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Dušan Zorica - One of the best experts on this subject based on the ideXlab platform.
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time distributed order Diffusion Wave equation ii applications of laplace and fourier transformations
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2009Co-Authors: Teodor M. Atanackovic, Stevan Pilipovic, Dušan ZoricaAbstract:A Cauchy problem for a time distributed-order multi-dimensional Diffusion-Wave equation containing a forcing term is reinterpreted in the space of tempered distributions, and a distributional Diffusion-Wave equation is obtained. The distributional equation is solved in the general case of weight function (or distribution). Solutions are given in terms of solution kernels (Green’s functions), which are studied separately for two cases. The first case is when the order of the fractional derivative is in the interval [0, 1], while, in the second case, the order of the fractional derivative is in the interval [0, 2]. Solutions of fractional DiffusionWave and fractional telegraph equations are obtained as special cases. Numerical experiments are also performed. An analogue of the maximum principle is also presented.
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Time distributed-order Diffusion-Wave equation. I. Volterra-type equation
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2009Co-Authors: Teodor M. Atanackovic, Stevan Pilipović, Dušan ZoricaAbstract:A single-order time-fractional Diffusion-Wave equation is generalized by introducing a time distributed-order fractional derivative and forcing term, while a Laplacian is replaced by a general linear multi-dimensional spatial differential operator. The obtained equation is (in the case of the Laplacian) called a time distributed-order Diffusion-Wave equation. We analyse a Cauchy problem for such an equation by means of the theory of an abstract Volterra equation. The weight distribution, occurring in the distributed-order fractional derivative, is specified as the sum of the Dirac distributions and the existence and uniqueness of solutions to the Cauchy problem, and the corresponding Volterra-type equation were proven for a general linear spatial differential operator, as well as in the special case when the operator is Laplacian.
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a Diffusion Wave equation with two fractional derivatives of different order
Journal of Physics A, 2007Co-Authors: Teodor M. Atanackovic, Stevan Pilipovic, Dušan ZoricaAbstract:We analyse a Diffusion Wave equation with two fractional derivatives of different order on bounded and unbounded spatial domains. Thus, our model represents a generalized telegraph equation. Solutions to signalling and Cauchy problems in terms of a series and integral representation are given. Classical Wave and heat conduction equations are obtained as limiting cases.
Teodor M. Atanackovic - One of the best experts on this subject based on the ideXlab platform.
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time distributed order Diffusion Wave equation ii applications of laplace and fourier transformations
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2009Co-Authors: Teodor M. Atanackovic, Stevan Pilipovic, Dušan ZoricaAbstract:A Cauchy problem for a time distributed-order multi-dimensional Diffusion-Wave equation containing a forcing term is reinterpreted in the space of tempered distributions, and a distributional Diffusion-Wave equation is obtained. The distributional equation is solved in the general case of weight function (or distribution). Solutions are given in terms of solution kernels (Green’s functions), which are studied separately for two cases. The first case is when the order of the fractional derivative is in the interval [0, 1], while, in the second case, the order of the fractional derivative is in the interval [0, 2]. Solutions of fractional DiffusionWave and fractional telegraph equations are obtained as special cases. Numerical experiments are also performed. An analogue of the maximum principle is also presented.
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Time distributed-order Diffusion-Wave equation. I. Volterra-type equation
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2009Co-Authors: Teodor M. Atanackovic, Stevan Pilipović, Dušan ZoricaAbstract:A single-order time-fractional Diffusion-Wave equation is generalized by introducing a time distributed-order fractional derivative and forcing term, while a Laplacian is replaced by a general linear multi-dimensional spatial differential operator. The obtained equation is (in the case of the Laplacian) called a time distributed-order Diffusion-Wave equation. We analyse a Cauchy problem for such an equation by means of the theory of an abstract Volterra equation. The weight distribution, occurring in the distributed-order fractional derivative, is specified as the sum of the Dirac distributions and the existence and uniqueness of solutions to the Cauchy problem, and the corresponding Volterra-type equation were proven for a general linear spatial differential operator, as well as in the special case when the operator is Laplacian.
