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Journal articles on the topic 'Modified CrankNicolson Method'

Author: Grafiati

Published: 7 June 2025

Last updated: 2 August 2025

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1

Tejaskumar, C. Sharma, P. Pathak Shreekant, and Trivedi Gargi. "Comparative Study of Crank-Nicolson and Modified Crank-Nicolson Numerical methods to solve linear Partial Differential Equations." Indian Journal of Science and Technology 17, no. 10 (2024): 924–31. https://doi.org/10.17485/IJST/v17i10.1776.

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Abstract <strong>Objectives:</strong> This paper aims to address the limitations of the Crank-Nicolson Finite Difference method and propose an improved version called the modified Crank-Nicolson method. <strong>Methods:</strong> Utilized implicit discretization in time and space, with parameters k = 0.001, h = 0.1, and γ = 0.1. Conducted extensive testing on various partial differential equations.<strong> Findings:</strong> Results, displayed in Table 1, showcase the method's stability and accuracy. Comparative analysis in Table 2 demonstrates the Modified Crank-

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2

Sharma, Tejaskumar C., Shreekant P. Pathak, and Gargi Trivedi. "Comparative Study of Crank-Nicolson and Modified Crank-Nicolson Numerical methods to solve linear Partial Differential Equations." Indian Journal Of Science And Technology 17, no. 10 (2024): 924–31. http://dx.doi.org/10.17485/ijst/v17i10.1776.

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Objectives: This paper aims to address the limitations of the Crank-Nicolson Finite Difference method and propose an improved version called the modified Crank-Nicolson method. Methods: Utilized implicit discretization in time and space, with parameters k = 0.001, h = 0.1, and γ = 0.1. Conducted extensive testing on various partial differential equations. Findings: Results, displayed in Table 1, showcase the method's stability and accuracy. Comparative analysis in Table 2 demonstrates the Modified Crank-Nicolson method consistently outperforming the traditional approach, reaffirming its superi

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3

WANI, SACHIN S. "Modified Crank Nicolson Type Method for Burgers Equation." Journal of Ultra Scientist of Physical Sciences Section A 28, no. 5 (2016): 259–77. http://dx.doi.org/10.22147/jusps-a/280505.

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4

Shi, Yao, Xiaozhen Liu, and Zhenyu Wang. "An Efficient Structure-Preserving Scheme for the Fractional Damped Nonlinear Schrödinger System." Fractal and Fractional 9, no. 5 (2025): 328. https://doi.org/10.3390/fractalfract9050328.

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This paper introduces a highly accurate and efficient conservative scheme for solving the nonlocal damped Schrödinger system with Riesz fractional derivatives. The proposed approach combines the Fourier spectral method with the Crank–Nicolson time-stepping scheme. To begin, the original equation is reformulated into an equivalent system by introducing a new variable that modifies both energy and mass. The Fourier spectral method is employed to achieve high spatial accuracy in this semi-discrete formulation. For time discretization, the Crank–Nicolson scheme is applied, ensuring conservation of

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5

Kurniawan, A. R., E. Suaebah, and Z. A. I. Supardi. "Modified Crank-Nicolson method for solving the time-dependent Schrödinger equation." Journal of Physics: Conference Series 2900, no. 1 (2024): 012009. https://doi.org/10.1088/1742-6596/2900/1/012009.

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Abstract The time-dependent Schrödinger equation is fundamental to quantum mechanics, describing the evolution of quantum systems. This study introduces a novel numerical approach that extends and modifies the Crank-Nicholson method to solve the time-dependent Schrödinger equation. This method is designed to enhance both the precision and performance of computational results, especially in the context of quantum tunneling phenomena and applications that require high accuracy over long periods. To assess the precision and performance of the modified Crank-Nicolson method, we benchmark it agains

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Arora, Geeta, Shubham Mishra, Homan Emaifar, and Masoumeh Khademi. "Numerical Simulation and Dynamics of Burgers’ Equation Using the Modified Cubic B-Spline Differential Quadrature Method." Discrete Dynamics in Nature and Society 2023 (March 29, 2023): 1–8. http://dx.doi.org/10.1155/2023/5102374.

