis 29, no. 2 (April 1992): 484–97. http://dx.doi.org/10.1137/0729029.

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36

Leykekhman, Dmitriy, and Boris Vexler. "Pointwise Best Approximation Results for Galerkin Finite Element Solutions of Parabolic Problems." SIAM Journal on Numerical Analysis 54, no. 3 (January 2016): 1365–84. http://dx.doi.org/10.1137/15m103412x.

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37

Pao, C. V. "Numerical Methods for Time-Periodic Solutions of Nonlinear Parabolic Boundary Value Problems." SIAM Journal on Numerical Analysis 39, no. 2 (January 2001): 647–67. http://dx.doi.org/10.1137/s0036142999361396.

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38

Luo, Yan, and Zhu Wang. "An Ensemble Algorithm for Numerical Solutions to Deterministic and Random Parabolic PDEs." SIAM Journal on Numerical Analysis 56, no. 2 (January 2018): 859–76. http://dx.doi.org/10.1137/17m1131489.

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39

McComb, Todd. "A Numerical Study of Very High Speed Flat Ship Theory." Journal of Ship Research 35, no. 01 (March 1, 1991): 63–72. http://dx.doi.org/10.5957/jsr.1991.35.1.63.

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This paper uses an asymptotic procedure to generate analytic solutions to the low-aspect-ratio problem of flat ship theory, in the limit for a very fast ship. The first two terms of the solution are worked out for hulls of parabolic planform and with 20 arbitrary constants in the expression for the draft. Optimizations are then performed for lift and drag on a smaller class of hulls. Analytic solutions were found by using symbolic computation, and the results are discussed. Optimal hulls are presented for various values of the ship's speed, optimized with both total lift and static lift held fixed. The optimization solution in the limit as the ship's speed goes to infinity gives independence of some constants in the expression for the hull.

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40

Caddick, Miles, and Endre Süli. "Numerical approximation of young-measure solutions to parabolic systems of forward-backward type." Applicable Analysis and Discrete Mathematics 13, no. 3 (2019): 649–96. http://dx.doi.org/10.2298/aadm190325026c.

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This paper is concerned with the proof of existence and numerical approximation of large-data global-in-time Young measure solutions to initial-boundaryvalue problems for multidimensional nonlinear parabolic systems of forward-backward type of the form ?tu - div(a(Du))+ Bu = F, where B ? Rmxm, Bv?v ? 0 for all v ? Rm, F is an m-component vector-function defined on a bounded open Lipschitz domain ? ? Rn, and a is a locally Lipschitz mapping of the form a(A)= K(A)A, where K: Rmxn ? R. The function a may have unequal lower and upper growth rates; it is not assumed to be monotone, nor is it assumed to be the gradient of a potential. We construct a numerical method for the approximate solution of problems in this class, and we prove its convergence to a Young measure solution of the system.

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DUCK, PETER W., SIMON R. STOW, and MANHAR R. DHANAK. "Non-similarity solutions to the corner boundary-layer equations (and the effects of wall transpiration)." Journal of Fluid Mechanics 400 (December 10, 1999): 125–62. http://dx.doi.org/10.1017/s0022112099006400.

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The incompressible boundary layer in the corner formed by two intersecting, semi-infinite planes is investigated, when the free-stream flow, aligned with the corner, is taken to be of the form U∞F(x), x representing the non-dimensional streamwise distance from the leading edge. In Dhanak & Duck (1997) similarity solutions for F(x) = xn were considered, and it was found that solutions exist for only a range of values of n, whilst for ∞ > n > −0.018, approximately, two solutions exist. In this paper, we extend the work of Dhanak & Duck to the case of non-90° corner angles and allow for streamwise development of solutions. In addition, the effect of transpiration at the walls of the corner is investigated. The governing equations are of boundary-layer type and as such are parabolic in nature. Crucially, although the leading-order pressure term is known a priori, the third-order pressure term is not, but this is nonetheless present in the leading-order governing equations, together with the transverse and crossflow viscous terms.Particular attention is paid to flows which develop spatially from similarity solutions. It turns out that two scenarios are possible. In some cases the problem may be treated in the usual parabolic sense, with standard numerical marching procedures being entirely appropriate. In other cases standard marching procedures lead to numerically inconsistent solutions. The source of this difficulty is linked to the existence of eigensolutions emanating from the leading edge (which are not present in flows appropriate to the first scenario), analogous to those found in the computation of some two-dimensional hypersonic boundary layers (Neiland 1970; Mikhailov et al. 1971; Brown & Stewartson 1975). In order to circumvent this difficulty, a different numerical solution strategy is adopted, based on a global Newton iteration procedure.A number of numerical solutions for the entire corner flow region are presented.

