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Electron J Math Anal Appl. Author manuscript; available in PMC 2015 September 28. Published in final edited form as: Electron J Math Anal Appl. 2013 January ; 1(1): 55–66.
THE FUNDAMENTAL SOLUTIONS FOR MULTI-TERM MODIFIED POWER LAW WAVE EQUATIONS IN A FINITE DOMAIN H. Jiang*, F. Liu**, M. M. Meerschaert***, and R. J. McGough**** H. Jiang: [email protected]; F. Liu: [email protected] *Department
of Mathematical, Qinghai Normal University, Xining 810008, China
**Mathematical
Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld.
Author Manuscript
4001, Australia ***Department
of Statistics and Probability, Michigan State University, East Lansing, Michigan
48824 ****Department
of Electrical and Computer Engineering, Michigan State University, East Lansing, Michigan 48824
Abstract Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development.
Author Manuscript
In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n) (n > 2), respectively. Analytical solutions of the multiterm modified power law wave equations are derived. These new techniques are based on Luchko’s Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi-term time-space fractional models including fractional Laplacian.
Keywords
Author Manuscript
Analytical solutions; the multi-term time-space fractional wave equations; Szabo wave equation; power law wave equation; Dirichlet boundary conditions
1. INTRODUCTION In recent years, a growing number of works by many authors from various fields of science and engineering deal with dynamical systems described by fractional partial differential equations (see Podlubny (1999); Liu et al (2004, 2007). These new models improve on the previously used integer-order models. Generalized fractional partial differential equations have been used for describing important physical phenomena Metzler (2000). However,
Jiang et al.
Page 2
Author Manuscript
studies of the multi-term time and space fractional wave equations are still under development. The effect of the attenuation plays a prominent role in many acoustic and ultrasound applications, for instance, the ultrasound second harmonic imaging and high intensity focused ultrasound beam for therapeutic surgery. The attenuation coefficient for biological tissue may be approximated by a power law (see Duck (1990)) over a wide range of frequencies used in ultrasonic imaging and thermal therapy applications. Frequencydependent loss and dispersion are typically modeled with a power-law attenuation coefficient, where the power-law exponent ranges from 0 to 2. To facilitate analytical solution, a fractional partial differential equation is derived that exactly describes power-law attenuation and the Szabo wave equation (see Szabo (1998)) is an approximation to this equation.
Author Manuscript
Measurements on the changes in phase velocity with frequency have been reported for at least 70 years, and much of this data, together with the results of more recent measurements, have been reported by Szabo (1995). He compared the experimental results with those predicted from absorption measurements based on causal dispersion relations. The Szabo wave equation was originally presented as an integro-differential equation for fractional power law media.
Author Manuscript
Stojanovic (2011) found solutions for the diffusion-wave problem in 1D with n-term time fractional derivatives whose orders belong to the intervals (0, 1), (1, 2) and (0, 2) respectively, using the method of the approximation of the convolution by Laguerre polynomials in the space of tempered distributions. Time domain wave-equations for lossy media obeying a frequency power-law (see Kelly et al (2008)). Mathematically, the powerlaw frequency dependence of the attenuation coefficient cannot be modeled with standard dissipative partial differential equations with integer-order derivatives. However, the multi-term time-space fractional wave equations successfully capture this power-law frequency dependence. Kelly et al (2008) considered a power law wave equation: (1)
The power law wave equation with the n-term time fractional derivatives whose orders belong to the intervals [2, 3) (if 1 < y < 3/2) and [2, 4) (if 3/2 < y < 2), respectively. The
Author Manuscript
Riemann-Liouville fractional derivative as (see Podlubny (1999)):
of order y (0 ≤ m − 1 < y < m) is defined
Kelly et al (2008) also considered the Szabo wave equation:
Electron J Math Anal Appl. Author manuscript; available in PMC 2015 September 28.
Jiang et al.
Page 3
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The Szabo wave equation with the n-term time fractional derivatives whose orders belong to the intervals (1, 2] (if 0 < y < 1) and [2, 3) (if 1 < y < 2), respectively. The third term is defined by the Riemann-Liouville fractional derivative of order y + 1.