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In the present work, a numerical approach using the Crank–Nicolson scheme along with the modified cubic B-spline differential quadrature (CN-MCDQ) method is proposed to find the numerical approximations to Burgers’ equation. After applying the well-known Crank–Nicolson technique, Burgers’ equation is solved in this study by using the differential quadrature approach to approximate the derivatives that lead to a system of equations to be solved. When compared to other methods for obtaining numerical solutions, the proposed method is shown to be efficient and easy to implement while still provid

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7

Wu, Chunya, Xinlong Feng, and Lingzhi Qian. "A Second-Order Crank–Nicolson Leap-Frog Scheme for the Modified Phase Field Crystal Model with Long-Range Interaction." Entropy 24, no. 11 (2022): 1512. http://dx.doi.org/10.3390/e24111512.

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In this paper, we construct a fully discrete and decoupled Crank–Nicolson Leap-Frog (CNLF) scheme for solving the modified phase field crystal model (MPFC) with long-range interaction. The idea of CNLF is to treat stiff terms implicity with Crank–Nicolson and to treat non-stiff terms explicitly with Leap-Frog. In addition, the scalar auxiliary variable (SAV) method is used to allow explicit treatment of the nonlinear potential, then, these technique combines with CNLF can lead to the highly efficient, fully decoupled and linear numerical scheme with constant coefficients at each time step. Fur

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8

Alanazi, A. A., Sultan Z. Alamri, S. Shafie, and Shazirawati Mohd Puzi. "Crank-Nicolson Scheme for Solving the Modified Nonlinear Schrodinger Equation." International Journal of Numerical Methods for Heat & Fluid Flow 31, no. 8 (2021): 2789–817. http://dx.doi.org/10.1108/hff-10-2020-0677.

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Purpose The purpose of this paper is to obtain the nonlinear Schrodinger equation (NLSE) numerical solutions in the presence of the first-order chromatic dispersion using a second-order, unconditionally stable, implicit finite difference method. In addition, stability and accuracy are proved for the resulting scheme. Design/methodology/approach The conserved quantities such as mass, momentum and energy are calculated for the system governed by the NLSE. Moreover, the robustness of the scheme is confirmed by conducting various numerical tests using the Crank-Nicolson method on different cases o

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9

Castillo, P. E., and S. A. Gómez. "Conservación de invariantes de la ecuación de Schrödinger no lineal por el método LDG." Revista Mexicana de Física E 64, no. 1 (2018): 52. http://dx.doi.org/10.31349/revmexfise.64.52.

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Conservation of the energy and the Hamiltonian of a general non linear Schr¨odinger equation is analyzed for the finite element method “Local Discontinuous Galerkin” spatial discretization. Conservation of the discrete analogue of these quantities is also proved for the fully discrete problem using the modified Crank-Nicolson method as time marching scheme. The theoretical results are validated on a series of problemsfor different nonlinear potentials.

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10

Ashyralyev, Allaberen, and Ali Sirma. "Modified Crank-Nicolson Difference Schemes for Nonlocal Boundary Value Problem for the Schrödinger Equation." Discrete Dynamics in Nature and Society 2009 (2009): 1–15. http://dx.doi.org/10.1155/2009/584718.

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The nonlocal boundary value problem for Schrödinger equation in a Hilbert space is considered. The second-order of accuracy -modified Crank-Nicolson difference schemes for the approximate solutions of this nonlocal boundary value problem are presented. The stability of these difference schemes is established. A numerical method is proposed for solving a one-dimensional nonlocal boundary value problem for the Schrödinger equation with Dirichlet boundary condition. A procedure of modified Gauss elimination method is used for solving these difference schemes. The method is illustrated by numerica

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11

Mei, Liquan, Yali Gao, and Zhangxin Chen. "A Galerkin Finite Element Method for Numerical Solutions of the Modified Regularized Long Wave Equation." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/438289.

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A Galerkin method for a modified regularized long wave equation is studied using finite elements in space, the Crank-Nicolson scheme, and the Runge-Kutta scheme in time. In addition, an extrapolation technique is used to transform a nonlinear system into a linear system in order to improve the time accuracy of this method. A Fourier stability analysis for the method is shown to be marginally stable. Three invariants of motion are investigated. Numerical experiments are presented to check the theoretical study of this method.

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12

Huang, Pengzhan, and Abdurishit Abduwali. "The Modified Local Crank–Nicolson method for one- and two-dimensional Burgers’ equations." Computers & Mathematics with Applications 59, no. 8 (2010): 2452–63. http://dx.doi.org/10.1016/j.camwa.2009.08.069.

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13

Luo, Guo, and Min Huang. "An Analytically Modified Finite Difference Scheme for Pricing Discretely Monitored Options." Mathematics 13, no. 2 (2025): 241. https://doi.org/10.3390/math13020241.

