Numerical Solutions of Linear Fractional Order BVP by Galerkin Residual Method with Differentiable Polynomials
Umme Ruman,
Md. Shafiqul Islam
Issue:
Volume 9, Issue 2, April 2020
Pages:
20-25
Received:
9 April 2020
Accepted:
3 May 2020
Published:
14 May 2020
Abstract: The main objective of this paper is to solve numerically the differential equations of fractional order with homogeneous boundary conditions by the Galerkin weighted residual method. In this method, linear combinations of some types of functions are used to find the approximate solutions which must satisfy the homogeneous boundary conditions. Such type of functions should be differentiated and integrated easily, so the piecewise polynomials, namely, Bernstein, Bernoulli and Modified Legendre polynomials are used as basis functions in this paper. The fractional derivatives are used in the conjecture of Riemann-Liouville and Caputo sense. Thus, we develop the Galerkin weighted residual formulation, in matrix form, to the linear fractional order boundary value problems, in details, which is easy to understand. The accuracy and applicability of the present method are demonstrated through few numerical examples. We observe that the approximate results converge monotonically to the exact solutions. In addition, we compare the approximate results with the exact solutions, and also with the existing solutions which are available in the literature. The absolute errors are depicted in tabular form as well as graphical representations, a reliable accuracy is achieved. The proposed method may be applied to fractional order partial differential equations also.
Abstract: The main objective of this paper is to solve numerically the differential equations of fractional order with homogeneous boundary conditions by the Galerkin weighted residual method. In this method, linear combinations of some types of functions are used to find the approximate solutions which must satisfy the homogeneous boundary conditions. Such ...
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Meandering Fractals in Water Resources Management
Issue:
Volume 9, Issue 2, April 2020
Pages:
26-29
Received:
29 April 2019
Accepted:
21 May 2019
Published:
19 May 2020
Abstract: Fractal dimension is a measure for the degree of complexity or that of fractals. An alternative to fractal dimension is ht-index, which quantifies complexity in a unique way. Back to your question, the physical meaning of fractal dimension is that many natural and social phenomena are nonlinear rather than linear, and are fractal rather than Euclidean. We need a new paradigm for studying our surrounding phenomena, Not Newtonian physics for simple systems, but complexity theory for complex systems, Not linear mathematics such as calculus, Gaussian statistics, and Euclidean geometry, but online mathematics including fractal geometry, chaos theory, and complexity science in general. A channel is characterized by its width, depth, and slope. The regime theory relates these characteristics to the water and sediment discharge transported bye the channel empirically. Empirical measurements are taken on channels and attempts are made to fit empirical equations to the observed data. The channel characteristics are related primarily to the discharge but allowance is also made for variations in other variables, such as sediment size.
Abstract: Fractal dimension is a measure for the degree of complexity or that of fractals. An alternative to fractal dimension is ht-index, which quantifies complexity in a unique way. Back to your question, the physical meaning of fractal dimension is that many natural and social phenomena are nonlinear rather than linear, and are fractal rather than Euclid...
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