Author Manuscript
The generalized time-fractional diffusion equation corresponds to a continuous time random walk model where the characteristic waiting time elapsing between two successive jumps diverge, but the accumulated jump length variance remains finite and is proportional to tα. The exponent α of the mean square displacement proportional to tα often does not remain constant and changes. To adequately describe these phenomena with fractional models, multi-term modified power law wave equations and several approaches have been suggested in the literature (for example, Coimdra (2003); Gorenflo et al (1998, 2009); Lorenzo et al (2002); Luchko (2010, 2011); Metzler (2000); Metzler (2004); Pedro et al (2008); Jiang et al (2012)). However, studies of the multi-term modified power law wave equations are still limited.
Author Manuscript
In this paper, a method of separating variables is used to solve the multi-term modified power law wave equations in a finite domain. It is found that these models can be recast with the Riemann-Liouville fractional derivative or Caputo fractional derivative for noninteger power. Then, the basic strategy of this study is that the Riemann-Liouville fractional derivative is replaced by the Caputo fractional derivative to derive a modified Szabo model, where the initial conditions can be easily prescribed. We discuss and derive the analytical solutions of these equations with nonhomogeneous Dirichlet boundary conditions. The rest of this paper is organized as follows. In section 2, we give some relevant definitions and lemmas. The analytical solutions of the multi-term modified power law wave equations with nonhomogeneous Dirichlet boundary conditions are derived in sections 3. Some special cases are considered in section 4. Finally, we summarise the main research findings of our work in section 5.
2. BACKGROUND THEORY
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For convenience, we introduce the following definitions and theorems, which are used throughout this paper. A modified time fractional wave equation can be written in the following form: (2)
where x and t are the space and time variables, k is an arbitrary positive constant, , (Δ) is Laplacian, f(x, t) is a sufficiently smooth function, 0 < α ≤ 4 and is a Caputo fractional derivative of order α defined as Podlubny (1999)
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(3)
where m − 1 < α < m, α = m ∈ N. When 1 < α < 2, Eq. (2) is the time-fractional wave equation and when 0 < α < 1, Eq. (2) is a fractional diffusion equation. When α = 2, it represents a traditional wave equation; while if α = 1, it represents a traditional diffusion equation. In some practical situations the underlying processes cannot be described by Eq.(2), but by its modified multi-term time-fractional wave and diffusion equations that are given by Luchko (2011), namely
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(4)
where 0 < x < L, t > 0,
(5)
0 ≤ αn < … < αh1+1 ≤ 1 < αh1< · · · < αhm−1+1 ≤ r − 1 < αhm−1 < · · · < α1 < α ≤ m, and ai, i = 1, …, n, n ∈ N.
is a Caputo fractional derivative of order αi with respect to t.
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In this paper, we will discuss and derive the analytical solution of (4) with nonhomogeneous Dirichlet boundary conditions using the method of separation of variables, respectively. The initial conditions are
(6)
where 0 < x < L. Definition 1
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(See Dimovski (1982)). A real or complex-valued function f(x), x > 0, is said to be in the space Cα, α ∈ R, if there exists a real number p > α such that (7)
for f1(x) ∈ C[0, ∞].
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Definition 2
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(See Luchko (1999)). A function f(x), x > 0, is said to be in the space if and only if fm ∈ Cα.
