Nicolson Finite Difference Research Articles

A local fractional modified Crank–Nicolson scheme for fractal LWR model of traffic flow

This research proposes a modified Crank–Nicolson finite difference scheme with local fractional derivatives to approximate the solutions of local fractional LWR traffic flow model. The stability and consistency of the scheme are examined. Further, convergence of the scheme is assured by using Lax’s equivalence theorem. Some exemplary instances are discussed along with their simulations to validate the proposed method. The obtained numerical solutions show the dynamical evolution of traffic density with respect to time and space. The results derived using the proposed numerical scheme establish that they are quite effective in obtaining the numerical solution to the fractal vehicular traffic flow problem.

Identification of Time-Wise Thermal Diffusivity, Advection Velocity on the Free-Boundary Inverse Coefficient Problem

This paper is concerned with finding solutions to free-boundary inverse coefficient problems. Mathematically, we handle a one-dimensional non-homogeneous heat equation subject to initial and boundary conditions as well as non-localized integral observations of zeroth and first-order heat momentum. The direct problem is solved for the temperature distribution and the non-localized integral measurements using the Crank–Nicolson finite difference method. The inverse problem is solved by simultaneously finding the temperature distribution, the time-dependent free-boundary function indicating the location of the moving interface, and the time-wise thermal diffusivity or advection velocities. We reformulate the inverse problem as a non-linear optimization problem and use the lsqnonlin non-linear least-square solver from the MATLAB optimization toolbox. Through examples and discussions, we determine the optimal values of the regulation parameters to ensure accurate, convergent, and stable reconstructions. The direct problem is well-posed, and the Crank–Nicolson method provides accurate solutions with relative errors below 0.006% when the discretization elements are M=N=80. The accuracy of the forward solutions helps to obtain sensible solutions for the inverse problem. Although the inverse problem is ill-posed, we determine the optimal regularization parameter values to obtain satisfactory solutions. We also investigate the existence of inverse solutions to the considered problems and verify their uniqueness based on established definitions and theorems.

Retrieving the time-dependent blood perfusion coefficient in the thermal-wave model of bio-heat transfer

PurposeIn order to include the non-negligible lag relaxation time feature that is characteristic of heat transfer in biological bodies, the classical Fourier's law of heat conduction has to be generalized as the Maxwell–Cattaneo law resulting in the thermal-wave model of bio-heat transfer. The purpose of the paper is to retrieve the unknown time-dependent blood perfusion coefficient in such a thermal-wave model of bio-heat transfer from (non-intrusive) measurements of the temperature on an accessible sub-portion of the boundary that may be taken with an infrared scanner.Design/methodology/approachThe nonlinear and ill-posed problem is reformulated as a nonlinear 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. Numerical results for a benchmark test example are presented and thoroughly discussed, shedding light on the performance and effectiveness of the proposed methodology.FindingsThe inverse problem of obtaining the time-dependent blood perfusion coefficient and the temperature in the thermal-wave model of bio-heat transfer from extra boundary temperature measurement has been solved. In particular, the uniqueness of the solution to this inverse problem has been established. Furthermore, our proposed computational method demonstrated successful attainment of the perfusion coefficient and temperature, even when dealing with noisy data.Originality/valueThe originalities of the present paper are to account for such a more representative thermal-wave model of heat transfer in biological bodies and to investigate the possibility of determining its time-dependent blood perfusion coefficient from non-intrusive boundary temperature measurements.

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.

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.

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.

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.

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.

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.

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.

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with a weakly singular kernel is of concern in this paper. The scheme is targeted to reduce the computation time and to improve the accuracy of the scheme developed by Xu et al. (Appl Numer Math 152:169–184, 2020). The constructed scheme is armed by three steps: First, a small nonlinear system is solved on the coarse grid using a fix-point iteration. Second, Lagrange’s linear interpolation formula is used to arrive at some auxiliary values for the analysis of the fine grid. Finally, a linearized Crank–Nicolson finite difference system is solved on the fine grid. Moreover, the algorithm uses a central difference approximation for the spatial derivatives. In the time direction, the time derivative and integral term are approximated by the Crank–Nicolson technique and product integral rule, respectively. By means of the discrete energy method, stability and space-time second-order convergence of the proposed approach are obtained in $$L^2$$ -norm. Finally, the numerical verification is fulfilled as the numerical results of the given numerical experiments agree with the theoretical analysis and verify the effectiveness of the algorithm.