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a Diffusion Wave equation with two fractional derivatives of different order
Journal of Physics A, 2007Co-Authors: Teodor M. Atanackovic, Stevan Pilipovic, Dušan ZoricaAbstract:We analyse a Diffusion Wave equation with two fractional derivatives of different order on bounded and unbounded spatial domains. Thus, our model represents a generalized telegraph equation. Solutions to signalling and Cauchy problems in terms of a series and integral representation are given. Classical Wave and heat conduction equations are obtained as limiting cases.
Francesco Mainardi - One of the best experts on this subject based on the ideXlab platform.
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Cauchy and Signaling Problems for the Time-Fractional Diffusion-Wave Equation
Journal of Vibration and Acoustics, 2014Co-Authors: Yuri Luchko, Francesco MainardiAbstract:In this paper, some known and novel properties of the Cauchy and signaling problems for the one-dimensional time-fractional Diffusion-Wave equation with the Caputo fractional derivative of order b;1 � b � 2 are investigated. In particular, their response to a localized disturbance of the initial data is studied. It is known that, whereas the Diffusion equation describes a process where the disturbance spreads infinitely fast, the propagation velocity of the disturbance is a constant for the Wave equation. We show that the time-fractional Diffusion-Wave equation interpolates between these two different responses in the sense that the propagation velocities of the maximum points, centers of gravity, and medians of the fundamental solutions to both the Cauchy and the signaling problems are all finite. On the other hand, the disturbance spreads infinitely fast and the time-fractional Diffusion-Wave equation is nonrelativistic like the classical Diffusion equation. In this paper, the maximum locations, the centers of gravity, and the medians of the fundamental solution to the Cauchy and signaling problems and their propagation velocities are described analytically and calculated numerically. The obtained results for the Cauchy and the signaling problems are interpreted and compared to each other. [DOI: 10.1115/1.4026892]
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Propagation speed of the maximum of the fundamental solution to the fractional Diffusion-Wave equation
Computers & Mathematics With Applications, 2013Co-Authors: Yuri Luchko, Francesco Mainardi, Yuriy PovstenkoAbstract:In this paper, the one-dimensional time-fractional Diffusion-Wave equation with the fractional derivative of order @a,1
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Some properties of the fundamental solution to the signalling problem for the fractional Diffusion-Wave equation
Central European Journal of Physics, 2013Co-Authors: Yuri Luchko, Francesco MainardiAbstract:In this paper, the one-dimensional time-fractional Diffusion-Wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the Diffusion and the Wave equations behave quite differently regarding their response to a localized disturbance. Whereas the Diffusion equation describes a process where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the Wave equation. We show that the time-fractional Diffusion-Wave equation interpolates between these two different responses and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.
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wright functions as scale invariant solutions of the Diffusion Wave equation
Journal of Computational and Applied Mathematics, 2000Co-Authors: Rudolf Gorenflo, Yuri Luchko, Francesco MainardiAbstract:The time-fractional Diffusion-Wave equation is obtained from the classical Diffusion or Wave equation by replacing the first- or second-order time derivative by a fractional derivative of order α (0 <α ≤ 2). Using the similarity method and the method of the Laplace transform, it is shown that the scale-invariant solutions of the mixed problem of signalling type for the time-fractional Diffusion-Wave equation are given in terms of the Wright function in the case 0 <α< 1 and in terms of the generalized Wright function in the case 1 <α< 2. The reduced equation for the scale-invariant solutions is given in terms of the Caputo type modification of the
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Wright functions as scale-invariant solutions of the Diffusion-Wave equation
Journal of Computational and Applied Mathematics, 2000Co-Authors: Rudolf Gorenflo, Yuri Luchko, Francesco MainardiAbstract:The time-fractional Diffusion-Wave equation is obtained from the classical Diffusion or Wave equation by replacing the first- or second-order time derivative by a fractional derivative of order α (0
Vijay P. Singh - One of the best experts on this subject based on the ideXlab platform.
