• Journal of applied mathematics & informatics
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• 제42권4호
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• pp.899-913
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• 2024

In this article, the Series Solution Method (SSM) is employed to solve the linear or non-linear Volterra integro-differential equations. Numerous examples have been presented to explain the numerical results, which is the comparison between the exact solution and the numerical solution, and it is found through the tables. The amount of error between the exact solution and the numerical solution is very small and almost nonexistent, and it is also illustrated through the graph how the exact solution completely applies to the numerical solution. This proves the accuracy of the method, which is the Series Solution Method (SSM) for solving the linear or non-linear Volterra integro-differential equations using Mathematica. Furthermore, this approach yields numerical results with remarkable accuracy, speed, and ease of use.

• 대한수학회지
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• 제45권3호
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• pp.631-644
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• 2008

In this paper, we present a new method for solving a nonlinear two-point boundary value problem with finitely many singularities. Its exact solution is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximation $u_n(x)$ to the exact solution u(x) is obtained and is proved to converge to the exact solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.

• 한국항공운항학회지
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• 제16권2호
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• pp.21-27
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• 2008

An inverse method is introduced to construct the problem for the error analysis of the numerical solution of initial value problem. These problems constructed through this method have a known exact solution, even though analytical solutions are generally not obtainable. The process leading to the exact solution makes use of an initially available approximate numerical solution. A smooth interpolation of the approximate solution is forced to exactly satisfy the differential equation by analytically deriving a small forcing function to absorb all of the errors in the interpolated approximate solution. Using this special case exact solution, it is possible to investigate the relationship between global errors of a candidate numerical solution process and the associated tuning parameters for a given problem. Under the assumption that the original differential equation is well-posed with respect to the small perturbations, we thereby obtain valuable information about the optimal choice of the tuning parameters and the achievable accuracy of the numerical solution.

• Structural Engineering and Mechanics
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• 제35권2호
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• pp.205-215
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• 2010

Recently, earthquake proof technology has been widely applied to both new and existing structures and bridges. The analysis of bridge systems equipped with structural control devices, which possess large degrees of freedom and nonlinear characteristics, is a result in time-consuming task. Therefore, a piecewise exact solution is proposed in this study to simplify the seismic mitigation analysis process for bridge systems equipped with sliding-type isolators. In this study, the simplified system having two degrees of freedom, to reasonably represent the large number of degrees of freedom of a bridge, and is modeled to obtain a piecewise exact solution for system responses during earthquakes. Simultaneously, we used the nonlinear finite element computer program to analyze the bridge responses and verify the accuracy of the proposed piecewise exact solution for bridge systems equipped with sliding-type isolators. The conclusions derived by comparing the results obtained from the piecewise exact solution and nonlinear finite element analysis reveal that the proposed solution not only simplifies the calculation process but also provides highly accurate seismic responses of isolated bridges under earthquakes.

• Journal of applied mathematics & informatics
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• 제33권3_4호
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• pp.387-399
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• 2015

In this paper, a new smoothing approximation to the l₁ exact penalty function for constrained optimization problems (COP) is presented. It is shown that an optimal solution to the smoothing penalty optimization problem is an approximate optimal solution to the original optimization problem. Based on the smoothing penalty function, an algorithm is presented to solve COP, with its convergence under some conditions proved. Numerical examples illustrate that this algorithm is efficient in solving COP.

• 한국소음진동공학회논문집
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• 제21권3호
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• pp.234-240
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• 2011

This study deals with progressive collapse of a structure retrofitted with MR dampers. In order to assess their effect of mitigation which prevents progressive collapse, control force ratio is defined by friction force of MR dampers divided by external force. First, simple model of a structure with MR dampers is suggested. Using the model, exact solution with the control force ratio is obtained. When and where the system is stopped is predicted by the derived solution. Through the dissipated energy by MR dampers during collapse event, equivalent damping ratio is derived. Finally, comparison of exact and equivalent solutions is presented.

• International Journal of Aeronautical and Space Sciences
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• 제3권2호
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• pp.46-58
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• 2002

An inverse method is introduced to construct benchmark problems for the numerical solution of initial value problems. Benchmark problems constructed through this method have a known exact solution, even though analytical solutions are generally not obtainable. The solution is constructed such that it lies near a given approximate numerical solution, and therefore the special case solution can be generated in a versatile and physically meaningful fashion and can serve as a benchmark problem to validate approximate solution methods. A smooth interpolation of the approximate solution is forced to exactly satisfy the differential equation by analytically deriving a small forcing function to absorb all of the errors in the interpolated approximate solution. A multi-variable orthogonal function expansion method and computer symbol manipulation are successfully used for this process. Using this special case exact solution, it is possible to directly investigate the relationship between global errors of a candidate numerical solution process and the associated tuning parameters for a given code and a given problem. Under the assumption that the original differential equation is well-posed with respect to the small perturbations, we thereby obtain valuable information about the optimal choice of the tuning parameters and the achievable accuracy of the numerical solution. Illustrative examples show the utility of this method not only for the ordinary differential equations (ODEs) but for the partial differential equations (PDEs).

• Structural Engineering and Mechanics
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• 제31권6호
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• pp.663-678
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• 2009

An exact solution is obtained for forced torsional vibration of a finite class 622 piezoelectric hollow cylinder with free-free ends subjected to dynamic shearing stress and time dependent electric potential at both internal and external surfaces. The solution is first expanded in axial direction with trigonometric series and the governing equations for the new variables about radial coordinate r and time t are derived with the aid of Fourier series expansion technique. By means of the superposition method and the separation of variables technique, the solution for torsional vibration is finally obtained. Natural frequencies and the transient torsional responses for finite class 622 piezoelectric hollow cylinder with free-free ends are computed and illustrated.

• Geomechanics and Engineering
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• 제38권3호
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• pp.261-273
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• 2024

This study presents a groundbreaking analytical approach to find an exact solution for the bearing capacity of strip footings on reinforced slopes, utilizing the two-phase approach and slip line method. The two-phase approach is considered as a generalized homogenization technique. The slip line method is leveraged to derive the stress field as a lower bound solution and the velocity field as an upper bound solution, thereby facilitating the attainment of an exact solution. The key finding points out the variation of the bearing capacity factor N_γ with influencing factors including the backfill soil friction angle, the footing setback distance from the slope crest edge, slope angle, strength, and volumetric fraction of inclusion layers. The results are evaluated by comparing them with those of relevant studies in the literature considering analytical and experimental studies. Through the application of the two-phase approach, it becomes feasible to determine the tensile loads mobilized along the inclusion layers associated with the failure zone. It is attempted to demonstrate the results by utilizing non-dimensional graphs to clearly illustrate variable impacts on reinforced soil stability. This research contributes significantly to advancing geotechnical engineering practices, specifically in the realm of static design considerations for reinforced soil structures.

• Structural Engineering and Mechanics
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• 제6권5호
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• pp.517-527
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• 1998

This paper describes an exact method of analysis for natural vibration of diagonal networks by considering an equivalent cyclic periodic structure and adopting the double U-transformation technique. Both a lumped mass system and a distributed mass system are considered to investigate the diagonal networks. The exact solution for the frequency equations and the natural modes of the networks can be derived. As numerical examples, square diagonal cable networks with different meshes are worked out.