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Finite difference methods are commonly used in the pricing of discretely monitored exotic options in the Black–Scholes framework, but they tend to converge slowly due to discontinuities contained in terminal conditions. We present an effective analytical modification to existing finite difference methods that greatly enhances their performance on discretely monitored options with non-smooth terminal conditions. We apply this modification to the popular Crank–Nicolson method and obtain highly accurate option pricing results with significantly reduced CPU cost. We also introduce an adaptive mesh

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14

Koroglu, Canan, and Ayhan Aydin. "An Unconventional Finite Difference Scheme for Modified Korteweg-de Vries Equation." Advances in Mathematical Physics 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/4796070.

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A numerical solution of the modified Korteweg-de Vries (MKdV) equation is presented by using a nonstandard finite difference (NSFD) scheme with theta method which includes the implicit Euler and a Crank-Nicolson type discretization. Local truncation error of the NSFD scheme and linear stability analysis are discussed. To test the accuracy and efficiency of the method, some numerical examples are given. The numerical results of NSFD scheme are compared with the exact solution and a standard finite difference scheme. The numerical results illustrate that the NSFD scheme is a robust numerical too

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15

Ali, Ajmal, Thabet Abdeljawad, Azhar Iqbal, Tayyaba Akram, and Muhammad Abbas. "On Unconditionally Stable New Modified Fractional Group Iterative Scheme for the Solution of 2D Time-Fractional Telegraph Model." Symmetry 13, no. 11 (2021): 2078. http://dx.doi.org/10.3390/sym13112078.

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In this study, a new modified group iterative scheme for solving the two-dimensional (2D) fractional hyperbolic telegraph differential equation with Dirichlet boundary conditions is obtained from the 2h-spaced standard and rotated Crank–Nicolson FD approximations. The findings of new four-point modified explicit group relaxation method demonstrates the rapid rate of convergence of proposed method as compared to the existing schemes. Numerical tests are performed to test the capability of the group iterative scheme in comparison with the point iterative scheme counterparts. The stability of the

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16

Zheng, Yunying, and Zhengang Zhao. "A Fully Discrete Galerkin Method for a Nonlinear Space-Fractional Diffusion Equation." Mathematical Problems in Engineering 2011 (2011): 1–20. http://dx.doi.org/10.1155/2011/171620.

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The spatial transport process in fractal media is generally anomalous. The space-fractional advection-diffusion equation can be used to characterize such a process. In this paper, a fully discrete scheme is given for a type of nonlinear space-fractional anomalous advection-diffusion equation. In the spatial direction, we use the finite element method, and in the temporal direction, we use the modified Crank-Nicolson approximation. Here the fractional derivative indicates the Caputo derivative. The error estimate for the fully discrete scheme is derived. And the numerical examples are also incl

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17

Erfanifar, Raziyeh, Khosro Sayevand, Nasim Ghanbari та Hamid Esmaeili. "A modified Chebyshev ϑ ‐weighted Crank–Nicolson method for analyzing fractional sub‐diffusion equations". Numerical Methods for Partial Differential Equations 37, № 1 (2020): 614–25. http://dx.doi.org/10.1002/num.22543.

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18

Ishwarya A., Susan, and Rachna Bhatia. "Numerical solution of fourth order generalised Kuramoto-Sivashinsky equation using modified quintic B-spline differential quadrature method." Structural Integrity and Life 25, Special Issue A (2025): S57—S65. https://doi.org/10.69644/ivk-2025-sia-0057.

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In this paper, numerical solutions of the nonlinear generalised Kuramoto-Sivashinsky equation are presented using a modified quintic B-spline differential quadrature method. The Crank-Nicolson and forward finite difference schemes are applied for discretization, while the Rubin and Graves approach is utilised for linearization. The matrix stability approach is used to analyse the method’s stability. Numerical examples demonstrate the accuracy of the method. The computed results are presented in tables and graphs along with a comparative analysis with previous results. The obtained numerical re

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19

Long, Xiaohan, and Chuanjun Chen. "General Formulation of Second-Order Semi-Lagrangian Methods for Convection-Diffusion Problems." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/763630.

