His solution depends upon using the finite sine transform technique to convert fractional diffusion wave equation from a space domain to a wave. Error analysis of a finite difference method on graded. Numerical algorithm based on bernstein polynomials for. Solving fractional diffusion and wave equations by modified homotopy perturbation method. A note on the inverse problem for a fractional parabolic equation erdogan, abdullah said and uygun, hulya, abstract and applied analysis, 2012. Fractional diffusion equation can be derived from the continuoustime random walk ctrw. In this paper, we consider cauchy problem of spacetime fractional diffusion wave equation. The laplace transform with respect to time and the finite sinfourier transform with respect to the spatial coordinate are employed. Research article sincchebyshev collocation method for a class of fractional diffusionwave equations zhimao, 1,2 aiguoxiao, 1 zuguoyu, 1 andlongshi 1 hunan key laboratory for computation and simulation in science and engineering and key. Pdf an approximate solution for a fractional diffusionwave. Solving fractional diffusion and wave equation by modified. Solving linear and nonlinear fractional diffusion and wave equations by adomian decomposition. Numerical methods for partial differential equations 34.
Pdf solving linear and nonlinear fractional diffusion. For some anomalous diffusive subdiffusive particles, the pdf of finding them at a given place at a given time follows a fractional i. Highaccuracy finite element method for 2d time fractional diffusionwave equation on anisotropic meshes. Fractional diffusion and wave equations are obtained by letting. Mainardi and others published on the initial value problem for the fractional diffusionwave equation find, read and cite all the. Manohar, numerical solution of fractional diffusion wave equation with two space variables by matrix method, fractional calculus and applied analaysis, 2 2010 191207. A bspline collocation method for solving fractional diffusion and fractional diffusionwave equations esen, a. Fractional diffusionwave equation with distributions andrzej lopushansky, halyna lopushanska communicated by mokhtar kirane abstract. A bspline collocation method for solving fractional. Many authors tried to model diffusion and wave equations from the.
We prove some properties of its solution and give some examples. Two fully discrete schemes for fractional diffusion and. The secondorder accuracy in the time direction has not been achieved in previous studies. Diffusion and wave equations together with appropriate initial conditions are rewritten as integrodifferential equations with time derivatives replaced by convolution with t1. This book systematically presents solutions to the linear timefractional diffusionwave equation. From newtons equation to fractional diffusion and wave. The main purpose of this paper is to derive numerical solutions of the multiterm time fractional wave diffusion equations with nonhomogeneous dirichlet boundary conditions. We prove a jump relation and solve an integral equation for an unknown. Fractionalorder diffusionwave equation springerlink. Povstenko 1,2 1 institute of mathematics and computer science, jan dlugosz university, 42200 cze.
A highorder adi scheme for the twodimensional time. In this paper, a fractional generalization of the wave equation that describes propagation of damped waves is considered. Solutions to timefractional diffusionwave equation in. In contrast to the fractional diffusionwave equation, the fractional wave equation contains fractional derivatives of the same order. The time fractional diffusion equation with appropriate initial and boundary conditions in an ndimensional wholespace and halfspace is considered. We consider initial valueboundary value problems for fractional diffusionwave equation. A wavelet approach for the multiterm time fractional. The multiterm time fractional derivatives are defined in the caputo sense, whose orders belong to the intervals 0,1, 1,2, 0,2, 0,3, 2,3 and 2,4, respectively. For the fractional diffusion equation, the l1 discretization formula of the fractional derivative is employed, whereas the l2 discretization formula is used for the fractional diffusion wave equation. The time fractional diffusionwave equation springerlink.
In this paper, the multiterm timefractional wavediffusion equations are considered. We consider initialboundary value problems for the subdiffusion and diffusion wave equations involving a caputo fractional derivative in time. Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator d,and of the integration operator j. Stochastic processes related to the timefractional diffusionwave equation introduction the fundamental solution and the mwright function the fundamental solution gx. Illustrative examples are solved to demonstrate the efficiency of the method. The time fractional diffusion equation with mass absorption in a sphere is considered under harmonic impact on the surface of a sphere. Approximation of fractional diffusion wave equation 68 fig. Numerical solutions of the multiterm timefractional wavediffusion equations mttfwde with the fractional orders lying in 0, nn 2 are still limited. Linear fractional diffusionwave equation for scientists and.
