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1 – 10 of over 23000Zoeljana Nikolic´, Ante Mihanovic´ and Pavao Marovic´
Presents a procedure for obtaining an improved finite element solution of boundary problems by estimating the principle of exact displacement method in the finite element…
Abstract
Presents a procedure for obtaining an improved finite element solution of boundary problems by estimating the principle of exact displacement method in the finite element technique. The displacement field is approximated by two types of functions: the shape functions satisfying the homogeneous differential equilibrium equation and the full clamping element functions as a particular solution of the differential equation between the nodes. The full clamping functions represent the solution of the full clamping state on finite elements. An improved numerical solution of displacements, strains, stresses and internal forces, not only at nodes but over the whole finite element, is obtained without an increase of the global basis, because the shape functions are orthogonal with the full clamping functions. This principle is generally applicable to different finite elements. The contribution of introducing two types of functions based on the principle of the exact displacement method is demonstrated in the solution procedure of frame structures and thin plates.
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Kang-Jia Wang, Guo-Dong Wang and Feng Shi
The fractal and fractional calculus have obtained considerable attention in the electrical and electronic engineering since they can model many complex phenomena that the…
Abstract
Purpose
The fractal and fractional calculus have obtained considerable attention in the electrical and electronic engineering since they can model many complex phenomena that the traditional integer-order calculus cannot. The purpose of this paper is to develop a new fractional pulse narrowing nonlinear transmission lines model within the local fractional calculus for the first time and derive a novel method, namely, the direct mapping method, to seek for the nondifferentiable (ND) exact solutions.
Design/methodology/approach
By defining some special functions via the Mittag–Leffler function on the Cantor sets, a novel approach, namely, the direct mapping method is derived via constructing a group of the nonlinear local fractional ordinary differential equations. With the aid of the direct mapping method, four groups of the ND exact solutions are obtained in just one step. The dynamic behaviors of the ND exact solutions on the Cantor sets are also described through the 3D graphical illustration.
Findings
It is found that the proposed method is simple but effective and can construct four sets of the ND exact solutions in just one step. In addition, one of the ND exact solutions becomes the exact solution of the classic pulse narrowing nonlinear transmission lines model for the special case 9 = 1, which strongly proves the correctness and effectiveness of the method. The ideas in the paper can be used to study the other fractal partial differential equations (PDEs) within the local fractional derivative (LFD) arising in electrical and electronic engineering.
Originality/value
The fractional pulse narrowing nonlinear transmission lines model within the LFD is proposed for the first time in this paper. The proposed method in the work can be used to study the other fractal PDEs arising in electrical and electronic engineering. The findings in this work are expected to shed a light on the study of the fractal PDEs arising in electrical and electronic engineering.
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María José Pujol, Francisco A. Pujol, Fidel Aznar, Mar Pujol and Ramón Rizo
In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that…
Abstract
Purpose
In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that appears in the study of the thermal behaviour of a spherical cloud and of the gravitational potential of a polytropic fluid at hydrostatic equilibrium.
Design/methodology/approach
In their work, the authors first review Emden's equation and its possible solutions using the Frobenius and power series methods; then, Adomian polynomials are introduced. Afterwards, Emden's equation is solved using Adomian's decomposition method and, finally, they conclude with a comparison of the solution given by Adomian's method with the solution obtained by the other methods, for certain cases where the exact solution is known.
Findings
Solving Emden's equation for n in the interval [0, 5] is very interesting for several scientific applications, such as astronomy. However, the exact solution is known only for n=0, n=1 and n=5. The experiments show that Adomian's method achieves an approximate solution which overlaps with the exact solution when n=0, and that coincides with the Taylor expansion of the exact solutions for n=1 and n=5. As a result, the authors obtained quite satisfactory results from their proposal.
Originality/value
The main classical methods for obtaining approximate solutions of Emden's equation have serious computational drawbacks. The authors make a new, efficient numerical implementation for solving this equation, constructing iteratively the Adomian polynomials, which leads to a solution of Emden's equation that extends the range of variation of parameter n compared to the solutions given by both the Frobenius and the power series methods.
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P. Léger and E.L. Wilson
The evaluation of linear dynamic response analysis of large structures by vector superposition requires, in its traditional formulation, the solution of a large and expensive…
Abstract
The evaluation of linear dynamic response analysis of large structures by vector superposition requires, in its traditional formulation, the solution of a large and expensive eigenvalue problem. A method of solution based on a Ritz transformation to a reduced system of generalized coordinates using load dependent vectors generated from the spatial distribution of the dynamic loads is shown to maintain the high expected accuracy of modern computer analysis and significantly reduces the execution time over eigensolution procedures. New computational variants to generate load dependent vectors are presented and error norms are developed to control the convergence characteristics of load dependent Ritz solutions. Numerical applications on simple structural systems are used to show the relative efficiency of the proposed solution procedures.