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The general formulation of the second-order semi-Lagrangian methods was presented for convection-dominated diffusion problems. In view of the method of lines, this formulation is in a sufficiently general fashion as to include two-step backward difference formula and Crank-Nicolson type semi-Lagrangian schemes as particular ones. And it is easy to be extended to higher-order schemes. We show that it maintains second-order accuracy even if the involved numerical characteristic lines are first-order accurate. The relationship between semi-Lagrangian methods and the modified method of characteris

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20

YAN, J. L., L. H. ZHENG, L. ZHU, and F. Q. LU. "LINEARLY IMPLICIT ENERGY-PRESERVING FOURIER PSEUDOSPECTRAL SCHEMES FOR THE COMPLEX MODIFIED KORTEWEG–DE VRIES EQUATION." ANZIAM Journal 62, no. 3 (2020): 256–73. http://dx.doi.org/10.1017/s1446181120000218.

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AbstractWe propose two linearly implicit energy-preserving schemes for the complex modified Korteweg–de Vries equation, based on the invariant energy quadratization method. First, a new variable is introduced and a new Hamiltonian system is constructed for this equation. Then the Fourier pseudospectral method is used for the space discretization and the Crank–Nicolson leap-frog schemes for the time discretization. The proposed schemes are linearly implicit, which is only needed to solve a linear system at each time step. The fully discrete schemes can be shown to conserve both mass and energy

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21

Li, Han Ling, Lei Hou, Jun Jie Zhao, De Zhi Lin, and Lin Qiu. "Dual Scaled Stochastic FEM Simulation of Porous Honeycomb Material." Applied Mechanics and Materials 443 (October 2013): 48–52. http://dx.doi.org/10.4028/www.scientific.net/amm.443.48.

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In this paper, a dual-scaled method is introduced to simulate the nonlinear property of porous honeycomb material. In microscopic scale, stochastic analysis upon a detailed representation of the hexagonal cells is applied. In macroscopic scale, coupled fluid-solid PDEs with a modified stochastic item are used to describe the rheology of non-Newtonian property of honeycomb. Semi-discrete finite element method (FEM) is applied to solve the PDEs. Comparison of stochastic dynamic system with definite dynamic system is introduced. Numerical Results of Euler explicit time scheme and Crank-Nicolson s

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22

Yan, J. L., L. H. Zheng, L. Zhu, and F. Q. Lu. "Linearly implicit energy-preserving Fourier pseudospectral schemes for the complex modified Korteweg-de Vries equation." ANZIAM Journal 62 (February 4, 2021): 256–73. http://dx.doi.org/10.21914/anziamj.v62.15467.

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We propose two linearly implicit energy-preserving schemes for the complex modified Korteweg–de Vries equation, based on the invariant energy quadratization method. First, a new variable is introduced and a new Hamiltonian system is constructed for this equation. Then the Fourier pseudospectral method is used for the space discretization and the Crank–Nicolson leap-frog schemes for the time discretization. The proposed schemes are linearly implicit, which is only needed to solve a linear system at each time step. The fully discrete schemes can be shown to conserve both mass and energy in the d

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23

J., Omowo B., Ogunbanwo S. T., Adeniran A. O., Olanegan O. O., and Longe I. O. "A convergence analysis of a modified Crank-Nicolson scheme for parabolic partial differential equations." Open Journal of Mathematical Sciences 9 (May 31, 2025): 172–81. https://doi.org/10.30538/oms2025.0252.

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This paper presents a new approach to the derivation of an existing numerical scheme for solving the one-dimensional heat equation. Two theorems are proposed and proven to establish the local truncation error and consistency of the scheme. The scheme’s accuracy is further investigated through step size refinement, and the stability of the scheme is rigorously analyzed using the matrix method. The scheme is implemented in a MATLAB environment, and its performance is evaluated by comparing the numerical solutions with the exact solution. Results demonstrate the superior accuracy of the proposed

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24

Pochai, Nopparat. "Numerical Treatment of a Modified MacCormack Scheme in a Nondimensional Form of the Water Quality Models in a Nonuniform Flow Stream." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/274263.

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Two mathematical models are used to simulate water quality in a nonuniform flow stream. The first model is the hydrodynamic model that provides the velocity field and the elevation of water. The second model is the dispersion model that provides the pollutant concentration field. Both models are formulated in one-dimensional equations. The traditional Crank-Nicolson method is also used in the hydrodynamic model. At each step, the flow velocity fields calculated from the first model are the input into the second model as the field data. A modified MacCormack method is subsequently employed in t

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25

Nong, Lijuan, and An Chen. "Fast High-Order Difference Scheme for the Modified Anomalous Subdiffusion Equation Based on Fast Discrete Sine Transform." Journal of Function Spaces 2021 (April 27, 2021): 1–9. http://dx.doi.org/10.1155/2021/9918955.

