eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-01-01
7
1
1
15
8080
Finite integration method with RBFs for solving time-fractional convection-diffusion equation with variable coefficients
Jafar Biazar
biazar@guilan.ac.ir
1
Mohammad Ali Asadi
eng.asadi@gmail.com
2
Department of Applied Mathematics, Faculty of Mathematical Science, University of Guilan, Rasht, Iran
Department of Applied Mathematics, Faculty of Mathematical Science, University of Guilan, Rasht, Iran
In this paper, a modification of finite integration method (FIM) is combined with the radial basis function (RBF) method to solve a time-fractional convection-diffusion equation with variable coefficients. The FIM transforms partial differential equations into integral equations and this creates some constants of integration. Unlike the usual FIM, the proposed method computes constants of integration by using initial condi\-tions. This leads to fewer computations rather than the standard FIM. Also, a product Simpson method is used to overcome the singularity included in the definition of fractional derivatives, and an integration matrix is obtained. Some numerical examples are provided to show the efficiency of the method. In addition, a comparison is made between the proposed method and the previous ones.
https://cmde.tabrizu.ac.ir/article_8080_c37433bd17731ac7c5b7f4f16952e0d6.pdf
Time-fractional convection-diffusion equation
Radial basis functions
Finite integration method
Product Simpson integration method
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-01-01
7
1
16
27
8246
Nordsieck representation of high order predictor-corrector Obreshkov methods and their implementation
Behnaz Talebi
b.talebi93@tabrizu.ac.ir
1
Ali Abdi
a_abdi@tabrizu.ac.ir
2
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
Predictor-corrector (PC) methods for the numerical solution of stiff ODEs can be extended to include the second derivative of the solution. In this paper, we consider second derivative PC methods with the three-step second derivative Adams-Bashforth as predictor and two-step second derivative Adams-Moulton as corrector which both methods have order six. Implementation of the proposed PC method is discussed by providing Nordsieck representation of the method and preparing an starting procedure, an estimate for local truncation error and a formula for changing stepsize. Efficiency and capability of the method are shown by some numerical experiments.
https://cmde.tabrizu.ac.ir/article_8246_ca3fd9da3259703da20ce1c8d0124535.pdf
PEC methods
Adams methods
Nordsieck representation
Local error estimation
Variable stepsize
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-01-01
7
1
28
41
8090
The modified BFGS method with new secant relation for unconstrained optimization problems
Razieh Dehghani
r.dehghani9080@yahoo.com
1
Mohammad Mehdi Hosseini
hosse_m@yazd.ac.ir
2
Narges Bidabadi
n_bidabadi@yazd.ac.ir
3
Faculty of Mathematics, Yazd University, Yazd, Iran
Faculty of Mathematics, Yazd University, Yazd, Iran
Faculty of Mathematics, Yazd University, Yazd, Iran
Using Taylor's series we propose a modified secant relation to get a more accurate approximation of the second curvature of the objective function. Then, based on this modified secant relation we present a new BFGS method for solving unconstrained optimization problems. The proposed method make use of both gradient and function values while the usual secant relation uses only gradient values. Under appropriate conditions, we show that the proposed method is globally convergent without needing convexity assumption on the objective function. Comparative results show computational efficiency of the proposed method in the sense of the Dolan-More performance profiles.
https://cmde.tabrizu.ac.ir/article_8090_f8473c5db12eeeba6e9cd9b694efaaeb.pdf
Unconstrained optimization
Modified BFGS method
Global convergence
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-01-01
7
1
42
53
8076
A new reduced mathematical model to simulate the action potential in end plate of skeletal muscle fibers
Samaneh Shahi
samanesh7@gmail.com
1
Hossein Kheiri
h-kheiri@tabrizu.ac.ir
2
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
Usually mathematicians use Hodgkin-Huxley model or FitzHug-Nagumo model to simulate action potentials of skeletal muscle fibers. These models are electrically excitable, but skeletal muscle fibers are stimulated chemically. To investigate skeletal muscle fibers we use a model with six ordinary differential equations. This dynamical system is sensitive to initial value of some variables so it is more realistic. Studying qualitative behavior and propagation of action potential through a cell with this model is time consuming .In this paper we try to use properties of variables of this model to reduced dimension of this dynamical model. We study qualitative behavior of obtained model and illustrate that this new model treats like the original model.
