scholarly journals Numerical Solution of Direct and Inverse Problems for Time-Dependent Volterra Integro-Differential Equation Using Finite Integration Method with Shifted Chebyshev Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 497 ◽  
Author(s):  
Ratinan Boonklurb ◽  
Ampol Duangpan ◽  
Phansphitcha Gugaew
Keyword(s):  
Differential Equation ◽  
Inverse Problems ◽  
Chebyshev Polynomials ◽  
Integration Method ◽  
Computational Cost ◽  
Main Tool ◽  
Time Dependent ◽  
Tikhonov Regularization Method ◽  
Direct And Inverse Problems ◽  
Integro Differential Equation

In this article, the direct and inverse problems for the one-dimensional time-dependent Volterra integro-differential equation involving two integration terms of the unknown function (i.e., with respect to time and space) are considered. In order to acquire accurate numerical results, we apply the finite integration method based on shifted Chebyshev polynomials (FIM-SCP) to handle the spatial variable. These shifted Chebyshev polynomials are symmetric (either with respect to the point x = L 2 or the vertical line x = L 2 depending on their degree) over [ 0 , L ] , and their zeros in the interval are distributed symmetrically. We use these zeros to construct the main tool of FIM-SCP: the Chebyshev integration matrix. The forward difference quotient is used to deal with the temporal variable. Then, we obtain efficient numerical algorithms for solving both the direct and inverse problems. However, the ill-posedness of the inverse problem causes instability in the solution and, so, the Tikhonov regularization method is utilized to stabilize the solution. Furthermore, several direct and inverse numerical experiments are illustrated. Evidently, our proposed algorithms for both the direct and inverse problems give a highly accurate result with low computational cost, due to the small number of iterations and discretization.

scholarly journals Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1201
Author(s):  
Ampol Duangpan ◽  
Ratinan Boonklurb ◽  
Tawikan Treeyaprasert
Keyword(s):  
Chebyshev Polynomials ◽  
Fractional Derivatives ◽  
Integration Method ◽  
Nonlinear Partial Differential Equations ◽  
Computational Cost ◽  
Numerical Algorithms ◽  
Approximate Solutions ◽  
Two Dimensions ◽  
Burgers Equations ◽  
Forward Difference

The Burgers’ equation is one of the nonlinear partial differential equations that has been studied by many researchers, especially, in terms of the fractional derivatives. In this article, the numerical algorithms are invented to obtain the approximate solutions of time-fractional Burgers’ equations both in one and two dimensions as well as time-fractional coupled Burgers’ equations which their fractional derivatives are described in the Caputo sense. These proposed algorithms are constructed by applying the finite integration method combined with the shifted Chebyshev polynomials to deal the spatial discretizations and further using the forward difference quotient to handle the temporal discretizations. Moreover, numerical examples demonstrate the ability of the proposed method to produce the decent approximate solutions in terms of accuracy. The rate of convergence and computational cost for each example are also presented.

scholarly journals Evaluation of oceanic and atmospheric trajectory schemes in the TRACMASS trajectory model v6.0

2016 ◽  
Author(s):  
Kristofer Döös ◽  
Bror Jönsson ◽  
Joakim Kjellsson
Keyword(s):  
Differential Equation ◽  
General Circulation ◽  
Computational Cost ◽  
Circulation Model ◽  
General Circulation Models ◽  
Velocity Fields ◽  
Time Dependent ◽  
Computational Time ◽  
Time Step ◽  
Large Ensembles

Abstract. Two different trajectory schemes for oceanic and atmospheric general circulation models are compared in two different experiments. The theories of the two trajectory schemes are presented showing the differential equations they solve and why they are mass conserving. One scheme assumes that the velocity fields are stationary for a limited period of time and solves the trajectory path from a differential equation only as a function of space, i.e. "stepwise stationary". The second scheme uses a continuous linear interpolation of the fields in time and solves the trajectory path from a differential equation as a function of both space and time, i.e. "time-dependent". A special case of the "stepwise-stationary" scheme, when velocities are assumed constant between GCM outputs, is also considered, named "fixed GCM time step". The trajectory schemes are tested "off-line", i.e. using the already integrated and stored velocity fields from a GCM. The first comparison of the schemes uses trajectories calculated using the velocity fields from an eddy-resolving ocean general circulation model in the Agulhas region. The second comparison uses trajectories calculated using the wind fields from an atmospheric reanalysis. The study shows that using the "time-dependent" scheme over the "stepwise-stationary" scheme greatly improves accuracy with only a small increase in computational time. It is also found that with decreasing time steps the "stepwise-stationary" scheme becomes more accurate but at increased computational cost. The "time-dependent" scheme is therefore preferred over the "stepwise-stationary" scheme. However, when averaging over large ensembles of trajectories the two schemes are comparable, as intrinsic variability dominates over numerical errors. The "fixed GCM time step" is found to be less accurate than the "stepwise-stationary" scheme, even when considering averages over large ensembles.