, m ∈ N0 = N ∪ {0},
Definition 3 (See Luchko (1999)). A multivariate Mittag-Leffer function (n dimensional cases) is defined as
(8)
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in which b > 0, ai > 0, |zi| < ∞, i = 1, …, n. Lemma 1 Let μ > μ1 > · · · > μn ≥ 0, mi − 1 < μi ≤ mi, mi ∈ N0 = N ∪ {0}, λi ∈ R, i = 1, …, n. The initial value problem
(9)
where the function g(x) is assumed to lie in C−1, if μ ∈ N, in
Author Manuscript
function y(x) is to be determined in the space
, if μ ∉ N, and the unknown
, has the solution
(10)
where
(11)
and
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(12)
fulfills the initial conditions
, k, l = 0, …, m − 1. The function
(13)
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The natural numbers lk, k = 0, …, m − 1, are determined from the condition
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(14)
In the case mi ≤ k, i = 1, …, n, we set lk:= 0, and if mi ≥ k + 1, i = 1, …, n, then lk:= n. Proof (See Luchko (1999); Jiang et al (2012)). Definition 4 (see Ilić et al (2005)). Suppose the Laplacian (−Δ) has a complete set of orthonormal
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eigenfunction ϕn corresponding to eigenvalues
on a bounded region
on a bounded region ; (ϕ) = 0 on ∂ three homogeneous boundary conditions. Let
, i.e.,
, where (ϕ) is one of the standard
(15)
then for any f ∈
, (−Δ) is defined by
(16)
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Lemma 2 ( See Braun (1975); Ilić et al (2005)). Suppose the one-dimensional Laplacian (−Δ) has a complete set of orthonormal eigenfunctions ϕn1 corresponding to eigenvalues boundary region Ω = [0, L], if
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The eigenvalues are
on a
on Dirichlet boundary conditions, i.e.,
for n = 1, 2, · · · and the corresponding eigenfunctions are
nonzero constant multiples of
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3. A MODIFIED MULTI-TERM TIME POWER LAW WAVE EQUATION WITH NONHOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS Firstly, we consider the modified multi-term time power law wave equation (4) with the initial condition (6) and the Dirichlet boundary condition: (17)
where μ1(t), μ2(t) are nonzero smooth functions with order-one continuous derivatives. In order to solve the problem with nonhomogeneous boundary conditions, we firstly transform the nonhomogeneous condition into a homogeneous boundary condition. Let
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(18)
where
(19)
satisfies the boundary conditions
(20)
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and the function W1(x, t) is the solution of the problem:
(21)
with
(22)
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where 0 ≤ x ≤ L, t ≥ 0, i = 1, 2, · · ·, m − 1. Using Definition 4, we set
(23)
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where ϕn1 are eigenfunctions that given in Lemma 2 corresponding to eigenvalues . We expand f1(x, t) in a Fourier series in the interval [0, L] by the eigenfunctions sin λn1x, namely
(24)
where
(25)
where
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(26)
Using the initial condition
(27)
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We assume that φ1(x), ψ1(x), ω1(x) are continuous functions with one-order derivative. Multiplying (27) by sin λn1x, and integrating from 0 to L respect x, we obtain
(28)
where bn is given in (26). Based on Definition 4 and substituting (23) and (25) into (21) gives the following equation: (29)
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where λn1 are given in Lemma 2. According to Lemma 1, we have
(30)
in which
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(31)
(32)
in which s = 1, 2, · · ·, m − 1, and are given in (28). Therefore we obtain the analytical solution of the modified multi-term time power law wave equation with Dirichlet boundary condition as
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(33)
where cn1 is given in (30), in which cn1(0) and
are given in (28).
4. SPECIAL CASES In this section, we give some special cases using the results in Section 3. Case 1 We consider a modified power law wave equation:
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(34)
where the fractional derivatives are defined by the Caputo fractional derivatives and whose ). The initial conditions are given by (6) with orders belong to the interval [2,3) (if m = 3 and boundary conditions are given by (17). Firstly, we consider the interval [2,3) (if
).
We rewrite (34) as the following form: (35)
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Let
, then Eq. (35) becomes
(36)
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Comparing Eq. (36) with Eq. (4) (let a3 = · · · = an = 0, α = 2y, α1 = y + 1, α2 = 2, f (x, t) = 0), we obtain the analytical solution of (36) with initial conditions (6) and Dirichlet boundary conditions (17) as
(37)
in which
(38)
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(39)
where
(40)
(41)
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(42)
(43)
, V (x, t) are given in (26) and(19), cn1(0), (28).
and
are given in
Case 2
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Similarly, we obtain the analytical solution of the modified power law wave equation (34) in the interval [2,4) (if ). The initial conditions are given by (6) with m = 4 and boundary conditions are given by (17).
