Crank-Nicolson Finite Difference Research Articles

On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics

In recent years, various complex systems and real-world phenomena have been shown to include memory and hereditary properties that change with respect to time, space, or other variables. Consequently, fractional partial differential equations containing variable-order fractional operators have been extensively resorted for modeling such phenomena accurately. In this paper, we consider the two-dimensional fractional cable equation with the Caputo variable-order fractional derivative in the time direction, which is preferable for describing neuronal dynamics in biological systems. A point-wise scheme, namely, the Crank–Nicolson finite difference method, along with a group-wise scheme referred to as the explicit decoupled group method are proposed to solve the problem under consideration. The stability and convergence analyses of the numerical schemes are provided with complete details. To demonstrate the validity of the proposed methods, numerical simulations with results represented in tabular and graphical forms are given. A quantitative analysis based on the CPU timing, iteration counting, and maximum absolute error indicates that the explicit decoupled group method is more efficient than the Crank–Nicolson finite difference scheme for solving the variable-order fractional equation.

Quadratic American Strangle Options in Light of Two-Sided Optimal Stopping Problems

The aim of this paper is to examine some American-style financial instruments that lead to two-sided optimal hitting problems. We pay particular attention to derivatives that are similar to strangle options but have a quadratic payoff function. We consider these derivatives in light of much more general payoff structures under certain conditions which guarantee that the optimal strategy is an exit from a strip. Closed-form formulas for the optimal boundaries and the fair price are derived when the contract has no maturity constraints. We obtain the form of the optimal boundaries under the finite maturity horizon and approximate them by maximizing the financial utility of the derivative holder. The Crank–Nicolson finite difference method is applied to the pricing problem. The importance of these novel financial instruments is supported by several features that are very useful for financial practice. They combine the characteristics of the power options and the ordinary American straddles. Quadratic strangles are suitable for investors who need to hedge strongly, far from the strike positions. In contrast, the near-the-money positions offer a relatively lower payoff than the ordinary straddles. Note that the usual options pay exactly the overprice; no more, no less. In addition, the quadratic strangles allow investors to hedge the positions below and above the strike together. This is very useful in periods of high volatility when large market movements are expected but their direction is unknown.

Numerical Solution of Two-Dimensional Nonlinear Unsteady Advection-Diffusion-Reaction Equations with Variable Coefficients

The advection-diffusion-reaction (ADR) equation is a fundamental mathematical model used to describe various processes in many different areas of science and engineering. Due to wide applicability of the ADR equation, finding accurate solution is very important to better understand a physical phenomenon represented by the equation. In this study, a numerical scheme for solving two-dimensional unsteady ADR equations with spatially varying velocity and diffusion coefficients is presented. The equations include nonlinear reaction terms. To discretize the ADR equations, the Crank–Nicolson finite difference method is employed with a uniform grid. The resulting nonlinear system of equations is solved using Newton’s method. At each iteration of Newton’s method, the Gauss–Seidel iterative method with sparse matrix computation is utilized to solve the block tridiagonal system and obtain the error correction vector. The consistency and stability of the numerical scheme are investigated. MATLAB codes are developed to implement this combined numerical approach. The validation of the scheme is verified by solving a two-dimensional advection-diffusion equation without reaction term. Numerical tests are provided to show the good performances of the proposed numerical scheme in simulation of ADR problems. The numerical scheme gives accurate results. The obtained numerical solutions are presented graphically. The result of this study may provide insights to apply numerical methods in solving comprehensive models of physical phenomena that capture the underlying situations.

A finite difference study of radiative mixed convection MHD heat propagating Casson fluid past an accelerating porous plate including viscous dissipation and Joule heating effects

A finite difference numerical simulation scrutiny is executed to evaluate the combined impacts of heat generation, buoyancy forces, viscous dissipation and Joule heating in unsteady hydro-magnetic mixed convective chemically reactive and radiative Casson fluid flowing along an exponentially accelerating permeable vertical plate engrossed in a porous media by considering ramp surface concentration and temperature. The dimensionless non-linear coupled PDEs describing the flow model are dealt numerically by adopting the competent implicit Crank-Nicolson finite difference procedure. The variance of velocity, temperature, and concentration distributions are exposed via graphical representations due to the dissimilarity of the flow restrained parameters. Computational outcomes of the skin-friction, Nusselt and the Sherwood numbers are portrayed in the tabular pattern. The final outcomes of the research exposed that the impacts of thermal radiation, viscous dissipation, and heat production parameters enlarges the temperature and velocity distributions. The fluid motion deflates for growing Casson parameter and magnetic field intensity. The rising chemical reaction parameter suppresses the concentration and velocity distributions. Very importantly it is distinguished that fluid momentum, temperature, and concentration are quicker in the instance of isothermal plate temperature than ramp wall temperature. This kind of research may find specific industrial and medical utilizations such as glass manufacturing, crude oil purification, lubrication, paper production, blood transport study in cardiovascular design, etc.