https://cmde.tabrizu.ac.ir/article_8076_7e1f99e57151a2613f90d573653b3907.pdf
Dynamical systems
Skeletal muscle
Qualitative behavior
Action potential
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-01-01
7
1
54
68
8082
Symmetry group, Hamiltonian equations and conservation laws of general three-dimensional anisotropic non-linear sourceless heat transfer equation
Seyed Reza Hejazi
ra.hejazi@gmail.com
1
Elaheh Saberi
saberi.elaheh@gmail.com
2
Elham Lashkarian
lashkarianelham@yahoo.com
3
Department of mathematical sciences, Shahrood university of technology, Shahrood, Semnan, Iran
Department of mathematical sciences, Shahrood university of technology, Shahrood, Semnan, Iran
Department of mathematical sciences, Shahrood university of technology, Shahrood, Semnan, Iran
In this paper Lie point symmetries, Hamiltonian equations and conservation laws of general three-dimensional anisotropic non-linear sourceless heat transfer equation are investigated. First of all Lie symmetries are obtained by using the general method based on invariance condition of a system of differential equations under a prolonged vector field. Then the structure of symmetry operators as a Lie algebra are clarified and the classification of subalgebras under adjoint transformation is given. Hamiltonian equations including Hamiltonian symmetry are obtained. Finally a modified virsion of Noether’s method including the direct method are applied in order to find local conservation laws of the equation.
https://cmde.tabrizu.ac.ir/article_8082_ce59954b9985ed275b6bcae07e350a29.pdf
Heat transfer equation
Lie symmetry
Partial differential equation
Hamiltonian equations
Conservation laws
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-01-01
7
1
69
85
8078
An efficient improvement of the Newton method for solving nonconvex optimization problems
Tayebeh Dehghan Niri
ta.dehghan18175@gmail.com
1
Mohammad Mehdi Hosseini
hosse_m@yazd.ac.ir
2
Mohammad Heydari
m.heydari@yazd.ac.ir
3
Department of Mathematics, Yazd University, P. O. Box 89195-74, Yazd, Iran
Department of Mathematics, Yazd University, P. O. Box 89195-74, Yazd, Iran
Department of Mathematics, Yazd University, P. O. Box 89195-74, Yazd, Iran
Newton method is one of the most famous numerical methods among the line search methods to minimize functions. It is well known that the search direction and step length play important roles in this class of methods to solve optimization problems. In this investigation, a new modification of the Newton method to solve unconstrained optimization problems is presented. The significant merit of the proposed method is that the step length $\alpha_k$ at each iteration is equal to 1. Additionally, the convergence analysis for this iterative algorithm is established under suitable conditions. Some illustrative examples are provided to show the validity and applicability of the presented method and a comparison is made with several other existing methods.
https://cmde.tabrizu.ac.ir/article_8078_186d3a4203be520076ab8a6d2c5b80df.pdf
Unconstrained optimization
Newton method
Line search methods
Convergence analysis
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-01-01
7
1
86
95
8233
Multi-soliton of the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff equation and KdV equation
Harun-Or- Roshid
harunorroshidmd@gmail.com
1
Department of Mathematics, Pabna University of Science and Technology, Bangladesh
A direct rational exponential scheme is offered to construct exact multi-soliton solutions of nonlinear partial differential equation. We have considered the Calogero–Bogoyavlenskii–Schiff equation and KdV equation as two concrete examples to show efficiency of the method. As a result, one wave, two wave and three wave soliton solutions are obtained. Corresponding potential energy of the soliton solutions are also found. Furthermore, three-dimensional plots of the wave solutions and its potential functions are given to visualize the dynamics of the model and their energy. We also provided the corresponding density plot of the solutions to understand the real direction and particles density in the waves which help to realize the elastic situations of the achieved solutions.
https://cmde.tabrizu.ac.ir/article_8233_2f1e949b8477c91b50ac8bbcb9d780b0.pdf
Direct rational exponential scheme
Calogero–Bogoyavlenskii–Schiff equation
KdV equation
Multi-soliton solutions
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-01-01
7
1
96
104
8308
On the solving of matrix equation of Sylvester type
Fikret Aliev
f_aliev@yahoo.com
1
Vladimir Larin
vblarin@gmail.com
2
Institute of Applied Mathematics, Baku State University, Baku, Azerbaijan
Institute of Mechanics of the Academy of Sciences of Ukraine, Ukraine, Kiev
A solution of two problems related to the matrix equation of Sylvester type is given. In the first problem, the procedures for linear matrix inequalities are used to construct the solution of this equation. In the second problem, when a matrix is given which is not a solution of this equation, it is required to find such solution of the original equation, which most accurately approximates the given matrix. For this, an algorithm for constructing a general solution of the Sylvester matrix equation is used. The effectiveness of the proposed approaches is illustrated on the examples.
