Journal of Mathematical Modeling
Volume:7 Issue: 1, Winter 2019

In the present paper we consider initial value problems for Hilfer fractional differential equations and for system of Hilfer fractional differential equations. By using equivalent integral equations and some fixed point theorems, we study the local existence of solutions. We extend these local existence results globally with the help of continuation theorems and generalized Gronwall inequality.

Since it is well-known that the Vandermonde matrix is ill-conditioned, this paper surveys the choices of other bases. These bases are data-dependent and are categorized into discretely $ell^2$-orthonormal  and continuously $L^2$-orthonormal bases. The first one is defined via a decomposition of the Vandermonde matrix while the latter is given by a decomposition of the Gramian matrix corresponding to monomial bases. A discussion of various matrix decomposition (e.g. Cholesky, QR and SVD) provides a variety of different bases with different properties. Special attention is given to duality. Numerical results show that the matrices of values of the new bases have smaller condition numbers than the common monomial bases. It can also be pointed out that the new introduced bases are good candidates for interpolation.

In this paper, a new two-parameter lifetime distribution called ``the exponentiated Shanker distribution" is suggested. The new distribution has an increasing, decreasing and bathtub-shaped hazard rate function (hrf) for modeling lifetime data. Various mathematical and statistical properties of the proposed distribution including its hrf, complete and incomplete moments, skewness and kurtosis, mean deviations, Bonferroni and Lorenz curves are discussed. Estimation of its parameters is also discussed using the method of maximum likelihood estimation and a simulation study is given. Finally, two applications of the new distribution are presented using two real data sets. The results also confirmed the suitability of the proposed model for the real data sets.

We propose and analyze a mathematical model to study the dynamics of smoking behavior under the influence of educational and media programs. Proposed mathematical model subdivides the total population into potential smokers, smokers and those smokers who quit smoking permanently. The biologically feasible equilibrium points are computed and their stability is analyzed and discussed. The theoretical analysis of the model reveals that the smoking-free equilibrium is stable when a threshold, termed as the smokers-generation number, is less than unity, and unstable if this threshold value is greater than unity. Moreover, number of smokers may be effectively controlled by keeping the smokers generation number less than unity. Analytical findings are justified by numerical simulation.

The aim of this research is to apply the stochastic arithmetic (SA) for validating the Sinc-collocation method (S-CM) with single or double exponentially decay to find the numerical solution of second kind Fredholm integral equation (IE). To this end, the CESTAC(Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are applied. Using this method, the optimal iteration of S-CM, the optimal approximation, the absolute error and the numerical instabilities can be determined. A theorem is proved which shows the accuracy of the S-CM by means of the concept of common significant digits. Some IEs are presented and the numerical results of comparison between the single exponentially decay (SE) and the double exponentially decay (DE) are demonstrated in the tables.

In this paper, we consider a time-fractional inverse heat conduction problem with an unknown function in the nonlinear boundary condition. First, ill-posedness of this problem is shown. Thus, we will apply the mollification regularization method with Gauss kernel to regularize the problem, then the space marching finite difference method is considered to solve numerically the mollified problem. The generalized cross-validation choice rule is used to find a suitable regularization parameter. The numerical scheme is completely described and the stability and convergence of the solutions are investigated. Finally, some numerical examples are presented to illustrate the validity and effectiveness of the proposed algorithm.

In this article, using the properties of the rationalized Haar (RH) wavelets and the matrix operator, a method is presented for calculating the numerical approximation of the first  Painlev'e equations solution. Also, an upper bound of the error is given and by applying the Banach fixed point theorem  the convergence analysis of the method is stated. Furthermore, an algorithm to solve the first Painlev'e equation is proposed. Finally, the reported results are compared with some other methods to show the effectiveness of the proposed approach.

In this paper, an economic group delay model is established. Dynamical behavior and Basic influence number of the proposed system are studied. Asymptotic stability analysis is carried out for the steady-states. The critical value of the delay $tau$ is determined. It is observed that for the interior steady-state remains stable if the adoption delay for the low-income group is less than the threshold value, i.e., $tau<tau_{0}^+$. If $tau$ crosses its threshold, system perceives oscillating behavior, and Hopf bifurcation occurs. Moreover, sensitivity analysis is performed for the system parameter used in the interior steady-state. Finally, numerical simulations are conducted to support our analytical findings.

In this paper, we study the existence, uniqueness and stability of solutions of a nonlocal Cauchy problem for nonlinear fractional integro-differential equations with positive constant coefficient. The results heavily depend on the Banach contraction principle, Schaefer's fixed point theorem and Pachpatte's integral inequality. In the last, results are illustrated with suitable example.