(44)
in which
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(45)
(46)
(47)
(48)
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(49)
is same as are given in (28).
, cn1(0),
and
Case 3 We consider the following modified Szabo wave equation:
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whose orders belong to the interval (2,3] ( if 1 < y ≤ 2). The initial conditions are given by (6) with m = 3 and boundary conditions are given by (17). Similarly, we rewrite this equation as
(50)
where
.
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Comparing (50) with Eq. (4) (let a2 = · · · = an = 0, α = y + 1, α1 = 2, f (x, t) = 0 ), we obtain the analytical solution of (50) with initial conditions (6) with m = 3 and Dirichlet boundary conditions (17) as
(51)
in which Electron J Math Anal Appl. Author manuscript; available in PMC 2015 September 28.
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(52)
(53)
(54)
(55)
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(56)
(57)
, V (x, t) are given in (26) and (19), cn1(0), where given in (28).
and
are
Case 4
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The time-fractional telegraph equation: (58)
(59)
(60)
where 1/2 < α < 1. From Eq. (33), the analytical solution of time-fractional telegraph equation (58), (59) and (60) can be written as the following form:
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(61)
in which
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(62)
(63)
(64)
(65)
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(66)
where , V(x, t) are given in (26) and(19), cn1(0), given in (28).
and
are
5. CONCLUSIONS
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In this paper, we have proposed some new solution techniques to derive analytic solutions to multi-term modified power law wave equations with nonhomogeneous Dirichlet boundary conditions in a finite domain. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n)(n > 2), respectively. These techniques are based on Luchko’s Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. By using these techniques, analytical solutions of the modified power law wave equations and a modified Szabo wave equation are also derived. These methods and techniques can also be extended to other kinds of the multi-term time-space fractional models including fractional Laplacian.
Acknowledgments The first author wishes to acknowledge the funding received by the China Scholarship Council to support this work. They also thank QUT for the resources made available to them during the visit. Research of MMM was partially supported by NSF grants DMS-1025486, DMS-0803360, and NIH grant R01-EB012079. RJM was supported in part by NIH grant R01-EB012079.
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References Braun, M. Applied Mathematical Sciences. Vol. 15. Springer-Verlag; New York, Heidelberg, Berlin: 1975. Differential equations and their applications. Chen W, Zhang X, Cai X. A study on modified Szabo’s wave equation modeling of frequencydependent dissipation in ultrasonic medical imaging. J Acoust Soc Am. 2009; 114(5):2570–2574. [PubMed: 14649993] Coimdra CFM. Mechanics with variable-order differential opearators. Annalen der Physik. 2003; 12:692–703.
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Dimovski, IH. Convolution calculus. Bulgarian Academy of Science; Sofia: 1982. Duck, FA. Physical Properties of Tissue. 1. Academic Press; London: 1990. p. 99-124. Gorenflo R, Mainardi F. Signalling problem and Dirichlet-Neumann map for time-fractional diffusionwave equation. Matimyas Matematika. 1998; 21:109–118. Gorenflo R, Mainardi F. Some recent advances in theory and simulation of fractional diffusion processes. J Comp Appl Math. 2009; 299:400–415. Ilić M, Liu F, Turner I, Anh V. Numerical approximation of a fractional-in-space diffusion equation. Fractional Calculus and Applied Analysis. 2005; 8(3):323–341. Jiang H, Liu F, Turner I, Burrage K. Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain. J Math Anal Appl. 2012; 389:1117– 1127. Kelly JF, McGough RJ, Meerschaert MM. Analytical time-domain Green’s function for power-law media. J Acoust Soc Am. 2008; 124(5):2861–2872. [PubMed: 19045774] Liu F, Anh V, Turner I. Numerical solution of the space Fokker-Planck equation. J Comput Appl Math. 2004; 166:290–219. Liu F, Zhuang P, Anh V, Turner I, Burrage K. Stability and convergence of difference methods for the space-time fractional advection-diffusion equation. Appl Math Comp. 2007; 191:12–20. Lorenzo CF, Hartley TT. Variable order and distributed order fractional operators. Nonlinear Dynamics. 