NUMERICAL SIMULATION OF FRACTIONAL BIOHEAT EQUATION IN HYPERTHERMIA TREATMENT

2014 ◽  
Vol 14 (02) ◽  
pp. 1450018 ◽  
Author(s):  
R. S. DAMOR ◽  
SUSHIL KUMAR ◽  
A. K. SHUKLA
Keyword(s):  
Numerical Simulation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Fractional Derivative ◽  
Thermal Damage ◽  
Heating Time ◽  
Bioheat Equation ◽  
Hyperthermia Treatment ◽  
Implicit Finite Difference ◽  
Time Fractional Derivative

This paper deals with the study of fractional bioheat equation for hyperthermia treatment in cancer therapy with external electromagnetic (EM) heating. Time fractional derivative is considered as Caputo fractional derivative of order α ∈ (0, 1]. Numerical solution is obtained by implicit finite difference method. The effect of anomalous diffusion in tissue has been studied. The temperature profile and thermal damage over the entire affected region are obtained for different values of α.

scholarly journals A Fast Implicit Finite Difference Method for Fractional Advection-Dispersion Equations with Fractional Derivative Boundary Conditions

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Taohua Liu ◽  
Muzhou Hou
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Difference Method ◽  
Computational Cost ◽  
First Order ◽  
Dispersion Equations ◽  
Advection Dispersion ◽  
Implicit Finite Difference

Fractional advection-dispersion equations, as generalizations of classical integer-order advection-dispersion equations, are used to model the transport of passive tracers carried by fluid flow in a porous medium. In this paper, we develop an implicit finite difference method for fractional advection-dispersion equations with fractional derivative boundary conditions. First-order consistency, solvability, unconditional stability, and first-order convergence of the method are proven. Then, we present a fast iterative method for the implicit finite difference scheme, which only requires storage of O(K) and computational cost of O(Klog⁡K). Traditionally, the Gaussian elimination method requires storage of O(K2) and computational cost of O(K3). Finally, the accuracy and efficiency of the method are checked with a numerical example.

scholarly journals Simulasi Numerik Model Matematika Arus Lalu Lintas Berbasis Fungsi Velositas Underwood

2021 ◽  
Vol 4 (1) ◽  
pp. 9
Author(s):  
Muh. Isbar Pratama ◽  
Dian Firmayasari ◽  
Nur Ahniyanti Rasyid ◽  
H. Harianto
Keyword(s):  
Numerical Simulation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Equivalence Theorem ◽  
Experimental Result ◽  
Velocity Function ◽  
Lwr Model ◽  
Implicit Finite Difference ◽  
Implicit Finite Difference Method

Abstrak.Model matematika arus lalu lintas pertama kali dikembangkan oleh Lighthill, Whitham dan Richards pada tahun 1956 yang dikenal dengan model (LWR). Dalam model LWR, fungsi kecepatan adalah unsur yang terpenting. Dalam makalah ini digunakan fungsi kecepatan underwood karena memiliki tingkat kesesuaian yang terbaik dibadingkan dengan fungsi kecepatan lainnya. Metode beda hingga implisit digunakan untuk menemukan solusi numerik model LWR dengan model kecepatan Underwood. Konvergensi metode beda hingga implisit dibuktikan dengan menggunakan teorema Ekuivalensi Lax. Simulasi numerik jalan raya satu lajur sepanjang 10 km dilakukan selama 1 jam menggunakan metode beda hingga implisit berdasarkan data awal dan batas yang dibuat secara artifisial. Simulasi numerik dilakukan dengan dua parameter berbeda. Hasil eksperimen menujukkan bahwa semakin tinggi rata-rata kepadatan kendaraan pada suatu laju mengakibatkan rata-rata kecepatan kendaraan akan berkurang. Kata kunci: Metode Beda Hingga Implisit, Model LWR, Arus Lalu Lintas, Fungsi Felositas Underwood, Simulasi Numerik.Kata kunci : Abstract. Mathematical traffic flow model was first developed by Lighthill, Whitham and Richards in 1956, known as (LWR) model. In LWR model, velocity function was most important. In this paper, Underwood velocity function was used. Implicit finite difference method used to found the numerical solution of LWR model with Underwood velocity model. Convergence the implicit finite difference method proved using the Lax equivalence theorem. The numerical simulation of 10 km highway of single lane was performed for 1 hours using the implicit finite difference method based on artificially generated initial and boundary data. Numerical simulation performed with two different parameters. An experimental result for the stability condition of the numerical scheme was also presented. Density, velocity, and fluks for 1 hours was experimental result of numerical simulation.Keywords: Implicit finite difference method, Lax equivalence theorem, LWR model, Traffic flow, Under-wood velocity Function, Numerical simulation.

scholarly journals A Banded Preconditioning Iteration Method for Time-Space Fractional Advection-Diffusion Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Min-Li Zeng ◽  
Guo-Feng Zhang
Keyword(s):  
Finite Difference ◽  
Fractional Derivative ◽  
Linear Systems ◽  
Iteration Method ◽  
Diffusion Equations ◽  
Finite Difference Schemes ◽  
Advection Diffusion ◽  
Time Space ◽  
Implicit Finite Difference ◽  
Time Fractional Derivative

In this paper, we concentrate on the efficient solvers for the time-space fractional advection-diffusion equations. Firstly, the implicit finite difference schemes with the shifted Grünwald-Letnikov approximations for spatial fractional derivative and unshifted Grünwald-Letnikov approximations for time fractional derivative are employed to discretize time-space fractional advection-diffusion equations. The discretization results in a series of large dense linear systems. Then, a banded preconditioner is proposed and some theoretical properties for the preconditioning matrix are studied. Numerical implementations show that the banded preconditioner may lead to satisfactory experimental results when we choose appropriate bandwidth in the preconditioner.

Diffraction Waves About an Advancing Wedge Model in Deep Water

1990 ◽  
Vol 34 (02) ◽  
pp. 105-122
Author(s):  
Hideaki Miyata ◽  
Makoto Kanai ◽  
Noriaki Yoshiyasu ◽  
Yohichi Furuno
Keyword(s):  
Numerical Simulation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Wedge Angle ◽  
Wave Pattern ◽  
Regular Waves ◽  
Nonlinear Characteristics ◽  
Diffracted Waves ◽  
Nonlinear Features ◽  
Incident Waves

The diffraction of regular waves by advancing wedge models is studied both experimentally and numerically. The nonlinear features of diffracted waves are visualized by wave pattern pictures and the formation is analyzed by the grid-projection method. The experimental observation indicates that the diffracted waves have a number of nonlinear characteristics similar to shock waves due to the interaction of incident waves with the advancing obstacle in the flow-field caused by the advancing motion. Bow waves of both oblique type and normal detached type are observed at remarkably lower Froude numbers than in the case of a ship in steady advance motion. Their occurrence systematically depends on the Froude number and the wedge angle. The numerical simulation of this phenomenon by a finite-difference method shows approximate agreement with the experimental results.

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