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Accuracy of kinematic Wave and Diffusion Wave approximations for time-independent flows
Hydrological Processes, 2006Co-Authors: Vijay P. Singh, V. AravamuthanAbstract:Time-independent (or steady-state) cases of planar (overland) flow were treated. Errors of the kinematic-Wave and Diffusion-Wave approximations were derived for three types of boundary conditions: zero flow at the upstream end, and critical flow depth and zero depth-gradient at the downstream end. The Diffusion Wave approximation was found to be in excellent agreement with the dynamic Wave approximation, with error in the range of 1–2% for values ofKF 0 2 (≥7.5). Even for small values ofKF 2 0 (e.g.,KF 2 0 =0.75), the errors were typically in the range of 11–15%. The accuracy of the Diffusion Wave approximation was greatly influenced by the downstream boundary condition. The error of the kinematic Wave approximation was found to vary from 7 to 13% in the regions 0.05≤x≤0.95 forKF 0 2 =0.75 and was greater than 30% forKF 0 2 =0.75.
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Water Encyclopedia - Kinematic Wave and Diffusion Wave Theories
Water Encyclopedia, 2005Co-Authors: Vijay P. SinghAbstract:A wide range of phenomena, natural as well as man-made, in physical, chemical, and biological hydrology exhibit characteristics similar to those of kinematic or Diffusion Waves. The wide range of phenomena suggests that these Waves are very pervasive. This study presents theories that are considered fundamental to advancing the state of the art of hydrology. The term “Wave” implies a disturbance traveling upstream, downstream, or remaining stationary. We can visualize a water Wave propagating where the water itself stays very much where it was before the Wave was produced. We witness other Waves that travel as well, such as heat Waves, pressure Waves, and sound Waves. There is obviously the motion of matter, but there can also be the motion of form and other properties of matter. Keywords: Diffusion Wave theory; flood hydrograph; flow routing; flux law; hydrology; kinematic shock; kinematic Wave theory; nonuniform flow; numerical scheme; steady-state flow; uniform flow; unsteady-state flow; Wave
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dem based modelling of surface runoff using Diffusion Wave equation
Journal of Hydrology, 2005Co-Authors: Manoj Jain, Vijay P. SinghAbstract:A digital elevation model (DEM)-based overland flow routing model was developed for computation of surface runoff for isolated storm events. The model operates on a grid or cell basis and routes the rainfall excess generated over the cells, following the DEM-derived drainage paths, to the catchment outlet. The rainfall excess for each cell of the catchment was computed using the Philip two-term infiltration model utilizing the physical properties of the cell. The overland flow was described by a finite volume-based numerical solution of the Diffusion Wave approximation of the St Venant equations. The cell physical properties, such as topographic characteristics, land use, soil, etc., were extracted from published maps for discretized cells of the catchment using a Geographic Information System. The results of model application indicate that the model satisfactorily predicted the runoff hydrograph. The cell-based structure of the model allowed for generation of spatially distributed catchment information in terms of the model-computed variables, such as the depth of flow and discharge.
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Errors of kinematic-Wave and Diffusion-Wave approximations for steady-state overland flows
Catena, 1996Co-Authors: Vijay P. Singh, V. AravamuthanAbstract:Abstract Steady-state flows occur in a variety of geophysical processes. Their mathematical treatment is usually based on either the dynamic-Wave equations or their approximations—kinematic Wave and Diffusion Wave. Errors of these approximations were derived for steady-state cases of overland flow under the zero depth-gradient at the downstream boundary condition. Although both the Diffusion-Wave and the kinematic-Wave approximations were found to be in good agreement with the dynamic Wave representation for a range of flow conditions, their error greatly depended on the parameter KF 0 2 where K is the kinematic-Wave number and F 0 is the Froude number. In general, the Diffusion-Wave approximation was superior to the kinematic Wave approximation.