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The modified anomalous subdiffusion equation plays an important role in the modeling of the processes that become less anomalous as time evolves. In this paper, we consider the efficient difference scheme for solving such time-fractional equation in two space dimensions. By using the modified L1 method and the compact difference operator with fast discrete sine transform technique, we develop a fast Crank-Nicolson compact difference scheme which is proved to be stable with the accuracy of O τ min 1 + α , 1 + β + h 4 . Here, α and β are the fractional orders which both range from 0 to 1, and τ

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26

Smagin, V. V. "Strong-Norm Error Estimates for the Projective-Difference Method for Parabolic Equations with Modified Crank--Nicolson Scheme." Mathematical Notes 74, no. 5/6 (2003): 864–73. http://dx.doi.org/10.1023/b:matn.0000009023.01722.00.

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27

Akram, Tayyaba, Muhammad Abbas, Azhar Iqbal, Dumitru Baleanu, and Jihad H. Asad. "Novel Numerical Approach Based on Modified Extended Cubic B-Spline Functions for Solving Non-Linear Time-Fractional Telegraph Equation." Symmetry 12, no. 7 (2020): 1154. http://dx.doi.org/10.3390/sym12071154.

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The telegraph model describes that the current and voltage waves can be reflected on a wire, that symmetrical wave patterns can form along a line. A numerical study of these voltage and current waves on a transferral line has been proposed via a modified extended cubic B-spline (MECBS) method. The B-spline functions have the flexibility and high order accuracy to approximate the solutions. These functions also preserve the symmetrical property. The MECBS and Crank Nicolson technique are employed to find out the solution of the non-linear time fractional telegraph equation. The time direction i

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28

Çelikkaya, İhsan. "A New Numerical Simulation for Modified Camassa-Holm and Degasperis-Procesi Equations via Trigonometric Quintic B-spline." Fundamentals of Contemporary Mathematical Sciences 5, no. 2 (2024): 143–58. http://dx.doi.org/10.54974/fcmathsci.1398394.

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In this study, the soliton solutions of the modified Camassa-Holm (mCH) and Degasperis-Procesi (mDP) equations, called modified b-equations with important physical properties, were obtained. The soliton waves' movement and positions formed by solving the mCH and mDP equations were calculated. Ordinary differential equation systems were obtained using trigonometric quintic B-spline bases for position and time direction derivatives in the equations to obtain numerical solutions. An algebraic equation system was then created by writing Crank-Nicolson type approximations for time and position-depe

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29

Norazrizal, Norazrizal, and Norazrizal Aswad bin Abdul Rahman. "Modified Compact Finite Difference Methods for Solving Fuzzy Time Fractional Wave Equation in Double Parametric Form of Fuzzy Number." International Journal of Neutrosophic Science 26, no. 2 (2025): 192–203. https://doi.org/10.54216/ijns.260214.

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Fuzzy fractional partial differential equations have become a powerful approach to handle uncertainty or imprecision in real-world modeling problems. In this article, two compact finite difference schemes, the compact Crank-Nicolson and the compact center time center space methods, were developed and used to obtain a numerical solution for fuzzy time fractional wave equations in the double parametric form. The principles of fuzzy set theory are utilized to perform a fuzzy analysis and formulate the proposed numerical schemes. The Caputo formula is used to define the time-fractional derivative

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Badri, Mohammadali, and Fereidoun Sabetghadam. "A Transformation for Imposing the Rigid Bodies on the Solution of the Vorticity-Stream Function Formulation of Incompressible Navier–Stokes Equations." International Journal of Computational Methods 17, no. 09 (2019): 1950063. http://dx.doi.org/10.1142/s0219876219500634.

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A new penalization method is proposed for implementing the rigid bodies on the solution of the vorticity-stream function formulation of the incompressible Navier–Stokes equations. The method is based upon an active transformation of dependent variables. The transformation may be interpreted as time dilation. In this interpretation, the rigid body is considered as a region where the time is dilated infinitely, that is, time is stopped. The transformation is introduced in the vorticity and stream function equations to achieve a set of modified equations. The, in the modified equations, the time

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31

Quinga, Santiago, Wilson Pavon, Nury Ortiz, Héctor Calvopiña, Gandhy Yépez, and Milton Quinga. "Numerical Solution of the Nonlinear Convection–Diffusion Equation Using the Fifth Order Iterative Method by Newton–Jarratt." Mathematics 13, no. 7 (2025): 1164. https://doi.org/10.3390/math13071164.

