• Journal of applied mathematics & informatics
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• v.35 no.5_6
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• pp.439-447
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• 2017

This paper investigates the inverse problem of determining an unknown heat radiative coefficient, which is only time-dependent. This is an ill-posed problem, that is, small errors in data may cause huge deviations in determining solution. In this paper, the existence and uniqueness of the problem is established by the second Volterra integral equation theory, and the method of trace-type functional formulation combined with finite difference scheme is studied. One typical numerical example using the proposed method is illustrated and discussed.

• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.35-51
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• 1993

This paper concerns collocation methods for integro-differential equations in which memory kernels have a singularity at t = 0. There has been extensive research in recent years on Volterra integral and integro-differential equations for physical systems with memory effects in which the stabilty and asymtotic stability of solutionsl have been the main interest. We will study a class of hereditary equations with singular kernels which interpolate between well known model equations as the order of singularity varies. We are also concerned with the smoothing effect of singular kernels, but we use energy methods and our results involve fractional time in fixed spatial norms. Galerkin methods for our models was studied and existence, uniqueness and stability results was obtained in [4]. Our major goal is to study collocation methods.

• Communications of the Korean Mathematical Society
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• v.31 no.4
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• pp.869-878
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• 2016

This paper presents an ecient fractional shifted Legendre polynomial method to solve the fractional Volterra's model for population growth model. The fractional derivatives are described based on the Caputo sense by using Riemann-Liouville fractional integral operator. The theoretical analysis, such as convergence analysis and error bound for the proposed technique has been demonstrated. In applications, the reliability of the technique is demonstrated by the error function based on the accuracy of the approximate solution. The numerical applications have provided the eciency of the method with dierent coecients of the population growth model. Finally, the obtained results reveal that the proposed technique is very convenient and quite accurate to such considered problems.

• Structural Engineering and Mechanics
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• v.19 no.4
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• pp.441-457
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• 2005

An analytical method is presented to solve the problem of transient wave propagation in a transversely isotropic piezoelectric hollow sphere subjected to thermal shock and electric excitation. Exact expressions for the transient responses of displacements, stresses, electric displacement and electric potentials in the piezoelectric hollow sphere are obtained by means of Hankel transform, Laplace transform, and inverse transforms. Using Hermite non-linear interpolation method solves Volterra integral equation of the second kind involved in the exact expression, which is caused by interaction between thermo-elastic field and thermo-electric field. Thus, an analytical solution for the problem of transient wave propagation in a transversely isotropic piezoelectric hollow sphere is obtained. Finally, some numerical results are carried out, and may be used as a reference to solve other transient coupled problems of thermo-electro-elasticity.

• International Journal of Automotive Technology
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• v.2 no.1
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• pp.17-23
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• 2001

Measurements of the initial values lead to an inverse and mathematically unprecisely formulated problem. A precise definition of an inverse problem is possible. It is to state a mathematical model of a physical process with clearly defined initial and exit values for the system behind the process. One can grasp the idea of an inverse problem by considering the tire as a copy of the objects of nature in a room with observations. Interpretation of nature is generally a result of an inverse problem. On one hand, the tire may be represented through the sensory organs and the nervous system as well as the experiences of the developer's existing apparatus of the projection of reality. On the other hand, it may be represented by a physical law or a model that can be confirmed or is to be refuted with the help of suitable measurements. During reconstruction of a measuring signal and the identification of a black box that can be assumed to be linear and causal, the tire becomes a first type Volterra integral equation of the convolution type. But measurements of the initial values are always fuzzy, the errors grow and the system behavior can no longer be forecasted. Thus, we have to deal with a chaotic system. This chaos produces fractals in a natural way. These are self-similar geometric structures. This self-similarity is clearly visible in the design.