scholarly journals Problem of Determining a Multidimensional Kernel in One Parabolic Integro–differential Equation

2021 ◽  
pp. 117-127
Author(s):  
Durdimurod K. Durdiev ◽  
Zhavlon Z. Nuriddinov
Keyword(s):  
Differential Equation ◽  
Inverse Problem ◽  
Cauchy Problem ◽  
Direct Problem ◽  
Contractive Mapping ◽  
Equivalent System ◽  
Direct And Inverse Problems ◽  
Integro Differential Equation ◽  
The Cauchy Problem ◽  
The Right

The multidimensional parabolic integro-differential equation with the time-convolution in- tegral on the right side is considered. The direct problem is represented by the Cauchy problem for this equation. In this paper it is studied the inverse problem consisting in finding of a time and spatial dependent kernel of the integrated member on known in a hyperplane xn = 0 for t > 0 to the solution of direct problem. With use of the resolvent of kernel this problem is reduced to the investigation of more convenient inverse problem. The last problem is replaced with the equivalent system of the integral equations with respect to unknown functions and on the bases of contractive mapping principle it is proved the unique solvability to the direct and inverse problems

scholarly journals INVESTIGATION OF THE NONLINEAR FLUTTER OF VISCOELASTIC PLATES IN A SUPERSONIC FLOW

2014 ◽  
pp. 42-45
Author(s):  
Bakhtiyar Khudayarov
Keyword(s):  
Differential Equation ◽  
Supersonic Flow ◽  
Differential Equations ◽  
Integration Method ◽  
Basic Direction ◽  
Partial Derivatives ◽  
Integro Differential Equation ◽  
Plate Flutter ◽  
Nonlinear Flutter ◽  
Nonlinear Viscoelastic

In this paper the flutter of nonlinear viscoelastic plates in a supersonic flow is investigated. The basic direction of work is consisted in taking into account of viscoelastic material’s properties at supersonic speeds. Quasi-steady aerodynamic panel loadings are determined using piston theory. The vibration equations relatively of deflection are described by Integrо-differential equations in partial derivatives. The plate nonlinear partial integro-differential equation is transformed info a set of nonlinear ordinary IDE through a Bubnov-Galerkin’s approach. The resulting system of IDE is solved through the Badalov-Eshmatov integration method. Critical speeds for plate flutter are defined.

scholarly journals Random walk algorithm for the Dirichlet problem for parabolic integro-differential equation

2021 ◽  
Author(s):  
G. Deligiannidis ◽  
S. Maurer ◽  
M. V. Tretyakov
Keyword(s):  
Differential Equation ◽  
Dirichlet Problem ◽  
Computational Cost ◽  
Activity Level ◽  
Currency Options ◽  
Euler Approximation ◽  
Integro Differential Equation ◽  
Time Stepping ◽  
Depth Analysis ◽  
Adaptive Time

AbstractWe consider stochastic differential equations driven by a general Lévy processes (SDEs) with infinite activity and the related, via the Feynman–Kac formula, Dirichlet problem for parabolic integro-differential equation (PIDE). We approximate the solution of PIDE using a numerical method for the SDEs. The method is based on three ingredients: (1) we approximate small jumps by a diffusion; (2) we use restricted jump-adaptive time-stepping; and (3) between the jumps we exploit a weak Euler approximation. We prove weak convergence of the considered algorithm and present an in-depth analysis of how its error and computational cost depend on the jump activity level. Results of some numerical experiments, including pricing of barrier basket currency options, are presented.

scholarly journals Unipotent differential algebraic groups as parameterized differential Galois groups

2013 ◽  
Vol 13 (4) ◽  
pp. 671-700 ◽  
Author(s):  
Andrey Minchenko ◽  
Alexey Ovchinnikov ◽  
Michael F. Singer
Keyword(s):  
Differential Equation ◽  
Inverse Problems ◽  
Linear Differential Equation ◽  
Galois Group ◽  
Algebraic Groups ◽  
Galois Groups ◽  
Direct And Inverse Problems ◽  
Unipotent Groups ◽  
Differential Type ◽  
Linear Differential

AbstractWe deal with aspects of direct and inverse problems in parameterized Picard–Vessiot (PPV) theory. It is known that, for certain fields, a linear differential algebraic group (LDAG) $G$ is a PPV Galois group over these fields if and only if $G$ contains a Kolchin-dense finitely generated group. We show that, for a class of LDAGs $G$, including unipotent groups, $G$ is such a group if and only if it has differential type $0$. We give a procedure to determine if a parameterized linear differential equation has a PPV Galois group in this class and show how one can calculate the PPV Galois group of a parameterized linear differential equation if its Galois group has differential type $0$.

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