2002; 29:57–98. Luchko, Y. Maximum Principle for the fractional differential equations and its application. Proceedings of FDA’10, The 4th IFAC Workshop Fractional Differentiation and Applications; Badajoz, Spain. October 18–22, 2010; Luchko Y. Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. J Math Anal Appl. 2011; 374:538–548. Luchko Y, Gorenflo R. An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math Vietnam. 1999; 24:207–233. Metzler R, Klafter J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep. 2000; 339:1–77. Metzler R, Klafter J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J Phys A: Math Gen. 2004; 37:161– 208. Pedro HTC, Kobayashi MH, Pereira JMC, Coimbra CFM. Variable order modelling of diffusiveconvective effects on the oscillatory flow past a sphere. Journal of Vibration and Control. 2008; 14:1659–1672. Podlubny, I. Fractional differential equations. Academic Press; San Diego: 1999. Szabo TL. Time-domain wave-equations for lossy media obeying a frequency power-law. J Acoust Soc Am. 1994; 96:491–500. Szabo TL. Causal theories and data for acoustic attenuation obeying a frequency power law. J Acoust Soc Am. 1995; 97:1424. Stojanovic M. Numerical method for solving diffusion-wave phenomena. Journal of Computational and Applied Mathematics. 2011; 235(10, 15):3121–3137.
Author Manuscript Electron J Math Anal Appl. Author manuscript; available in PMC 2015 September 28.
Electron J Math Anal Appl. Author manuscript; available in PMC 2015 September 28. Published in final edited form as: Electron J Math Anal Appl. 2013 January ; 1(1): 55–66.
THE FUNDAMENTAL SOLUTIONS FOR MULTI-TERM MODIFIED POWER LAW WAVE EQUATIONS IN A FINITE DOMAIN H. Jiang*, F. Liu**, M. M. Meerschaert***, and R. J. McGough**** H. Jiang: [email protected]; F. Liu: [email protected] *Department
of Mathematical, Qinghai Normal University, Xining 810008, China
**Mathematical
Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld.
Author Manuscript
4001, Australia ***Department
of Statistics and Probability, Michigan State University, East Lansing, Michigan
48824 ****Department
of Electrical and Computer Engineering, Michigan State University, East Lansing, Michigan 48824
Abstract Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development.
Author Manuscript
In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n) (n > 2), respectively. Analytical solutions of the multiterm modified power law wave equations are derived. These new techniques are based on Luchko’s Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi-term time-space fractional models including fractional Laplacian.
Keywords
Author Manuscript
Analytical solutions; the multi-term time-space fractional wave equations; Szabo wave equation; power law wave equation; Dirichlet boundary conditions
1. INTRODUCTION In recent years, a growing number of works by many authors from various fields of science and engineering deal with dynamical systems described by fractional partial differential equations (see Podlubny (1999); Liu et al (2004, 2007). These new models improve on the previously used integer-order models. Generalized fractional partial differential equations have been used for describing important physical phenomena Metzler (2000). However,
Jiang et al.
Page 2
Author Manuscript
studies of the multi-term time and space fractional wave equations are still under development. The effect of the attenuation plays a prominent role in many acoustic and ultrasound applications, for instance, the ultrasound second harmonic imaging and high intensity focused ultrasound beam for therapeutic surgery. The attenuation coefficient for biological tissue may be approximated by a power law (see Duck (1990)) over a wide range of frequencies used in ultrasonic imaging and thermal therapy applications. Frequencydependent loss and dispersion are typically modeled with a power-law attenuation coefficient, where the power-law exponent ranges from 0 to 2. To facilitate analytical solution, a fractional partial differential equation is derived that exactly describes power-law attenuation and the Szabo wave equation (see Szabo (1998)) is an approximation to this equation.
Author Manuscript
Measurements on the changes in phase velocity with frequency have been reported for at least 70 years, and much of this data, together with the results of more recent measurements, have been reported by Szabo (1995). He compared the experimental results with those predicted from absorption measurements based on causal dispersion relations. The Szabo wave equation was originally presented as an integro-differential equation for fractional power law media.