An explicit difference method is considered for solving fractional diffusion and fractional diffusion wave equations where the time derivative is a fractional derivative in the caputo form. The problem is to nd a solution continuous in time in generalized sense of the direct. Mitkowski, approximation of fractional diffusion wave equation, acta mechanica et automatica, 5 2011 6568. Nonaxisymmetric solutions to time fractional diffusion wave equation with a source term in cylindrical coordinates are obtained for an infinite medium. It is known see1 that the fundamental solution of the distributed order fractional diffusion equation. 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. The secondkind chebyshev wavelets collocation method is applied for solving a class of time fractional diffusion wave equation. Concluding remarks in this paper we consider the selection of the fractional differential equations. Boundary value problems for fractional diffusionwave equation article pdf available in australian journal of mathematical analysis and applications 31 january 2006 with 524 reads.
Fractional wave equations with attenuation have been proposed by caputo, szabo, chen and holm, and kelly et al. Fractional diffusion equation an overview sciencedirect. Research article numerical algorithms for the fractional diffusionwave equation with reaction term hengfeidingandchangpinli department of mathematics, shanghai university, shanghai, china. Linear fractional diffusionwave equation for scientists. Identifying a fractional order and a space source term in. Numerical solution of fractional diffusionwave equation with two space variables by matrix method mridula garg, pratibha manohar abstract in the present paper we solve spacetime fractional di. An explicit difference method is considered for solving fractional diffusion and fractional diffusionwave equations where the time derivative is a fractional derivative in the caputo form. It introduces the integral transform technique and discusses the properties of the mittagleffler. Existence of solution of spacetime fractional diffusionwave. Mitkowski approximation of fractional diffusion wave equation acta mechanica et automatica 5 2011 6568. The fundamental solution of the distributed order fractional wave equation in one space dimension is a probability density introduction fourierlaplace solution remains to show that always and everywhere gx. The corresponding greens functions are obtained in closed form for arbitrary space dimensions in terms of fox functions and their properties are exhibited. Solutions to timefractional diffusionwave equation in spherical coordinates 108 solutions to timefractional diffusionwave equatio n in spherical coordinates yuriy povstenko, institute of mathematics and computer science, jan dlugosz university in cz estochowa, al.
In this paper, a compact alternating direction implicit finite difference scheme for the twodimensional time fractional diffusionwave equation is developed, with temporal and spatial accuracy order equal to two and four, respectively. Solving fractional diffusion and wave equation by modified homotopy perturbation method article in physics letters a 37056. Some computationally effective numerical methods are proposed for simulating the multiterm timefractional wavediffusion. The green function is sought in terms of a doublelayer potential of the equation under consideration. In this paper, a galerkin method based on the second kind chebyshev wavelets skcws is established for solving the multiterm time fractional diffusionwave equation. Recall from chapter 8 that ctrw is a random walk that permits intervals between successive walks to be independent and identically distributed. Mahdy an efficient numerical method for solving the fractional diffusion equation journal of applied mathematics and bioinformatics 2 2011 112. And in the studies available, the accuracy of the temporal direction for the subdi usion and fractional di usion wave equations is o en less than. Research article numerical algorithms for the fractional.
Timefractional diffusionwave equation with mass absorption. Fractional integral formula of a single chebyshev wavelet in the riemannliouville sense is derived by means of shifted chebyshev polynomials of the second kind. We study the inverse cauchy problem for a time fractional di usionwave equation with distributions in righthand sides. Solution for a fractional diffusionwave equation defined in a. When one solves this equation by means of standard finite difference methods, the cpu time and computer memory consumption scale as time. Pdf solving fractional diffusion and wave equations by. Boundary value problems for fractional diffusionwave equation. In this article, the adomian decomposition method has been used to obtain solutions of fourth. Moreover, convergence and accuracy estimation of the secondkind chebyshev wavelets expansion of two. Fractional diffusion wave equation with distributions andrzej lopushansky, halyna lopushanska communicated by mokhtar kirane abstract.
