European Journal of Mathematical Analysis 

Abstract: In this paper, we will represent relation of graph which bring different type of topological structure to the graph [2], then, consider certain properties of the graph. We will discuss mainly blood circulation in lungs and some different diseases of it [4] and relate them with graph and make topologies [8]. Moreover, certain applications in medical field will be represent. We can also use results in real life [11].

Abstract: A geometric based approach for specifying approximations to the Lambert W function, which can achieve any set relative error bound over the interval [0, ∞), is detailed. Approximations that can achieve arbitrarily high accuracy for the interval [-1/e, 0], based on a two point spline approximation, are specified. Iterative methods can be used to improve the accuracy of the approximations.Applications include, first, analytical expressions, with set relative error bounds, for the Lambert W function over the interval [0, ∞). Second, approximations, with an arbitrarily low relative error, for upper and lower bounds for the Lambert W function. Third, analytical expressions for the evaluation of and the integral of ⌊W(y)⌋, for y∈[0, ∞), without knowledge of W(y). Fourth, a direct approach for evaluating the Lambert W function to achieve a prior set error constraint.

Abstract: In this study, we explore the possibility that the Drought Monitor database belongs to class of fractal process which can be characterized using a single scaling exponent. The Drought Monitor map identifies areas of drought and labels them by intensity: D0 abnormally dry, D1 moderate drought, D2 severe drought, D3 extreme drought, and D4 exceptional drought. The vibration analysis using power spectral densities (PSD) method has been carried out to discover whether some type of power-law scaling exists for various statistical moments at different scales of this database. We perform multi-fractal analysis to estimate the multi-fractal spectrum of each group. We apply Higuchi algorithm to find the fractal complexity of each group and then compare them for different time intervals. Our findings reveal that we have a wide range of exponents for D0-D4. Therefore, D0-D4 belong to class of multi-fractal process for which a large number of scaling exponents are required to characterize the scaling structure.

Abstract: Fractional calculus is a new approach for modeling biological and physical phenomena with memory effects. Fractional calculus uses differential and integral operators including non-integer orders to study the non-linear behavior of physical and biological systems with some degrees of fractionality or fractality. Since the long memory properties of neuronal responses can be better explained using fractional derivative, in this study we generalize the integer-order Morris-Lecar model in the fractional-order domain to better modeling of neuron dynamics. To investigate the complex spiking patterns of fractional-order Morris-Lecar neural system the fractional calculus has been applied to build this new mathematical model. We compare the results with integer-order Morris-Lecar model. The analytical solutions of these equations cannot explicitly be obtained. Therefore, to find the dynamical behaviors of solutions, we used approximation and numerical schemes. Depending on the different parameters values for 0<η≤1, the fractional-order Morris-Lecar reproduces quiescent, spiking and bursting activities the same as its original model but for higher input current. We numerically discover the hopf bifurcation, saddle node bifurcation of limit cycle and homoclinic bifurcation for this model for different input current and derivative orders. Taking the advantages of the fractional order derivative, for a variety of orders, we define different classes of this model which helps to better extract all the complicated dynamics of this single neuron model.