Author Manuscript
Stojanovic (2011) found solutions for the diffusion-wave problem in 1D with n-term time fractional derivatives whose orders belong to the intervals (0, 1), (1, 2) and (0, 2) respectively, using the method of the approximation of the convolution by Laguerre polynomials in the space of tempered distributions. Time domain wave-equations for lossy media obeying a frequency power-law (see Kelly et al (2008)). Mathematically, the powerlaw frequency dependence of the attenuation coefficient cannot be modeled with standard dissipative partial differential equations with integer-order derivatives. However, the multi-term time-space fractional wave equations successfully capture this power-law frequency dependence. Kelly et al (2008) considered a power law wave equation: (1)
The power law wave equation with the n-term time fractional derivatives whose orders belong to the intervals [2, 3) (if 1 < y < 3/2) and [2, 4) (if 3/2 < y < 2), respectively. The
Author Manuscript
Riemann-Liouville fractional derivative as (see Podlubny (1999)):
of order y (0 ≤ m − 1 < y < m) is defined
Kelly et al (2008) also considered the Szabo wave equation:
Electron J Math Anal Appl. Author manuscript; available in PMC 2015 September 28.
Jiang et al.
Page 3
Author Manuscript
The Szabo wave equation with the n-term time fractional derivatives whose orders belong to the intervals (1, 2] (if 0 < y < 1) and [2, 3) (if 1 < y < 2), respectively. The third term is defined by the Riemann-Liouville fractional derivative of order y + 1.
Author Manuscript
The generalized time-fractional diffusion equation corresponds to a continuous time random walk model where the characteristic waiting time elapsing between two successive jumps diverge, but the accumulated jump length variance remains finite and is proportional to tα. The exponent α of the mean square displacement proportional to tα often does not remain constant and changes. To adequately describe these phenomena with fractional models, multi-term modified power law wave equations and several approaches have been suggested in the literature (for example, Coimdra (2003); Gorenflo et al (1998, 2009); Lorenzo et al (2002); Luchko (2010, 2011); Metzler (2000); Metzler (2004); Pedro et al (2008); Jiang et al (2012)). However, studies of the multi-term modified power law wave equations are still limited.
Author Manuscript
In this paper, a method of separating variables is used to solve the multi-term modified power law wave equations in a finite domain. It is found that these models can be recast with the Riemann-Liouville fractional derivative or Caputo fractional derivative for noninteger power. Then, the basic strategy of this study is that the Riemann-Liouville fractional derivative is replaced by the Caputo fractional derivative to derive a modified Szabo model, where the initial conditions can be easily prescribed. We discuss and derive the analytical solutions of these equations with nonhomogeneous Dirichlet boundary conditions. The rest of this paper is organized as follows. In section 2, we give some relevant definitions and lemmas. The analytical solutions of the multi-term modified power law wave equations with nonhomogeneous Dirichlet boundary conditions are derived in sections 3. Some special cases are considered in section 4. Finally, we summarise the main research findings of our work in section 5.
2. BACKGROUND THEORY
Author Manuscript
For convenience, we introduce the following definitions and theorems, which are used throughout this paper. A modified time fractional wave equation can be written in the following form: (2)
where x and t are the space and time variables, k is an arbitrary positive constant, , (Δ) is Laplacian, f(x, t) is a sufficiently smooth function, 0 < α ≤ 4 and is a Caputo fractional derivative of order α defined as Podlubny (1999)
Electron J Math Anal Appl. Author manuscript; available in PMC 2015 September 28.
Jiang et al.