Timespace fractional diffusion equation models anomalous diffusion processes in complex media can be well characterized by using fractional order diffusion equation models. Applying laplace transform and fourier transform, we establish the existence of solution in terms of mittagleffler function and prove its uniqueness in weighted sobolev space by use of mikhlin multiplier theorem. The time fractional diffusionwave equation is obtained from the classical diffusion or wave equation by replacing the first or secondorder time derivative by a fractional derivative of order 2. Pdf boundary value problems for fractional diffusionwave. Pdf an explicit difference method for solving fractional. The fundamental solutions for the fractional diffusionwave equation. A graphical representation of the obtained analytical solution for different. It introduces the integral transform technique and discusses the properties of the mittagleffler, wright, and mainardi functions that appear in the solutions. Numerical solutions of the multiterm time fractional wave diffusion equations mttfwde with the fractional orders lying in 0, nn 2 are still limited. A very important model is the fractional diffusion and wave equations. Its solution has been obtained in terms of green functions by schneider and wyss. Initial valueboundary value problems for fractional.
Stochastic processes related to the timefractional. For the fractional diffusion equation, the l1 discretization formula of the fractional derivative is employed, whereas the l2 discretization formula is used for the fractional diffusionwave. Nonaxisymmetric solutions to timefractional diffusionwave 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. The fundamental solution of the distributed order fractional.
Green functions of the first boundaryvalue problem for a. This book systematically presents solutions to the linear time fractional diffusion wave equation. The main purpose of this paper is to derive numerical solutions of the multiterm timefractional wavediffusion equations with nonhomogeneous dirichlet boundary conditions. Recall from chapter 8 that ctrw is a random walk that permits intervals between successive walks. The estimate of solution also shows the connections between the loss of regularity and the. The fractional diffusionwave equation is obtained by replacing the. Two numerical algorithms are derived to compute the fractional di usionwave equation with a reaction term. The fractionalorder diffusionwave equation is an evolution equation of order. In this paper, a compact alternating direction implicit finite difference scheme for the twodimensional time fractional diffusion wave equation is developed, with temporal and spatial accuracy order equal to two and four, respectively. The time fractional diffusionwave equation is obtained from the classical diffusion or wave equation by replacing the first or secondorder time derivative by a. The time fractional diffusion equation and the advection. Mar 01, 20 fractional wave equations with attenuation have been proposed by caputo, szabo, chen and holm, and kelly et al. Mitkowski approximation of fractional diffusionwave equation acta mechanica et automatica 5 2011 6568. Pdf nonhomogeneous fractional diffusionwave equation has been solved under linearnonlinear boundary conditions.
We show that this feature is a decisive factor for inheriting some crucial. Identifying a fractional order and a space source term in a. Numerical solution of timefractional diffusionwave. Pdf boundary value problems for fractional diffusionwave equation.
An approximate solution for a fractional diffusion wave equation using the decomposition method. We study the inverse cauchy problem for a time fractional di usion wave equation with distributions in righthand sides. The considerations have been illustrated by examples. Some efficient numerical schemes are proposed to solve onedimensional and twodimensional multiterm time fractional diffusionwave equation, by combining the compact difference approach for the spatial discretisation and an l1 approximation for the multiterm time caputo fractional derivatives. Superconvergence of a discontinuous galerkin method for. To do this, a new operational matrix of fractional integration for the skcws must be derived and in order to improve the computational efficiency, the hat functions. Agarwal 3 presented a general solution for a time fractional diffusion wave equation defined in a bounded space domain. The fundamental solution of a diffusionwave equation of. In the present paper, we extend the matrix method to solve spacetime fractional diffusionwave equations with two space. For the first and the second derivative terms, these expressions reduce to the ordinary diffusion and wave solutions. Research article sincchebyshev collocation method for a. Superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations.
Efficient numerical solution of the multiterm time. Advances in difference equations hindawi publishing corporation from newtons equation to fractional diffusion and wave equations luis va. Adaptive finite difference method with variable timesteps. These equations capture the powerlaw attenuation with frequency observed in many experimental settings when sound waves travel through inhomogeneous media. Solutions to timefractional diffusionwave equation in cylindrical coordinates y. We define a new fractional calculus negativedirection fractional calculus and study some of its properties. The considerations have been illustrated by a numerical examples. Numerical solution of fractional diffusion wave equation. We develop two fully discrete schemes based on the pi. Pdf on the initial value problem for the fractional diffusionwave. The corresponding greens functions are obtained in closed form for.
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