The Two-Dimensional Conduction Heat Transfer Equation on a Square Plate: Explicit vs. Crank-Nicolson Method in MS Excel Spreadsheet

In this paper, the two-dimensional conduction heat transfer equation on a square plate is analyzed using a finite difference method. We have developed both the forward time-centered space (FTCS) and Crank-Nicolson (CN) finite difference schemes for the two-dimensional heat equation, employing Taylor series. Subsequently, these schemes were employed to solve the governing equations. The primary objective of this study is to compare the efficiency of the two methods in solving the conduction heat transfer equation. This was accomplished by implementing Spreadsheet Excel instructions. The results are presented, highlighting a comparison between the exact and approximate solutions. Furthermore, to demonstrate the convergence of the numerical schemes, we estimated the error between the actual and approximate solutions for a specific numerical problem and presented the results graphically. The data utilized in this research included the thermal conductivity of the medium of the square plate concerning width, grid, and compliance with initial and boundary conditions. The findings indicate that the Crank-Nicolson method is more accurate than the forward time-centered space method, as it approaches the exact solution more effectively. Furthermore, this study confirms that the solution and simulation of the heat transfer equation on a square plate can be accurately performed using an Excel spreadsheet as well as other numerical software.

Comparative Study of Crank-Nicolson and Modified Crank-Nicolson Numerical methods to solve linear Partial Differential Equations

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 superiority in accuracy. Novelty: The modified Crank-Nicolson method offers a significant enhancement to the traditional Crank-Nicolson finite difference method, making it a valuable tool for effectively solving partial differential equations. Keywords: Crank­Nicolson Method, Modified Crank­Nicolson Method, Finite Difference, Partial Differential Equations, Parabolic Equations, Python Software

Investigation of Fractional Order Tumor Cell Concentration Equation Using Finite Difference Method

The fractional differential equation provides a powerful mathematical framework for modeling and understanding a wide range of complex, non-local and memory-dependent phenomena in various scientific, engineering and real-world applications. Liver metastasis is a secondary cancerous tumor that develops in the liver due to the spread of cancer cells from a primary cancer originating in another part of the human body. The primary focus of this study is to understand tumor growth in the human liver, both in the presence and absence of medication therapy. To achieve this, a temporal fractional-order parabolic partial differential equation is utilized, and its analysis is carried out using numerical methods. The Caputo derivative is employed to explore the impact of medication therapy on tumor growth. To numerically solve the mathematical model, the Crank-Nicolson finite difference method has been developed. This method is selected for its remarkable attributes, including unconditional stability and second-order accuracy in both spatial and temporal dimensions. The outcomes and insights derived from this study are effectively communicated through various graphical representations. These visual aids serve as invaluable tools for comprehending the profound impact of medication therapy on the growth of liver tumors. Through the medium of these graphical depictions, one can glean a clearer and more intuitive understanding of the complex dynamics at play. The numeric solution to this intricate problem is achieved through the implementation of algorithms meticulously crafted using versatile and powerful Python programming language. Python’s flexibility, extensive libraries, and robust numerical computing capabilities make it a good choice for handling the complexities of this study.

An improved approximate method for solving two-dimensional time-fractional-order Black-Scholes model: a finite difference approach

<abstract><p>In this paper, we considered the two-dimensional fractional-order Black-Scholes model in the Liouville-Caputo sense. The Black-Scholes model was an important tool in the financial market, used for determining option prices in the European-style market. However, finding a closed-form analytical solution for the fractional-order partial differential equation was challenging. To address this, we introduced an improved finite difference method for approximating the solution of the two-dimensional fractional-order Black-Scholes model in the Liouville-Caputo sense, based on the Crank-Nicolson finite difference method. This method combined the concepts of the finite difference method for solving the multidimensional Black-Scholes model and the finite difference method for solving the fractional-order heat equation. We analyzed the conditional stability and the order of convergence. Furthermore, numerical examples were provided to illustrate the determination of option prices.</p></abstract>