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Liang, Zongqi, and Huashui Zhan. "On the Cauchy Problem of a Quasilinear Degenerate Parabolic Equation." Discrete Dynamics in Nature and Society 2009 (2009): 1–12. http://dx.doi.org/10.1155/2009/827087.

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By Oleinik's line method, we study the existence and the uniqueness of the classical solution of the Cauchy problem for the following equation in[0,T]×R2:∂xxu+u∂yu−∂tu=f(⋅,u), provided thatTis suitable small. Results of numerical experiments are reported to demonstrate that the strong solutions of the above equation may blow up in finite time.

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Fernández, J. R., and M. Sofonea. "Variational and numerical analysis of the Signorini′s contact problem in viscoplasticity with damage." Journal of Applied Mathematics 2003, no. 2 (2003): 87–114. http://dx.doi.org/10.1155/s1110757x03202023.

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We consider the quasistatic Signorini′s contact problem with damage for elastic-viscoplastic bodies. The mechanical damage of the material, caused by excessive stress or strain, is described by a damage function whose evolution is modeled by an inclusion of parabolic type. We provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution of the model. We then introduce and study a fully discrete scheme for the numerical solutions of the problem. An optimal order error estimate is derived for the approximate solutions under suitable solution regularity. Numerical examples are presented to show the performance of the method.

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Shishkin, G. I. "Numerical Method for Singularly Perturbed Parabolic Equations in Unbounded Domains in the Case of Solutions Growing at Infinity." Computational Methods in Applied Mathematics 9, no. 1 (2009): 100–110. http://dx.doi.org/10.2478/cmam-2009-0006.

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AbstractAn initial-boundary value problem is considered in an unbounded do- main on the x-axis for a singularly perturbed parabolic reaction-diffusion equation. For small values of the parameter ε, a parabolic boundary layer arises in a neighbourhood of the lateral part of the boundary. In this problem, the error of a discrete solution in the maximum norm grows without bound even for fixed values of the parameter ε. In the present paper, the proximity of solutions of the initial-boundary value problem and of its numerical approximations is considered. Using the method of special grids condensing in a neighbourhood of the boundary layer, a special finite difference scheme converging ε-uniformly in the weight maximum norm has been constructed.

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Giménez, Ángel, Francisco Morillas, José Valero, and José María Amigó. "Stability and Numerical Analysis of the Hébraud-Lequeux Model for Suspensions." Discrete Dynamics in Nature and Society 2011 (2011): 1–24. http://dx.doi.org/10.1155/2011/415921.

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We study both analytically and numerically the stability of the solutions of the Hébraud-Lequeux equation. This parabolic equation models the evolution for the probability of finding a stressσin a mesoscopic block of a concentrated suspension, a non-Newtonian fluid. We prove a new result concerning the stability of the fixed points of the equation, and pose some conjectures about stability, based on numerical evidence.

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Xue, Guanyu, Yunjie Gong, and Hui Feng. "The Splitting Crank–Nicolson Scheme with Intrinsic Parallelism for Solving Parabolic Equations." Journal of Function Spaces 2020 (March 30, 2020): 1–12. http://dx.doi.org/10.1155/2020/8571625.

