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A Spectral Method Based on Hahn Polynomials for Numerical Solution of Fractional Integro-Differential Equations with Weakly Singular Kernel
Farideh Salehi, Habibollah Saeedi, Mahmoud Mohseni Moghadam,
Volume 6, Issue 1 (Vol. 6, No. 1 2020)
Abstract
Introduction
Despite wide applications of constant order fractional derivatives, some systems require the use of derivatives whose order changes with respect to other parameters. Samko and Ross produced an extension of the classical fractional calculus with a continuously varying order for differential and integral operators. Variable-order fractional (V-OF) calculus has applications in optimal control, processing of geographical data, diffusion processes, description of anomalous diffusion, heat-transfer problems, etc. Due to the V-OF operators which are non-local with singular kernels, finding the exact solutions of V-OF problems is difficult. Therefore, efficient numerical techniques are necessary to be developed. The numerical solution of V-OF differential equation has been considered in some papers.
    Recently, discrete orthogonal polynomials have been considered as basis functions instead of continuous orthogonal polynomials. Discrete orthogonal polynomials are orthogonal with respect to a weighted discrete inner product. These polynomials have important applications in chemical engineering, theory of random matrices, queuing theory and image coding. In this paper, we focus on a special class of discrete polynomials, called Hahn polynomials.
    In this work, first, a new operational matrix is obtained for V-OF integral of Hahn polynomials. Then, we use a spectral collocation technique combined with the associated operational matrices of V-OF integral for solving weakly singular fractional integro-differential equations.
Material and methods
In this scheme, the operational matrix of fractional integration of Hahn polynomials is calculated. This method converts the weakly singular fractional integro-differential equations into an algebraic system which can be solved by a technique of linear algebra.
Results and discussion
In this paper, some numerical examples are provided to show the accuracy and efficiency of the presented method. By using a small number of Hahn polynomials, significant results are achieved which are compared to other methods. A comparison to the numerical solutions by CAS and Haar wavelets and Adomain decomposition method, shows that this technique is accurate enough to be known as a powerful device.
Conclusion
The following results are obtained from this research.
  • The operational matrix of fractional integration of Hahn polynomials is presented for the first time.
  • The main advantage of approximating a continuous function by Hahn polynomials is that they have a spectral accuracy at interval [0,N], where N is the number of bases.
  • Furthermore, for estimating the coefficients of the expansion of approximate solution, we only have to compute a summation which is calculated exactly.
  • Using Hahn polynomials, the numerical results achieved only by a small number of bases, are accurate in a larger interval and significant results are achieved../files/site1/files/61/7.pdf

Existence of Two Solutions for Nonlinear Difference Equations Involving p(k)-Laplacian Operator and Boundary Value Conditions
Mohsen Khaleghi Moghadam, Yasser Khalili,
Volume 6, Issue 2 (Vol. 6, No. 2 2020)
Abstract
In this paper, we deal with the existence of at least 
two solutions for an anisotropic discrete non-linear problem involving p(k)-Laplacian with
Dirichlet boundary value conditions. The technical approach is based on  a two critical
points theorem for differentiable functionals. Two examples are inserted to illustrate the
importance of main results../files/site1/files/62/6Abstract.pdf
Nonparametric Estimation of Spatial Risk for a Mean Nonstationary Random Field}
Mohammad Moghadam, Prof. Mohsen Mohammadzadeh,
Volume 8, Issue 2 (Vol. 8,No. 2, 2022)
Abstract
Introduction
Estimating the spatial hazard, or in other words, the probability of exceeding a certain boundary is one of the important issues in environmental studies that are used to control the level of pollution and prevent damage from natural disasters. Risk zoning provides useful information to decision-makers; For example, in areas where spatial hazards are high, zoning is used to design preventive policies to avoid adverse effects on the environment or harm to humans.
Generally, the common spatial risk estimating methods are for stationary random fields. In addition, a parametric form is usually considered for the distribution and variogram of the random field. Whereas in practice, sometimes these assumptions are not realistic. For an example of these methods, we can point to the Indicator kriging, Disjunctive kriging, Geostatistical Markov Chain, and simple kriging.  In practice utilize the parametric spatial models caused unreliable results. In this paper, we use a nonparametric spatial model to estimate the unconditional probability or spatial risk:
rcs0=PZs0⩾c.                 (1)
Because the conditional distribution at points close to the observations has less variability than the unconditional probability, nonparametric spatial methods will be used to estimate the unconditional probability.