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This study presents a novel fifth-order iterative method for solving nonlinear systems derived from a modified combination of Jarratt and Newton schemes, incorporating a frozen derivative of the Jacobian. The method is applied to approximate solutions of the nonlinear convection–diffusion equation. A MATLAB script function was developed to implement the approach in two stages: first, discretizing the equation using the Crank–Nicolson Method, and second, solving the resulting nonlinear systems using Newton’s iterative method enhanced by a three-step Jarratt variant. A comprehensive analysis of

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Lv, Xin, Qingping Zou, D. E. Reeve, and Yong Zhao. "A Preconditioned Implicit Free-Surface Capture Scheme for Large Density Ratio on Tetrahedral Grids." Communications in Computational Physics 11, no. 1 (2012): 215–48. http://dx.doi.org/10.4208/cicp.170510.290311a.

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AbstractWe present a three dimensional preconditioned implicit free-surface capture scheme on tetrahedral grids. The current scheme improves our recently reported method [10] in several aspects. Specifically, we modified the original eigensystem by applying a preconditioning matrix so that the new eigensystem is virtually independent of density ratio, which is typically large for practical two-phase problems. Further, we replaced the explicit multi-stage Runge-Kutta method by a fully implicit Euler integration scheme for the Navier-Stokes (NS) solver and the Volume of Fluids (VOF) equation is

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Ma, Chupeng, and Liqun Cao. "A Crank--Nicolson Finite Element Method and the Optimal Error Estimates for the Modified Time-Dependent Maxwell--Schrödinger Equations." SIAM Journal on Numerical Analysis 56, no. 1 (2018): 369–96. http://dx.doi.org/10.1137/16m1085231.

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Haghighi, Ahmad Reza, and Shirin Pakrou. "Comparison of the LBM with the modified local Crank-Nicolson method solution of transient one-dimensional nonlinear Burgers' equation." International Journal of Computing Science and Mathematics 7, no. 5 (2016): 459. http://dx.doi.org/10.1504/ijcsm.2016.080084.

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Haghighi, Ahmad Reza, and Shirin Pakrou. "Comparison of the LBM with the modified local Crank-Nicolson method solution of transient one-dimensional nonlinear Burgers' equation." International Journal of Computing Science and Mathematics 7, no. 5 (2016): 459. http://dx.doi.org/10.1504/ijcsm.2016.10000910.

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Wang, Jinfeng, Baoli Yin, Yang Liu, Hong Li, and Zhichao Fang. "A Mixed Element Algorithm Based on the Modified L1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model." Fractal and Fractional 5, no. 4 (2021): 274. http://dx.doi.org/10.3390/fractalfract5040274.

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In this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupled system with three equations is constructed. Compared to the previous space–time high-order model, the derived system is a lower coupled equation with lower time derivatives and second-order space derivatives, which can be approximated by using many time discrete schemes. Here, the second-order Cr

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Zureigat, Hamzeh, Mohammad A. Tashtoush, Ali F. Al Jassar, Emad A. Az-Zo’bi, and Mohammad W. Alomari. "A Solution of the Complex Fuzzy Heat Equation in Terms of Complex Dirichlet Conditions Using a Modified Crank–Nicolson Method." Advances in Mathematical Physics 2023 (September 11, 2023): 1–8. http://dx.doi.org/10.1155/2023/6505227.

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Complex fuzzy sets (CFSs) have recently emerged as a potent tool for expanding the scope of fuzzy sets to encompass wider ranges within the unit disk in the complex plane. This study explores complex fuzzy numbers and introduces their application for the first time in the literature to address a complex fuzzy partial differential equation that involves a complex fuzzy heat equation under Hukuhara differentiability. The researchers utilize an implicit finite difference scheme, namely the Crank–Nicolson method, to tackle complex fuzzy heat equations. The problem’s fuzziness arises from the coeff

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Qian, Xu, Hao Fu, and Songhe Song. "Conservative modified Crank–Nicolson and time-splitting wavelet methods for modeling Bose–Einstein condensates in delta potentials." Applied Mathematics and Computation 307 (August 2017): 1–16. http://dx.doi.org/10.1016/j.amc.2017.02.037.