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In this paper, a splitting Crank–Nicolson (SC-N) scheme with intrinsic parallelism is proposed for parabolic equations. The new algorithm splits the Crank–Nicolson scheme into two domain decomposition methods, each one is applied to compute the values at (n + 1)th time level by use of known numerical solutions at n-th time level, respectively. Then, the average of the above two values is chosen to be the numerical solutions at (n + 1)th time level. The new algorithm obtains accuracy of the Crank–Nicolson scheme while maintaining parallelism and unconditional stability. This algorithm can be extended to solve two-dimensional parabolic equations by alternating direction implicit (ADI) technique. Numerical experiments illustrate the accuracy and efficiency of the new algorithm.

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Sinsoysal, Bahaddin. "The Analytical and a Higher-Accuracy Numerical Solution of a Free Boundary Problem in a Class of Discontinuous Functions." Mathematical Problems in Engineering 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/791026.

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A new method is suggested for obtaining the exact and numerical solutions of the initial-boundary value problem for a nonlinear parabolic type equation in the domain with the free boundary. With this aim, a special auxiliary problem having some advantages over the main problem and being equivalent to the main problem in a definite sense is introduced. The auxiliary problem allows us to obtain the weak solution in a class of discontinuous functions. Moreover, on the basis of the auxiliary problem a higher-resolution numerical method is developed so that the solution accurately describes all physical properties of the problem. In order to extract the significance of the numerical solutions obtained by using the suggested auxiliary problem, some computer experiments are carried out.

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Zhu, Liyong. "Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations." Advances in Applied Mathematics and Mechanics 9, no. 1 (October 11, 2016): 157–72. http://dx.doi.org/10.4208/aamm.2015.m1045.

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AbstractIn this paper we develop explicit fast exponential Runge-Kutta methods for the numerical solutions of a class of parabolic equations. By incorporating the linear splitting technique into the explicit exponential Runge-Kutta schemes, we are able to greatly improve the numerical stability. The proposed numerical methods could be fast implemented through use of decompositions of compact spatial difference operators on a regular mesh together with discrete fast Fourier transform techniques. The exponential Runge-Kutta schemes are easy to be adopted in adaptive temporal approximations with variable time step sizes, as well as applied to stiff nonlinearity and boundary conditions of different types. Linear stabilities of the proposed schemes and their comparison with other schemes are presented. We also numerically demonstrate accuracy, stability and robustness of the proposed method through some typical model problems.

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TOURIN, AGNÈS. "NUMERICAL SOLUTIONS FOR THE CHERIDITO-SONER-TOUZI SUPER-REPLICATION MODEL UNDER GAMMA CONSTRAINTS." International Journal of Theoretical and Applied Finance 09, no. 03 (May 2006): 401–14. http://dx.doi.org/10.1142/s0219024906003561.

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We solve, by using a monotone and stable approximation, the fully nonlinear degenerate parabolic equation derived by Cheridito, Soner and Touzi [8] from the stochastic control problem of super-replicating a contingent claim under gamma constraints. We present some numerical results.

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Pikulin, S. V. "ON SOLUTIONS OF TRAVELING WAVE TYPE FOR A NONLINEAR PARABOLIC EQUATION." Vestnik of Samara University. Natural Science Series 21, no. 6 (May 17, 2017): 110–16. http://dx.doi.org/10.18287/2541-7525-2015-21-6-110-116.

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We consider the Kolmogorov — Petrovsky — Piskunov equation which isa quasilinear parabolic equation of second order appearing in the flame propagationtheory and in modeling of certain biological processes. An analyticalconstruction of self-similar solutions of traveling wave kind is presented for thespecial case when the nonlinear term of the equation is the product of theargument and a linear function of a positive power of the argument. The approachto the construction of solutions is based on the study of singular pointsof analytic continuation of the solution to the complex domain and on applyingthe Fuchs — Kovalevskaya — Painlev´e test. The resulting representation of thesolution allows an efficient numerical implementation.

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