Page 4
Author Manuscript
(3)
where m − 1 < α < m, α = m ∈ N. When 1 < α < 2, Eq. (2) is the time-fractional wave equation and when 0 < α < 1, Eq. (2) is a fractional diffusion equation. When α = 2, it represents a traditional wave equation; while if α = 1, it represents a traditional diffusion equation. In some practical situations the underlying processes cannot be described by Eq.(2), but by its modified multi-term time-fractional wave and diffusion equations that are given by Luchko (2011), namely
Author Manuscript
(4)
where 0 < x < L, t > 0,
(5)
0 ≤ αn < … < αh1+1 ≤ 1 < αh1< · · · < αhm−1+1 ≤ r − 1 < αhm−1 < · · · < α1 < α ≤ m, and ai, i = 1, …, n, n ∈ N.
is a Caputo fractional derivative of order αi with respect to t.
Author Manuscript
In this paper, we will discuss and derive the analytical solution of (4) with nonhomogeneous Dirichlet boundary conditions using the method of separation of variables, respectively. The initial conditions are
(6)
where 0 < x < L. Definition 1
Author Manuscript
(See Dimovski (1982)). A real or complex-valued function f(x), x > 0, is said to be in the space Cα, α ∈ R, if there exists a real number p > α such that (7)
for f1(x) ∈ C[0, ∞].
Electron J Math Anal Appl. Author manuscript; available in PMC 2015 September 28.
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Page 5
Definition 2
Author Manuscript
(See Luchko (1999)). A function f(x), x > 0, is said to be in the space if and only if fm ∈ Cα.
, m ∈ N0 = N ∪ {0},
Definition 3 (See Luchko (1999)). A multivariate Mittag-Leffer function (n dimensional cases) is defined as
(8)
Author Manuscript
in which b > 0, ai > 0, |zi| < ∞, i = 1, …, n. Lemma 1 Let μ > μ1 > · · · > μn ≥ 0, mi − 1 < μi ≤ mi, mi ∈ N0 = N ∪ {0}, λi ∈ R, i = 1, …, n. The initial value problem
(9)
where the function g(x) is assumed to lie in C−1, if μ ∈ N, in
Author Manuscript
function y(x) is to be determined in the space
, if μ ∉ N, and the unknown
, has the solution
(10)
where
(11)
and
Author Manuscript
(12)
fulfills the initial conditions
, k, l = 0, …, m − 1. The function
(13)
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The natural numbers lk, k = 0, …, m − 1, are determined from the condition
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(14)
In the case mi ≤ k, i = 1, …, n, we set lk:= 0, and if mi ≥ k + 1, i = 1, …, n, then lk:= n. Proof (See Luchko (1999); Jiang et al (2012)). Definition 4 (see Ilić et al (2005)). Suppose the Laplacian (−Δ) has a complete set of orthonormal
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eigenfunction ϕn corresponding to eigenvalues
on a bounded region
on a bounded region ; (ϕ) = 0 on ∂ three homogeneous boundary conditions. Let
, i.e.,
, where (ϕ) is one of the standard
(15)
then for any f ∈
, (−Δ) is defined by
(16)
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Lemma 2 ( See Braun (1975); Ilić et al (2005)). Suppose the one-dimensional Laplacian (−Δ) has a complete set of orthonormal eigenfunctions ϕn1 corresponding to eigenvalues boundary region Ω = [0, L], if
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The eigenvalues are
on a
on Dirichlet boundary conditions, i.e.,
for n = 1, 2, · · · and the corresponding eigenfunctions are
nonzero constant multiples of
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3. A MODIFIED MULTI-TERM TIME POWER LAW WAVE EQUATION WITH NONHOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS Firstly, we consider the modified multi-term time power law wave equation (4) with the initial condition (6) and the Dirichlet boundary condition: (17)
where μ1(t), μ2(t) are nonzero smooth functions with order-one continuous derivatives. In order to solve the problem with nonhomogeneous boundary conditions, we firstly transform the nonhomogeneous condition into a homogeneous boundary condition. Let
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(18)
where
(19)
satisfies the boundary conditions
(20)
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and the function W1(x, t) is the solution of the problem:
(21)
with
(22)
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where 0 ≤ x ≤ L, t ≥ 0, i = 1, 2, · · ·, m − 1. Using Definition 4, we set
(23)
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where ϕn1 are eigenfunctions that given in Lemma 2 corresponding to eigenvalues . We expand f1(x, t) in a Fourier series in the interval [0, L] by the eigenfunctions sin λn1x, namely
(24)
where
(25)
where
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(26)
Using the initial condition
(27)
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We assume that φ1(x), ψ1(x), ω1(x) are continuous functions with one-order derivative. Multiplying (27) by sin λn1x, and integrating from 0 to L respect x, we obtain
(28)
where bn is given in (26). Based on Definition 4 and substituting (23) and (25) into (21) gives the following equation: (29)
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where λn1 are given in Lemma 2. According to Lemma 1, we have
(30)
in which
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(31)
(32)
in which s = 1, 2, · · ·, m − 1, and are given in (28). Therefore we obtain the analytical solution of the modified multi-term time power law wave equation with Dirichlet boundary condition as
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(33)
where cn1 is given in (30), in which cn1(0) and
are given in (28).