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Deng, Shuyan. "Thermally Fully Developed Electroosmotic Flow of Power-Law Nanofluid in a Rectangular Microchannel." Micromachines 10, no. 6 (2019): 363. http://dx.doi.org/10.3390/mi10060363.

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The hydrodynamic and thermal behavior of the electroosmotic flow of power-law nanofluid is studied. A modified Cauchy momentum equation governing the hydrodynamic behavior of power-law nanofluid flow in a rectangular microchannel is firstly developed. To explore the thermal behavior of power-law nanofluid flow, the energy equation is developed, which is coupled to the velocity field. A numerical algorithm based on the Crank–Nicolson method and compact difference schemes is proposed, whereby the velocity, temperature, and Nusselt number are computed for different parameters. A larger nanopartic

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Christain, Onugu,, Davies, Iyai, and Amad, Innocent Uchenna. "A Numerical Approximation on Black-Scholes Equation of Option Pricing." Asian Research Journal of Mathematics 19, no. 7 (2023): 92–105. http://dx.doi.org/10.9734/arjom/2023/v19i7682.

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This paper considered the notion of European option which is geared towards solving analytical and numerical solutions. In particular, we examined the Black-Scholes closed form solution and modified Black-Scholes (MBS) partial differential equation using Crank-Nicolson finite difference method. These partial differential equations were approximated to obtain Call and Put option prices. The explicit price of both options is found accordingly. The numerical solutions were compared to the closed form prices of Black-Scholes formula. More so, comparisons of other parameters were discussed for the

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Koroche, Kedir Aliyi. "Investigation the Accuracy of Crank Nicolson Methods and their Modified Schemes for One-Dimensional Linear Convection-Reaction-Diffusion Equations." Asian Journal of Mathematics and Computer Research 31, no. 1 (2024): 42–56. http://dx.doi.org/10.56557/ajomcor/2024/v31i18546.

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In this paper, Crank-Nicolson and Their Modified scheme are applied to find the solution of the convection-reaction-diffusion equation. First, the given solution domain is discretized. Then, using Taylor series expansion, we obtain the difference scheme of the model problem. By rearranging this scheme, we gain proposed techniques. To verify the validity of the proposed techniques, two model illustrations are considered. The stability and convergent analysis of the present scheme is worked by supporting the theoretical and numerical error bound. The accuracy of the present scheme has been shown

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Pochai, Nopparat. "A Numerical Treatment of Nondimensional Form of Water Quality Model in a Nonuniform Flow Stream Using Saulyev Scheme." Mathematical Problems in Engineering 2011 (2011): 1–15. http://dx.doi.org/10.1155/2011/491317.

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The stream water quality model of water quality assessment problems often involves numerical methods to solve the equations. The governing equation of the uniform flow model is one-dimensional advection-dispersion-reaction equations (ADREs). In this paper, a better finite difference scheme for solving ADRE is focused, and the effect of nonuniform water flows in a stream is considered. Two mathematical models are used to simulate pollution due to sewage effluent. The first is a hydrodynamic model that provides the velocity field and elevation of the water flow. The second is a advection-dispers

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Baker, Mark, and Budimir Rosic. "1D analytic and numerical analysis of multilayer laminates and thin film heat transfer gauges." Journal of the Global Power and Propulsion Society 6 (September 1, 2022): 238–53. http://dx.doi.org/10.33737/jgpps/151660.

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The impulse response method is widely used for heat transfer analysis in turbomachinery applications. Traditionally, the 1D method assumes a linear time invariant, isotropic, semi-infinite block and does not accurately model the behaviour of laminated materials. This paper evaluates the error introduced by the single layer assumption and outlines the required modifications for multilayer analysis. The analytic solution for an N layer, semi-infinite laminate is presented. Adapted multilayer basis functions are derived for the impulse response method and used to evaluate the impact of uniform, i

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Baker, Mark, and Budimir Rosic. "1D analytic and numerical analysis of thin film heat transfer gauges and infra-red cameras in non-planar applications." Journal of the Global Power and Propulsion Society 8 (June 19, 2024): 188–202. http://dx.doi.org/10.33737/jgpps/185746.

