Journal of Mathematical Modeling
Volume:10 Issue: 1, Winter 2022

The price variance of the associated fractal transmission mechanism was used to estimate the Black-Scholes fractional model of which a time-fractional derivative is $alpha$. In the current paper, the time-fractional Black-Scholes equation (TFBSE) that the temporal derivative is the Caputo fractional derivative is known by regulating the European option. At first, linear interpolation with a temporally $tau^{2-alpha}$ order accuracy is used for constructing the semi-discrete. Then, the spatial derivative terms are approximated with the help of the collocation approach centered on the  Chebyshev polynomials of the third form (CPTF). Finally, The unconditional stability and convergence order are analyzed by applying the energy method. To show the precision of the numerical  system, we solved two instances of the TFBSE. Numerical results and comparisons indicate the proposed approach is very reliable and efficient.

Recently, based on a singular value analysis on the Dai--Liao conjugate gradient method, Babaie-Kafaki and Aminifard  suggested a fixed point equation. The prominent feature of  the proposed equation is that its solutions may increase numerical stability of the method  while improving the global convergence. Here, the  same fixed point equation  is employed to upgrade previously proposed choices of the Dai--Liao parameter based on the well-known functional iteration method.  Global convergence analysis is conducted and numerical experiments are done to support our discussions.

We propose new tensor Krylov subspace methods  for ill-posed linear tensor problems such as color or video image restoration. Those methods are based on the tensor-tensor discrete cosine transform that gives fast tensor-tensor product computations. In particular, we will focus on the tensor discrete cosine versions of GMRES, Golub-Kahan bidiagonalisation and LSQR methods. The presented numerical tests show that the methods are very fast and give good accuracies when solving some linear tensor ill-posed problems.

We provide a general finite iterative approach for constructing factorizations of a matrix $A$   under a common framework of a general decomposition $A=BC$  based on the generalized Schur complement. The approach  applies  a zeroing process using  two index sets. Different choices of the index sets  lead to different real and integer matrix  factorizations.    We also provide the  conditions under which this approach is well-defined.

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Micromagnetics is a continuum theory describing magnetization patterns inside ferromagnetic media. The dynamics of a ferromagnetic material are governed by the Landau-Lifshitz equation. This equation is highly nonlinear and  has a non-convex constraint. In this work, a finite element approximation of a current-induced magnetization dynamics model is proposed. The model consists of a modified Landau-Lifshitz-Gilbert (LLG) equation incorporating spin transfer torque. The scheme preserves a non-convex constraint, requires only a linear solver at each time step and is easily applicable to the limiting cases.  As the time and space steps tend to zero, a proof of convergence of the numerical solution to a (weak) solution of the modified LLG equation is given. Numerical results are presented to show the effect of the injected current on magnetization switching.

The Hamilton-Jacobi-Bellman (HJB) equation, as a notable approach obtained from dynamic programming, is widely used in solving optimal control problems that results in a feedback control law. In this study, the HJB equation is first transformed into the Convection-Diffusion (CD) equation by adding a viscosity coefficient. Then, a novel numerical method is presented to solve the corresponding CD equation and to obtain a viscosity solution of the HJB. The proposed approach encompasses two well-known methods of Finite Volume Method (FVM) and Algebraic Multigrid (AMG). The former as a reliable method for solving parabolic PDEs and the latter as a powerful tool for acceleration. Finally, numerical examples illustrate the practical performance of the proposed approach.

In this paper, the exact expressions as well as recurrence relations for single and product moments of the generalized lower record values from generalized inverse Weibull distribution are obtained. Further, the characterization of the given distribution is carried out through recurrence relations and conditional moment.

Electrocardiogram (ECG) signals is widely used as one of the common procedures for heart's disease diagnose. Since electrical signals generated by biological sources have low level, they are destroyed by interference. Therefore, it is difficult to achieve high resolution electrical signals. A new approach based on non-polynomial cubic spline has been developed to approximate the ECG signal. The Efficiency of proposed method is analyzed by simulation results and filter evaluation metrics.

In this paper, a special kind of Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations is introduced. Since it is not possible to solve DGLAP integral equations analytically, the numerical solutions of these equations can be of interest. Here, the Tau spectral method is used for solving this integral equation and offer an approximate solution. Finally, this solution is compared  with solution obtained experimentally  for $Q_0^2=0.35 GeV^2$.

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Fuzzy delay differential equation driven by Liu's process is a type of functional differential equations. In this paper, we are going to provide and prove a novel existence and uniqueness theorem for the solutions of fuzzy delay differential equation under Local Lipschitz and linear growth conditions. Also the stability of the solutions for fuzzy delay differential is investigated. Finally, to illustrate the main results we give some examples.

It is well-known that the symplectic Lanczos method is an efficient tool for computing a few eigenvalues of large and sparse Hamiltonian matrices. A variety of block Krylov subspace methods were introduced by Lopez and Simoncini to compute an approximation of $exp(M)V$ for a given large square Hamiltonian matrix $M$ and a tall and skinny matrix $V$ that preserves the geometric property of $V$. For the same purpose, in this paper, we have proposed a new method based on a global version of the symplectic Lanczos algorithm, called the global $J$-Lanczos method ($GJ$-Lanczos). To the best of our knowledge, this is probably the first adaptation of the symplectic Lanczos method in the global case. Numerical examples are given to illustrate the effectiveness of the proposed approach.

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In this paper, we consider the 2-generator $p$-groups of nilpotency class 2.  We  find the Fibonacci length and the period of the generalized order $k$--Pell sequences of these groups.

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