4. SPECIAL CASES In this section, we give some special cases using the results in Section 3. Case 1 We consider a modified power law wave equation:
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(34)
where the fractional derivatives are defined by the Caputo fractional derivatives and whose ). The initial conditions are given by (6) with orders belong to the interval [2,3) (if m = 3 and boundary conditions are given by (17). Firstly, we consider the interval [2,3) (if
).
We rewrite (34) as the following form: (35)
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Let
, then Eq. (35) becomes
(36)
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Comparing Eq. (36) with Eq. (4) (let a3 = · · · = an = 0, α = 2y, α1 = y + 1, α2 = 2, f (x, t) = 0), we obtain the analytical solution of (36) with initial conditions (6) and Dirichlet boundary conditions (17) as
(37)
in which
(38)
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(39)
where
(40)
(41)
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(42)
(43)
, V (x, t) are given in (26) and(19), cn1(0), (28).
and
are given in
Case 2
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Similarly, we obtain the analytical solution of the modified power law wave equation (34) in the interval [2,4) (if ). The initial conditions are given by (6) with m = 4 and boundary conditions are given by (17).
(44)
in which
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(45)
(46)
(47)
(48)
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(49)
is same as are given in (28).
, cn1(0),
and
Case 3 We consider the following modified Szabo wave equation:
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whose orders belong to the interval (2,3] ( if 1 < y ≤ 2). The initial conditions are given by (6) with m = 3 and boundary conditions are given by (17). Similarly, we rewrite this equation as
(50)
where
.
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Comparing (50) with Eq. (4) (let a2 = · · · = an = 0, α = y + 1, α1 = 2, f (x, t) = 0 ), we obtain the analytical solution of (50) with initial conditions (6) with m = 3 and Dirichlet boundary conditions (17) as
(51)
in which Electron J Math Anal Appl. Author manuscript; available in PMC 2015 September 28.
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(52)
(53)
(54)
(55)
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(56)
(57)
, V (x, t) are given in (26) and (19), cn1(0), where given in (28).
and
are
Case 4
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The time-fractional telegraph equation: (58)
(59)
(60)
where 1/2 < α < 1. From Eq. (33), the analytical solution of time-fractional telegraph equation (58), (59) and (60) can be written as the following form:
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(61)
in which
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(62)
(63)
(64)
(65)
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(66)
where , V(x, t) are given in (26) and(19), cn1(0), given in (28).
and
are
5. CONCLUSIONS
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In this paper, we have proposed some new solution techniques to derive analytic solutions to multi-term modified power law wave equations with nonhomogeneous Dirichlet boundary conditions in a finite domain. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n)(n > 2), respectively. These techniques are based on Luchko’s Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. By using these techniques, analytical solutions of the modified power law wave equations and a modified Szabo wave equation are also derived. These methods and techniques can also be extended to other kinds of the multi-term time-space fractional models including fractional Laplacian.
Acknowledgments The first author wishes to acknowledge the funding received by the China Scholarship Council to support this work. They also thank QUT for the resources made available to them during the visit. Research of MMM was partially supported by NSF grants DMS-1025486, DMS-0803360, and NIH grant R01-EB012079. RJM was supported in part by NIH grant R01-EB012079.
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