Simulation of a Hybrid Thermoelectric-Magnetocaloric Refrigerator with a Magnetocaloric Material Having a First-Order Transition

A simple hybrid thermoelectric-magnetocaloric (TE-MC) system is analytically and numerically simulated using the working parameters of commercial Peltier cells and the properties of a material with a first-order and low-hysteresis magneto-structural phase transition as La(Fe,Mn,Si)13H1.65. The need for a new master equation of the heat diffusion is introduced to deal with these materials. The equation is solved by the Crank–Nicolson finite difference method. The results are compared with those corresponding to a pure TE system and a pure MC system with ideal thermal diodes. The MC material acts as a heat “elevator” to adapt its temperature to the cold or hot source making the TE system very efficient. The efficiency of the realistic hybrid system is improved by at least 30% over the pure Peltier system for the same current supply and is similar to the pure MC with ideal diodes for the same cooling power.

A non‐overlapping domain decomposition dual reciprocity method for solving the forward–backward heat equation in two‐dimension

AbstractWe apply a boundary element dual reciprocity method (DRBEM) to the numerical solution of the forward–backward heat equation in a two‐dimensional case. The method is employed for the spatial variable via the fundamental solution of the Laplace equation and the Crank–Nicolson finite difference scheme is utilized to treat the time variable. The physical domain is divided into two non‐overlapping subdomains resulting in two standard forward and backward parabolic equations. The subproblems are then treated by the underlying method assuming a virtual boundary in the interface and starting with an initial approximate solution on this boundary followed by updating the solution by an iterative procedure. In addition, we show that the time discrete scheme is unconditionally stable and convergent using the energy method. Furthermore, some computational aspects will be suggested to efficiently deal with the formulation of the proposed method. Finally, two forward–backward problems, for which the exact solution is available, will be numerically solved for two different domains to demonstrate the efficiency of the proposed approach.

A second-order accurate Crank–Nicolson finite difference method on uniform meshes for nonlinear partial integro-differential equations with weakly singular kernels

A second-order accurate Crank–Nicolson finite difference method on uniform meshes is proposed and analyzed for a class of nonlinear partial integro-differential equations with weakly singular kernels. The first-order time derivative is approximated by using a Crank–Nicolson time-stepping technique, and the singular integral term is treated by a product averaged integration rule, which preserves the positive semi-definite property of the singular integral operator. In order to obtain a fully discrete method, the standard central finite difference approximation is used to discretize the second-order spatial derivative, and a suitable second-order discretization is adopted for the nonlinear convection term. The solvability, stability and convergence of the method are rigorously proved by the discrete energy method, the positive semi-definite property of the associated quadratic form of the method and a perturbation technique. The error estimation shows that the method has the optimal second-order convergence in time and space for non-smooth solutions. Newton’s iterative method and its algorithm implementation are presented to solve the resulting nonlinear system. Numerical results confirm the theoretical convergence result and show the effectiveness of the method.

Entropy analysis for MHD blood flow of hybrid nanoparticles (Au–Al 2 O 3 /blood) of different shapes through an irregular stenosed permeable walled artery under periodic body acceleration: Hemodynamical applications

The current study investigates the entropy generation for hybrid nanoparticles (Au–Al2O3/blood) through a vertical irregular stenosed artery in the presence of an external magnetic field, Joule heating, viscous dissipation, and heat source considering two-dimensional pulsatile blood flow and periodic body acceleration. The blood flow is assumed to be unsteady, laminar, viscous, and incompressible, and the artery walls are considered as permeable. The Reynolds temperature-dependent viscosity model is used to determine the variable viscosity effects. The governing momentum and energy equations are solved using Crank–Nicolson finite difference method by employing an appropriate coordinate transformation to build an accurate mesh using rectangular mesh units. Outcomes of the work are represented graphically for velocity, wall shear stress (WSS), volumetric flow rate, resistance impedance, temperature, heat transfer coefficient, entropy generation, and Bejan number, respectively. Also, the results are validated for velocity and temperature profiles for a given set of values of the dimensionless parameters. The entropy generation increases with rising values of shape parameter (n) up to a specific radius ( x 1 ∗ = 0.84 $x_1^*=0.84$ ) and then changes its behavior in the vicinity of the stenotic zone, in which entropy generation decreases with increasing values of shape parameter (n). The current findings may be helpful for biomedical scientists those are interested to investigate the treatment of various cardiovascular diseases (CVDs).

A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation

Abstract In this article, we present a novel numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ (BBMB) equation using Atangana Baleanu Caputo (ABC) derivative. First, we apply a linearization technique to deal with the generalized non-linear expression, and then the Crank–Nicolson finite difference formula is used in the temporal direction. A reliable numerical technique is applied to discretize the time-fractional ABC derivative, and the central difference formulae are used to approximate the derivatives in the spatial direction. The method is shown unconditionally stable and second-order convergent in both directions through the Fourier analysis. The numerical results of two test problems are analyzed to validate the theoretical results.