in this manuscript a new method is introduced for solving fractional differential equations. the fractional derivative is described in the caputo sense. the main idea is to use fractional-order legendre wavelets and operational matrix of fractional-order integration. first the fractional-order legendre wavelets (flws) are presented. then a family of piecewise functions is proposed, based on whi...

introduction one of the critical issues associated with sustainability in the twenty-first century is how cities are growing and developing in the space. rapid urban growth and physical development of cities cause disintegration of the spatial organization of cities and consequently spatial inequality between urban spaces. this phenomenon can be considered as spatial segregation or fragmentatio...

The classical term by term integration technique used for obtaining asymptotic expansions of integrals requires the integrand to have an uniform asymptotic expansion in the integration variable. A modification of this method is presented in which the uniformity requirement is substituted by a much weaker condition. As we show in some examples, the relaxation of the uniformity condition provides...

Ever since the thermal generation units, once a foundation of large number power systems, were marked as one main emitters greenhouse gases and, therefore, undesirable in modern renewable energy sources are taking over their share mix. Although this process is going fast, it might still not be efficient enough to reach climate goals, set with short deadlines. In accordance that, every measure t...

In this paper, we propose a method to approximate the solution of a linear Fredholm integro-differential equation by using the Chebyshev wavelet of the first kind as basis. For this purpose, we introduce the first Chebyshev operational matrix of integration. Chebyshev wavelet approximating method is then utilized to reduce the integro-differential equation to a system of algebraic equations. Il...