Entropy-driven optimization of radiative Jeffrey tetrahybrid nanofluid flow through a stenosed bifurcated artery with Hall effects

Atherosclerosis, which causes the artery walls to thicken, the lumen to narrow, and the wall to thin in some places, is characterized by plaque accumulation in the arteries. These blood flow modifications can cause aneurysms and heart attacks if left unattended. Most of the arteries in the cardiovascular system are branched; therefore, a parent artery (main artery) with two daughter arteries (branched arteries) is considered in the present analysis. To examine the impact of various nanoparticle combinations on blood flow, four distinct nanoparticles, namely, gold (Au), graphene oxide (GO), copper (Cu), and tantalum (Ta), were injected into the blood to generate Au–GO–Cu–Ta/blood tetrahybrid nanofluid. In arteries with small diameters, blood behavior is regarded as non-Newtonian; therefore, blood behavior is governed by Jeffrey fluid in the present analysis. It has been investigated how Hall effects, Joule heating, radiation, and viscous dissipation affect blood flow through an artery that has an overlapping stenosis in the branches and a bell-shaped stenosis in the main artery. The approximation of mild stenosis is utilized to simplify and non-dimensionalize the governing equations. The Crank–Nicolson finite-difference scheme is used in MATLAB to solve the resulting equations. The results for velocity, temperature, wall shear stress, flow rate, and heat transfer rate are represented graphically. Furthermore, the entropy optimization has been performed for the specified problem. Enhancement in velocity with half of the bifurcation angle (η) can be observed from the velocity contours. The velocity of the tetrahybrid nanofluid increases with an increase in Jeffrey fluid parameter (λ1*) and shape parameter of the nanoparticles (n) as well. Introducing nanoparticles into the bloodstream can improve targeted drug delivery, allowing for more precise treatment at the cellular level. In addition, the tunable properties of nanoparticles offer possibilities for enhanced therapeutic and diagnostic treatments in a variety of medical disorders.

Turbulent Lorentz heat flow visualization in radiative boundary layer regime

Modern nuclear energy systems often employ the magneto-hydrodynamic and feature radiative heat transfer. Motivated by studying the near-wall transport phenomena in such applications, the present article examines the simultaneous influence of thermal radiative flux and magneto-hydrodynamics on two-dimensional electrically conducting natural convective turbulent heat transfer about a vertical surface under a transverse static magnetic field using the low Reynolds number kinetic energy and dissipation (k – ε) model. An optimized Crank–Nicolson finite difference method is applied to solve the nonlinear and coupled system of Reynolds-averaged boundary layer equations. Numerical computations are conducted to visualize the streamlines and heatlines in laminar and turbulent regimes via mathematical stream and heat functions based on the Bejan approach. A detailed parametric study of the effects of the magnetic field, radiative flux, and turbulent Reynolds number on flow average velocity, temperature, kinetic energy, and dissipation rate is conducted. The simulations reveal that an increase in the magnetic field intensity reduces the average velocity and dissipation rate, whereas an increase in thermal radiation decreases the time mean temperature. The study also includes contour plots of turbulence energy and dissipation rate, along with skin friction coefficient and Nusselt number. The obtained numerical outcomes are compared to previous literature, and a good agreement is found.

Characterization of Excitation Effects and Data Interpretation of Combined Time-Domain Multiwaveform Transmission Currents

With the development of electronic technology, time-domain electromagnetic (TDEM) transmitters are now capable of emitting various waveforms such as trapezoidal, triangular, and semi-sinusoidal waveforms, as well as their combinations. However, the application of multi-waveform combination detection methods is still immature, and modeling methods are required to analyze the characteristics of different waveforms and complete data interpretation. Based on the Crank-Nicolson finite difference time-domain (CN-FDTD) method for discretizing the electromagnetic wave equations, the current waveform can be converted into current surface density for source-included calculations, enabling high-precision modeling of the responses to different current waveforms. The responses to multiple transmission current waveforms were compared and analyzed using various models, and the results showed that selecting targeted excitation parameters based on the characteristics of the target can improve detection sensitivity. We applied the multi-waveform combination detection method to the actual exploration in the field, obtained the pure secondary field response by subtracting the primary field of the actual current waveform from the measured data, converted the responses of different transmission current waveforms into step wave responses by deconvolution method, and performed resistivity imaging. The results of modeling and data interpretation demonstrate the effectiveness of the multi-waveform combination detection method in improving geological exploration accuracy, and promoted the development of fine-scale geophysical exploration.