 Material and methods
Let Z=Zs1,…,ZsnT be an observation vector from the random field {Zs;s∈D⊆Rd} which is decomposed as follows
Zs=μs+εs,                       (2)
where μ(s) is the trend and ε(s) is the error term, that is a second-order stationary random field with zero mean and covariogram Ch=Covεs,εs+h. The local linear model for the trend is given by
μHs= e1TSsTWsSs-1 SsTWsZ≡  ϕTsZ,
where e1 is a vector with 1 in the first entry and all other entries 0, Ss is a matrix with ith row equal to (1, (si-s)T), Ws = diag {KHs1 – s,…,KH(sn-s)}, KHu=H-1K(H-1u), K is a triple multiplicative multivariate kernel function and H is a nonsingular symmetric d×d bandwidth matrix. In this model, the bandwidth matrix obtained from a bias corrected and estimated generalized cross-validation (CGCV).
From nonparametric residuals ε(s) = Z(s) -μ(s) a local linear estimate of the variogram 2 γ(⋅)is obtained as the solution of the following least-squares problem
minα.βinεi-εj2-α-βT si-sj-u2 KGsi-sj-u,
where G is the corresponding bandwidth matrix, that obtained from minimizing cross-validation relative squared error of semi-variogram estimate.
Algorithm1: Semiparametric Bootstrap
  1. Obtain estimates of the error covariance and nonparametric residuals covariance.
  2. Generate bootstrap samples with the estimated spatial trend μHs and adding bootstrap errors generated as a spatially correlated set of errors.
  3. Compute the kriging prediction Z*s0 at each unsampled location s0 from the bootstrap sample Z*s1,…,Z*sn.
  4. Repeat steps 2 and 3 a larger number times B. Therefore, for each un-sampled location s0, B bootstrap replications Z*(1)s0.…. Z*(B)(s0) are obtained.
  5. Calculate (1) at position s_0 by calculating the relative frequency of Bootstrap repetition as follows to estimate the unconditional probability of excess of boundary c.
rcs0= 1B j=1BIZ*js0≥ c
Results and discussion
To analysis the practical behavior of the proposed methods a simulation study is conducted under different scenarios. For N=150 samples and n=16×16 were generated on a regular grid in the unit square following model (2), with mean function
μs=2.5 + sin( 2π x1) + 4x2 - 0.5 2,
and random errors normally distributed with zero mean and isotropic exponential covariogram 
Ch= 0.04 + 2.01 1- exp-3 ∥ h∥0.5,   h∈ R2.
For comparing nonparametric spatial methods for estimate unconditional risk, conditional risk, and Indicator kriging, we considered 7 missing observations in certain situations. Empirical spatial risk and its estimates are presented in Table 1. The Indicator kriging is overestimating and estimate spatial risk larger than 1. Generally, an estimated risk with unconditional and conditional methods is near value to empirical value.

Table 1. Empirical spatial risk and its estimates
Locations
(0.13, 0) (0.87, 0.87) (0.80, 0.20) (0.94, 0.27) (0, 0.47) (074, .60) (0.34, 0.60) Methods
0.999 0.300 0.069 0.317 0.504 0.011 0.989 Empirical
0.998 0.351 0.054 0.347 0.494 0.057 0.954 Conditional
1.002 0.230 0.091 0.091 0.652 0.006 0.996 Indicator
1.000 0.388 0.418 0.481 0.602 0.024 0.994 Unconditional
The spatial risk mapping for the maximum temperature means of Iran in 364 stations in March 2018 is obtained. By applying Algorithm 1 final trend and semi-variogram estimates are smoother than the pilot version. 
The conditional and unconditional spatial risk with 150 bootstrap replicates for two values of threshold 25 and 31 on a 75×75 grid are estimated. The unconditional risk estimate is smoother than the conditional risk estimate. Because of this in the unconditional version, biased residual unused directly in the spatial prediction but in the conditional risk estimating, original residuals and simple kriging used.
Conclusion
The spatial risk estimated with the nonparametric spatial method. For the trend and variability of the random field, modeling applied a local linear nonparametric model. In the simulation study, this method better results than Indicator kriging. Because the flexibility of the nonparametric spatial method could apply for the construction of confidence or prediction intervals and hypothesis testing.
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