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The impulse response method is widely used for heat transfer analysis in turbomachinery applications. Traditionally, the 1D method assumes a: linear time invariant, isotropic, semi-infinite block with planar surfaces and does not accurately model the true geometric behaviour. This paper evaluates the error introduced by the planar assumption and outlines the required modifications for accurate freeform surface analysis. Adapted cylindrical basis functions are defined for the impulse response method and used to evaluate the impact of the 1D planar assumption. The analytic solutions for both con

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Babaei, Masoud, Faraz Kiarasi, Kamran Asemi, Rossana Dimitri, and Francesco Tornabene. "Transient Thermal Stresses in FG Porous Rotating Truncated Cones Reinforced by Graphene Platelets." Applied Sciences 12, no. 8 (2022): 3932. http://dx.doi.org/10.3390/app12083932.

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The present work studies an axisymmetric rotating truncated cone made of functionally graded (FG) porous materials reinforced by graphene platelets (GPLs) under a thermal loading. The problem is tackled theoretically based on a classical linear thermoelasticity approach. The truncated cone consists of a layered material with a uniform or non-uniform dispersion of GPLs in a metal matrix with open-cell internal pores, whose effective properties are determined according to the extended rule of mixture and modified Halpin–Tsai model. A graded finite element method (FEM) based on Rayleigh–Ritz ener

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Seaïd, Mohammed. "Semi-lagrangian integration schemes for viscous incompressible flows." Computational Methods in Applied Mathematics 2, no. 4 (2002): 392–409. http://dx.doi.org/10.2478/cmam-2002-0022.

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AbstractA new second-order accurate scheme for the computation of unsteady viscous incompressible flows is proposed. The scheme is based on the vorticity-stream function formulation along the characteristics and consists of combining the modified method of characteristics with an explicit scheme with an extended real stability interval. A comparison of the new method with the semi-Lagrangian Cranck-Nicolson and classical semi-Lagrangian Runge-Kutta schemes is presented. Numerical results are carried out on Navier-Stokes equations and this efficient second-order scheme has also made it possible

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Babajide Johnson, Omowo, and Longe Idowu Oluwaseun. "Crank-Nicolson and Modified Crank-Nicolson Scheme for One Dimensional Parabolic Equation." International Journal of Applied Mathematics and Theoretical Physics 6, no. 3 (2020): 35. http://dx.doi.org/10.11648/j.ijamtp.20200603.11.

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Khan, Amin, Muhammad Ahsan, Ebenezer Bonyah, et al. "Numerical Solution of Schrödinger Equation by Crank–Nicolson Method." Mathematical Problems in Engineering 2022 (April 14, 2022): 1–11. http://dx.doi.org/10.1155/2022/6991067.

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In this study, we implemented the well-known Crank–Nicolson scheme for the numerical solution of Schrödinger equation. The numerical results converge to the exact solution because the Crank–Nicolson scheme is unconditionally stable and accurate. We have compared the results for different parameters with analytical solution, and it is found that the Crank–Nicolson scheme is suitable for the numerical solution of Schrödinger equations. Three different problems are included to verify the accuracy, stability, and capability of the Crank–Nicolson scheme.

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Ajeel, Omar Abdullah, and Awni M. Gaftan. "Using Crank-Nicolson Numerical Method to solve Heat-Diffusion Problem." Tikrit Journal of Pure Science 28, no. 3 (2023): 101–4. http://dx.doi.org/10.25130/tjps.v28i3.1434.

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The current study aimed to use the Crank-Nicolson numerical method to solve Heat-Diffusion Problem in comparison with the ADI method. In this paper, the general formula of the Crank-Nicolson Numerical Method was derived and applied to solve the heat diffusion. The same problem then has been solved using ADI numerical method. The results of the Crank-Nicolson numerical method were compared with that of the ADI numerical method. The comparison results revealed that Crank-Nicolson is more accurate than the results of ADI at the initial steps of the problem solution.

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Ghani, Mohammad. "Numerical Results of Crank-Nicolson and Implicit Schemes to Laplace Equation with Uniform and Non-Uniform Grids." InPrime: Indonesian Journal of Pure and Applied Mathematics 3, no. 2 (2021): 122–35. http://dx.doi.org/10.15408/inprime.v3i2.20917.

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AbstractIn this paper, we investigate the numerical results between Implicit and Crank-Nicolson method for Laplace equation. Based on the numerical results obtained, we get the conclusion that the absolute error of Crank-Nicolson method is smaller than the absolute error of Implicit method for uniform and non-uniform grids which both refer to the analytical solution of Laplace equation obtained by separable variable method.Keywords: Crank-Nicolson; Implicit; Laplace equation; separable variable method; uniform and non-uniform grids. AbstrakDalam makalah ini, kami menyelidiki hasil numerik anta

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