Numerical solutions of free convective flow from a moving cone in the presence of non-uniform surface temperature

This computational study intends to determine the impact of non-isothermal surface temperature on the natural convective boundary layerflow of fluid along a moving vertical cone. The governing equation deals with continuity, momentum, and energy along with thermal fluctuations are reduced into dimensionless unsteady non-linear coupled PDE. The non-dimensional governing equations of the boundary layer are then illuminated by a systematic and unconditionally stable method of implicit finite difference Crank-Nicolson scheme. Analyzing the stability and convergence of the system has helped in the optimization of parameters and accuracy of the system. Using a tabular approach, we validate the present numerical methodology by comparing it to the previously discussed results. Through a table of values, the physical impacts of parameters are analyzed and graphical representations are provided. From the graphs, it is evident that an increment in Pr and n results in lower temperature and velocity profiles.

Reconstruction of Timewise Dependent Coefficient and Free Boundary in Nonlocal Diffusion Equation with Stefan and Heat Flux as Overdetermination Conditions

The problem of reconstruction of a timewise dependent coefficient and free boundary at once in a nonlocal diffusion equation under Stefan and heat Flux as nonlocal overdetermination conditions have been considered. A Crank–Nicolson finite difference method (FDM) combined with the trapezoidal rule quadrature is used for the direct problem. While the inverse problem is reformulated as a nonlinear regularized least-square optimization problem with simple bound and solved efficiently by MATLAB subroutine lsqnonlin from the optimization toolbox. Since the problem under investigation is generally ill-posed, a small error in the input data leads to a huge error in the output, then Tikhonov’s regularization technique is applied to obtain regularized stable results.

A Numerical Approximation on Black-Scholes Equation of Option Pricing

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 purpose of investment plans. The computational results shows: increase in stock volatility increases the value of options, when the initial stock price is equal to its strike price the values of call option is higher than the put option. This informs the investor about the behavior of stock prices for the purpose of decision making. Finally, all simulation results presented graphically using MATLAB.

Computational study on 2D three-phase lag bioheat model during cryosurgery using RBF meshfree method

Biological tissue has a multidimensional and non-homogeneous inner structure by nature. The temperature distribution and freezing front locations in biological tissue are crucial to optimizing the damage to tissue during cryosurgery. There is a need for a good mathematical model and effective simulation techniques to predict the effectiveness of the therapy. The present study concerns the numerical study of phase change phenomena during cryosurgery using the three-phase lag (TPL) bioheat model in arbitrary soft tissue domains, i.e., circular (Γ1), ameba-like (Γ2), and multiconnected (Γ3). We employ the effective heat capacity formulation to solve the nonlinear governing equation. The Gaussian radial basis function and Crank–Nicolson finite difference approximation are applied for spatial and time derivatives, respectively. Using the present algorithm, we study the impact of phase lag (τv) due to thermal displacement involved in the TPL model on phase change interface position and thermal distribution in all three domains. The obtained results may be beneficial in the field of oncology.

American strangle options with arbitrary strikes

AbstractThe so‐called American strangle options are examined in this paper. Their main characteristic is the combined put and call feature. The holder has the right to exercise prematurely choosing the option's style—put or call. We abandon the traditional assumption that the put strike is below the call one considering arbitrary values. We also assume that the put and call weights are different. The equations for the early exercise boundaries are derived in the perpetual case. After that we approximate numerically these boundaries for the finite maturity options maximizing the option holder's utility. On the basis of them we apply a Crank–Nicolson finite difference method to the corresponding Black–Scholes‐style partial differential equation to obtain the fair option price.

Coupled Effect of Chloride Corrosion and Repeated Uniaxial Compressive Loading on Unsaturated Concrete.