https://cmde.tabrizu.ac.ir/article_8308_8490b775c091e584aa7755c033bcd854.pdf
Matrix equation
Linear matrix inequalities (LMI)
Matrix Sylvester equation
Sylvester type matrix equation
Complex matrices
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-01-01
7
1
105
116
8250
$L^p$-existence of mild solutions of fractional differential equations in Banach space
Adel Jawahdou
adeljaw2002@yahoo.com
1
Carthage University, Department of Mathematics, Bizerte Preparatory Engineering Institute, Tunisia
We study the existence of mild solutions for semilinear fractional differential equations with nonlocal initial conditions in $L^p([0,1],E)$, where $E$ is a separable Banach space. The main ingredients used in the proof of our results are measure of noncompactness, Darbo and Schauder fixed point theorems. Finally, an application is proved to illustrate the results of this work.
https://cmde.tabrizu.ac.ir/article_8250_4a4a2db78f9752e2efa6089c2e99af92.pdf
Semilinear integro-differential fractional equations
Measure of noncompactness
Mild solutions
Schauder fixed point theorem
Darbo fixed point theorem
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-01-01
7
1
117
123
8178
Properties of utility function for Barles and Soner model
Mojtaba Ranjbar
ranjbar633@gmail.com
1
Somayeh Pourghanbar
s.pourghanbar@azaruniv.edu
2
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
The nonlinear Black-Scholes equation has been increasingly attracting interest over the last two decades, because it provides more accurate values by considering transaction costs as a viable assumption. In this paper we review the fully nonlinear Black-Scholes equation with an adjusted volatility which is a function of the second derivative of the price and then we prove two new theorems in this realistic model.
https://cmde.tabrizu.ac.ir/article_8178_eb961039a8ddf83ca42eeee832a76ae9.pdf
Nonlinear Black-Scholes equation
Transaction costs
Utility function
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-01-01
7
1
124
137
8319
The numerical values of the nodal points for the Sturm-Liouville equation with one turning point
Abdol Ali Neamaty
a.n.hosseinabady@gmail.com
1
Najibeh Yousefi
nah.yosofy@gmail.com
2
Abdol Hadi Dabbaghian
a.dabbaghian@umz.ac.ir
3
Department of Mathematics, University of Mazandaran, Babolsar, Iran
Department of Mathematics, University of Mazandaran, Babolsar, Iran
Department of Mathematics, Neka Branch, Islamic Azad University, Neka, Iran
An inverse nodal problem has first been studied for the Sturm-Liouville equation with one turning point. The asymptotic representation of the corresponding eigenfunctions of the eigenvalues has been investigated and an asymptotic of the nodal points is obtained. For this problem, we give a reconstruction formula for the potential function. Furthermore, numerical examples have been established and results have been illustrated in tables and graphics.
https://cmde.tabrizu.ac.ir/article_8319_5dd732d6ed7c5af8ff5e5945ed5f7fdd.pdf
Turning point
Inverse nodal problem
Nodal Points
eigenvalues
Eigenfunctions
eng
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
2019-01-01
7
1
138
151
8177
Stable Gaussian radial basis function method for solving Helmholtz equations
Jalil Rashidinia
rashidinia@iust.ac.ir
1
Manoochehr Khasi
m_khasi@iust.ac.ir
2
School of Mathematics, Iran University of Science and Technology, Tehran, Iran
School of Mathematics, Iran University of Science and Technology, Tehran, Iran
Radial basis functions (RBFs) are a powerful tool for approximating the solution of high-dimensional problems. They are often referred to as a meshfree method and can be spectrally accurate. In this paper, we analyze a new stable method for evaluating Gaussian radial basis function interpolants based on the eigenfunction expansion. We develop our approach in two-dimensional spaces for solving Helmholtz equations. In this paper, the eigenfunction expansions are rebuilt based on Chebyshev polynomials which are more suitable in numerical computations. Numerical examples are presented to demonstrate the effectiveness and robustness of the proposed method for solving two-dimensional Helmholtz equations.
https://cmde.tabrizu.ac.ir/article_8177_70b29d014600a6b7b05102f1660f70ee.pdf
Gaussian radial basis functions
Eigenfunction expansion
Helmholtz equations
Sylvester system