Sunil Kumar, Surath Ghosh, Shaher Momani and S. Hadid
The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species…
Abstract
Purpose
The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. This paper aims to propose a new Yang-Abdel-Aty-Cattani (YAC) fractional operator with a non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this study has explained the analytical methods, reduced differential transform method (RDTM) and residual power series method (RPSM) taking the fractional derivative as YAC operator sense.
Design/methodology/approach
This study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense.
Findings
This study has expressed the solutions in terms of Mittag-Leffler functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.
Research limitations/implications
The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this study, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.
Practical implications
The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation which is arised in biological population model. Here, this study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.
Social implications
The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this paper has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.
Originality/value
The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this paper has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.
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THIS report gives a general solution of the problem of the calculation of the Glauert loading of wings with discontinuities of incidence. The three existing variations of the…
Abstract
THIS report gives a general solution of the problem of the calculation of the Glauert loading of wings with discontinuities of incidence. The three existing variations of the original theory, the Glauert solution, the Gates' least squares solution, and the Lotz' solution, are not entirely satisfactory and may involve a considerable amount of labour. The present solution divides the Fourier series representing the circulation into two parts: (a) a standard solution representing the discontinuities, which includes the slowly convergent part of the solution, and which is expressible as a precise infinite series dependent only upon the position of discontinuities along the span, and (b) a secondary solution due to plan form, aspect ratio, slope of section lift curve, and so on, which is the quickly convergent part of the solution and usually requires a terminated scries of only from four to six terms. Once the standard solution has been computed, the remaining work is little more than for the standard Glauert solution for a flat wing.
Mehdi Dehghan and Jalil Manafian Heris
This paper aims to show that the variational iteration method (VIM) and the homotopy perturbation method (HPM) are powerful and suitable methods to solve the Fornberg‐Whitham…
Abstract
Purpose
This paper aims to show that the variational iteration method (VIM) and the homotopy perturbation method (HPM) are powerful and suitable methods to solve the Fornberg‐Whitham equation.
Design/methodology/approach
Using HPM the explicit exact solution is calculated in the form of a quickly convergent series with easily computable components. Also, by using VIM the analytical results of this equation have been obtained in terms of convergent series with easily computable components.
Findings
Numerical solutions obtained by these methods are compared with the exact solutions, revealing that the obtained solutions are of high accuracy.
Originality/value
Also the results show that the introduced methods are efficient tools for solving the nonlinear partial differential equations.
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This paper presents approximate analytical solutions for the diffusion problems of a cylindrical hole in an infinite medium and a slot in an infinite medium with properly…
Abstract
This paper presents approximate analytical solutions for the diffusion problems of a cylindrical hole in an infinite medium and a slot in an infinite medium with properly prescribed boundary conditions and initial conditions. These solutions have much simpler forms than those of exact analytical solutions, and asymptotically approach the exact solutions with increasing time or the material point moving away from the internal boundary. The approximate analytical solution for the diffusion problem of a slot in an infinite medium is applied to establish a shape function for the infinite elements. Good agreement is found in comparison of our results with those presented by Li and Huang and Cinco‐Ley et al. Finally, an example simulating a primary recovery procedure in hydraulic fracturing technique for an oil field is presented.
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Pankaj and Khalid Moin
Exact solutions for Mohr Coulomb elastoplasticity are developed. Using these solutions an exact stress increment for a given finite strain increment can be computed. The developed…
Abstract
Exact solutions for Mohr Coulomb elastoplasticity are developed. Using these solutions an exact stress increment for a given finite strain increment can be computed. The developed solutions are valid for perfect and linear hardening/softening plasticity using isotropic work hardening hypotheses. The solutions can be used to check computer codes and assess their ability to handle multiple active yield surfaces. Illustrative examples are included.
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Rambabu Vana and Karunakar Perumandla
To provide a new semi-analytical solution for the nonlinear Benjamin–Bona–Mahony (BBM) equation in the form of a convergent series. The results obtained through HPTM for BBM are…
Abstract
Purpose
To provide a new semi-analytical solution for the nonlinear Benjamin–Bona–Mahony (BBM) equation in the form of a convergent series. The results obtained through HPTM for BBM are compared with those obtained using the Sine-Gordon Expansion Method (SGEM) and the exact solution. We consider the initial condition as uncertain, represented in terms of an interval then investigate the solution of the interval Benjamin–Bona–Mahony (iBBM).
Design/methodology/approach
We employ the Homotopy Perturbation Transform Method (HPTM) to derive the series solution for the BBM equation. Furthermore, the iBBM equation is solved using HPTM to the initial condition has been considered as an interval number as the coefficient of it depends on several parameters and provides lower and upper interval solutions for iBBM.
Findings
The obtained numerical results provide accurate solutions, as demonstrated in the figures. The numerical results are evaluated to the precise solutions and found to be in good agreement. Further, the initial condition has been considered as an interval number as the coefficient of it depends on several parameters. To enhance the clarity, we depict our solutions using 3D graphics and interval solution plots generated using MATLAB.
Originality/value
A new semi-analytical convergent series-type solution has been found for nonlinear BBM and interval BBM equations with the help of the semi-analytical technique HPTM.
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