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Errors of kinematic-Wave and Diffusion-Wave approximations for time-independent flows
Water Resources Management, 1995Co-Authors: Vijay P. Singh, V. AravamuthanAbstract:Time-independent (or steady-state) cases of planar (overland) flow were treated. Errors of the kinematic-Wave and Diffusion-Wave approximations were derived for three types of boundary conditions: zero flow at the upstream end, and critical flow depth and zero depth-gradient at the downstream end. The Diffusion Wave approximation was found to be in excellent agreement with the dynamic Wave approximation, with error in the range of 1–2% for values ofKF 0 2 (≥7.5). Even for small values ofKF 2 0 (e.g.,KF 2 0 =0.75), the errors were typically in the range of 11–15%. The accuracy of the Diffusion Wave approximation was greatly influenced by the downstream boundary condition. The error of the kinematic Wave approximation was found to vary from 7 to 13% in the regions 0.05≤x≤0.95 forKF 0 2 =0.75 and was greater than 30% forKF 0 2 =0.75.
Yuriy Povstenko - One of the best experts on this subject based on the ideXlab platform.
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Solutions to the fractional Diffusion-Wave equation in a wedge
Fractional Calculus and Applied Analysis, 2014Co-Authors: Yuriy PovstenkoAbstract:The Diffusion-Wave equation with the Caputo derivative of the order 0 < α ≤ 2 is considered in polar coordinates in a domain 0 ≤ r < ∞, 0 < φ < φ0 under Dirichlet and Neumann boundary conditions. The Laplace integral transform with respect to time, the finite sin- and cos-Fourier transforms with respect to the angular coordinate, and the Hankel transform with respect to the radial coordinate are used. The numerical results are illustrated graphically.
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axisymmetric solutions to fractional Diffusion Wave equation in a cylinder under robin boundary condition
European Physical Journal-special Topics, 2013Co-Authors: Yuriy PovstenkoAbstract:The axisymmetric time-fractional Diffusion-Wave equation with the Caputo derivative of the order 0 < α ≤ 2 is considered in a cylinder under the prescribed linear combination of the values of the sought function and the values of its normal derivative at the boundary. The fundamental solutions to the Cauchy, source, and boundary problems are investigated. The Laplace transform with respect to time and finite Hankel transform with respect to the radial coordinate are used. The solutions are obtained in terms of Mittag-Leffler functions. The numerical results are illustrated graphically.
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Propagation speed of the maximum of the fundamental solution to the fractional Diffusion-Wave equation
Computers & Mathematics With Applications, 2013Co-Authors: Yuri Luchko, Francesco Mainardi, Yuriy PovstenkoAbstract:In this paper, the one-dimensional time-fractional Diffusion-Wave equation with the fractional derivative of order @a,1
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non axisymmetric solutions to time fractional Diffusion Wave equation in an infinite cylinder
Fractional Calculus and Applied Analysis, 2011Co-Authors: Yuriy PovstenkoAbstract:The time-fractional Diffusion-Wave equation is considered in an infinite cylinder in the case of three spatial coordinates r, ϕ and z. The Caputo fractional derivative of the order 0 < α ≤ 2 is used. Several examples of problems with Dirichlet and Neumann boundary conditions at a surface of the cylinder are solved using the integral transforms technique. Numerical results are illustrated graphically.
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solutions to time fractional Diffusion Wave equation in cylindrical coordinates
Advances in Difference Equations, 2011Co-Authors: Yuriy PovstenkoAbstract:Nonaxisymmetric solutions to time-fractional Diffusion-Wave equation with a source term in cylindrical coordinates are obtained for an infinite medium. The solutions are found using the Laplace transform with respect to time , the Hankel transform with respect to the radial coordinate , the finite Fourier transform with respect to the angular coordinate , and the exponential Fourier transform with respect to the spatial coordinate . Numerical results are illustrated graphically.