Concrete structure performance continuously deteriorates during operation, and the performance is simultaneously affected by chloride corrosion and repeated traffic loading. Repeated-loading-induced cracks have an impact on the rate of chloride corrosion. Chloride-induced concrete corrosion also affect the stress level of the structure under loading. Therefore, the coupled effect of repeated loading and chloride corrosion on the structural performance needs to be investigated. An upgraded test device was developed for chloride corrosion testing of unsaturated concrete structures under repeated loading. Based on the experimental results, considering the influence of repeated loading on the moisture diffusion coefficient and the chloride diffusion coefficient, a chloride transport model for unsaturated concrete under the coupled effect of repeated uniaxial compressive loading and corrosion was established. The chloride concentration under coupled loading was determined by the Crank-Nicolson finite difference method and the Thomas algorithm, and then chloride transport under the coupled effect of repeated loading and corrosion was analyzed. The results showed that the stress level and the repeated loading cycles directly affect the relative volumetric water content and chloride concentration in unsaturated concrete. The effect of chloride corrosion is more severe in unsaturated concrete compared to saturated concrete.

A modified Crank-Nicolson finite difference method preserving maximum-principle for the phase-field model

In this paper, we mainly study a new Crank-Nicolson finite difference (FD) method with a large time step for solving the nonlinear phase-field model with a small parameter disturbance. To this end, we first introduce an artificial stability term to build a modified Crank-Nicolson FD (MCNFD) scheme, and then prove that the MCNFD scheme satisfies the intrinsic maximum principle of the phase-field model. Secondly, we discuss that the MCNFD scheme satisfies the intrinsic energy stability of the phase-field model, and obtain the optimal error estimation of the MCNFD solution based on the conclusion of the discrete maximum principle. Finally, two numerical examples of one-dimensional and two-dimensional phase-field models are given to verify that the solution of the MCNFD scheme satisfies the above theoretical results.

Bejan’s numerical heat and mass flow visualization in turbulent boundary layer regime

Low Reynolds number k-ϵ model is employed to demonstrate the Bejan's numerical heat and mass flow visualization in turbulent boundary layer regime. The governing turbulence kinetic energy and its dissipation rate equations are included along with conservation equations. The turbulent flow equations are highly nonlinear and coupled in nature, hence, an unconditionally stable Crank–Nicolson finite difference approach is deployed. The computer-generated heatlines, masslines, streamlines, and isotherms for different control parameters are analyzed in the turbulent flow regime. Average momentum, thermal, and concentration diffusion rates are increased upon the increment in in laminar regime and are decreasing with rising in turbulent regime. The stream, heat and masslines deviate more in turbulent regime when compared to laminar regime. Further, the kinetic energy and dissipation rate contours are deviated more in turbulent heat and mass flow regime for different values of . Mainly, in this research paper, the authors made an attempt to demonstrate the turbulent heat and mass flow visualization about a vertical plate using boundary layer approximations through LRN k-ϵ turbulence model.

Determination of the space-dependent blood perfusion coefficient in the thermal-wave model of bio-heat transfer

PurposeWhen modeling heat propagation in biological bodies, a non-negligible relaxation time (typically between 15-30 s) is required for the thermal waves to accumulate and transfer, i.e. thermal waves propagate at a finite velocity. To accommodate for this feature that is characteristic to heat transfer in biological bodies, the classical Fourier's law has to be modified resulting in the thermal-wave model of bio-heat transfer. The purpose of the paper is to retrieve the space-dependent blood perfusion coefficient in such a thermal-wave model of bio-heat transfer from final time temperature measurements.Design/methodology/approachThe non-linear and ill-posed blood perfusion coefficient identification problem is reformulated as a non-linear minimization problem of a Tikhonov regularization functional subject to lower and upper simple bounds on the unknown coefficient. For the numerical discretization, an unconditionally stable direct solver based on the Crank–Nicolson finite difference scheme is developed. The Tikhonov regularization functional is minimized iteratively by the built-in routine lsqnonlin from the MATLAB optimization toolbox. Both exact and numerically simulated noisy input data are inverted.FindingsThe reconstruction of the unknown blood perfusion coefficient for three benchmark numerical examples is illustrated and discussed to verify the proposed numerical procedure. Moreover, the proposed algorithm is tested on a physical example which consists of identifying the blood perfusion rate of a biological tissue subjected to an external source of laser irradiation. The numerical results demonstrate that accurate and stable solutions are obtained.Originality/valueAlthough previous studies estimated the important thermo-physical blood perfusion coefficient, they neglected the wave-like nature of heat conduction present in biological tissues that are captured by the more accurate thermal-wave model of bio-heat transfer. The originalities of the present paper are to account for such a more accurate thermal-wave bio-heat model and to investigate the possibility of determining its space-dependent blood perfusion coefficient from temperature measurements at the final time.