Applications of the Sumudu transform to fractional

NONLINEAR STUDIES - www.nonlinearstudies.com Vol. 18, No. 1, pp. 99-112, 2011 c CSP - Cambridge, UK; I&S - Florida, USA, 2011 °

Applications of the Sumudu transform to fractional differential equations Qutaibeh Deeb Katatbeh 1 , Fethi Bin Muhammad Belgacem 2,∗ 1

Department of Mathematics and Statistics Jordan University of Science and Technology Irbid, 22110 Jordan. E-mail: [email protected] 2

Department of Mathematics Faculty of Basic Sciences, PAAET Shaamyia Block 9, Wahran St, Kuwait. E-mail: [email protected] ?

Corresponding Author. E-mail address: [email protected]

Abstract. In this work, Sumudu transform applications are extended to fractional integrals and derivatives. A table of a hundred instances of basic and special functions fractional integrals sumudi is provided. Some Sumudu properties are either generalized, or newly established. The Sumudu operator is then shown to help solve wide classes of fractional differential equations.

1 Introduction: Sumudu transform, fractional integrals and derivatives. In this paper, we apply the Sumudu transform to fractional integrals, derivatives, and use it to solve initial value fractional differential equations. In [1-7, 9, 16-19, 21-25], the authors studied many properties of the Sumudu transform in light of which they developed efficient and straightforward methodologies for treating ordinary and partial differential equations. There is evident interest in further studying this transform, and applying it to various mathematical and physical sciences problems. Being linear and bilateral, with scale and unit preserving properties [1-4, 7, 22-23], the Sumudu transform can be used to solve many types of difference and differential equations problems without resorting to a new frequency domain. Containing all functions of exponential order, possibly on both sides of the real line, the domain |t|

of admissible functions for the Sumudu transform, A = { f (t)/ ∃ M, t1 , t2 > 0, | f (t)| < Me t j , if 2010

Mathematics Subject Classification: 45A05, 44A15, 44A35, 44A99. Keywords: Fractional Derivatives, Fractional Differential Equations, Fractional Integrals, Mittag-Leffler functions, Sumudu, Sumudu Transform.

100

Qutaibeh Deeb Katatbeh, Fethi Bin Muhammad Belgacem

t ∈ (−1) j × [0, ∞)}, properly includes the domain of the Laplace transform, but is lager than it. On the other hand, the Sumudu transform, defined by, Z∞

f (ut)e−t dt, u ∈ (−t1 ,t2 ),

G(u) = S[ f (t)] =

(1.1)

0

is yielded from the Laplace transform by a simple variable change and a posterior division by the new variable, from which we have the (bilateral) Laplace-Sumudu Duality (LSD, [3,5, 7, 9]), F( 1u ) = uG(u), and, G( 1s ) = sF(s). While, F(s), stands for the Laplace transform of the function f (t), G(u), is the Sumudu transform, or simply the ”Sumudu” of f (t) (as connoted in [5]), with ”Sumudi” for the plural. The LSD is often used to translate, Laplace related results to Sumudu ones[1-3, 6-7, 17, 21-25]. Note that, G(0) = f (0), when defined. More generally, lim G (u) = lim f (t)), and, if f (t) is u→0

t→0

defined on the real interval, (c1 , c2 ), with c1 c2 < 0, where the numbers, c1 , c2 may be infinite, then lim G (u) = lim f (t).

u→±ti

t→±ci

The theory of fractional calculus plays an important role in many applied fields, such as modelling mechanical, electrical and rheological materials properties [10, 11, 13]. Viscoelasticity theories, and hereditary solid mechanics reinforced fractional calculus wide use [8, 12, 14, 15]. Fractional Calculus models rely on the concepts of fractional integrals, derivatives, and proper formulations of initial conditions. Fractional integrals and derivatives have various definitions and take various names in the literature, mostly because of variations in the initial conditions associated with them. While some are physically interpretable, others are simply more amenable to mathematical analysis. For t > 0, 0 < Re v ≤ 1, and f an integrable function on any subinterval of [0, ∞) and piecewise continuous on (0, ∞), the Riemann-Liouville fractional integral of f of order v, is defined by, D

−v

1 f (t) = Γ(v)

Zt

(t − ζ)v−1 f (ζ)dζ = 0

1 v−1 t ∗ f (t). Γ(v)

(1.2)

This integral is obviously improper. When it exists, Dt−v f (t), is then called the fractional integral of the function, f , of order v [12, 15]. The symbol, Γ, stands for the Gamma function, and, ∗, is the convolution operation. Taking n to be the smallest integer greater than Re w > 0, setting, v = n − w, implies that, 0 < Re v ≤ 1. So, the fractional derivative of f (t) of order w, is given by, Dw f (t) = Dn [D−v f (t)]

(1.3)

To solve some fractional calculus problems, we may need to use the Caputo fractional derivative, defined by, 1 D f (t) = Γ(w − n)

Zt

w

0

f (n) (τ) dτ, (n − 1 < w < n), (t − τ)w+1−n

(1.4)

while to solve others we may need sequential fractional derivatives of non-integer order α, generally defined by (see [15]), Dα f (t) = Dα1 Dα2 ...Dαn f (t) , with, α = α1 + α2 + ... + αn .

(1.5)

In particular, the Miller-Ross sequential fractional derivatives [12, 15], are defined by, Dσm = Dαm Dαm−1 ...Dα1 , with, σm = α1 + α2 + ... + αm where, 0 < α j ≤ 1, for, j = 1, 2, ..., m, (1.6)

Applications of the Sumudu transform to fractional differential equations

101

and such that, Dσm −1 = D−1 Dσm = Dαm −1 Dαm−1 ...Dα1 .

(1.7)

Many problems described by way of fractional differential equations have been solved in terms the Mittag-Leffler functions. The following three two-parameters functions, Ev,a (t) =

1 Cv,a (t) = Γ(v) and, 1 Sv,a (t) = Γ(v)

1 Γ(v)

Zt

ζv−1 ea(t−ζ) dζ = D−v eat , Re v > 0,

(1.8)

ζv−1 cos(a(t − ζ))dζ = D−v cos(at), Re v > 0,

(1.9)

ζv−1 sin(a(t − ζ))dζ = D−v sin(at), Re v > 0,

(1.10)

0

Zt 0

Zt 0

are respectively called the exponential, cosine, and sine Mittag-Leffler functions.

2 Sumudu transform of fractional integrals. To introduce the Sumdu transform to fractional differential equations, we have to generalize formulations for the Sumudu transforms of integer order derivatives, to those of fractional order. An attempt in this direction, albeit very brief, can be found in [17]. Here, we make results in this direction more concrete and focus on Sumudu theory developing and consequent applications. The first proofs of the following two results used the LSD [2, 7, 23, 25], and others used direct integrations methods [4-6]. Proposition 1. Let M(u) and N(u) be the Sumudi of f (t), and g(t), respectively, then the Sumudu of the convolution, Zt

( f ∗ g)(t) =

f (τ)g(t − τ)dτ,

(2.1)

0

is given by, S[( f ∗ g)(t)] = uM(u)N(u).

(2.2)

In particular, if g(t) u 1, then the Sumudu of the anti-derivative of the function, f (t), is Zt

S[( f ∗ 1)(t)] = S[

f (τ)dτ] = uM(u).

(2.3)

0

Theorem 2. If G(u) is the Sumudu of the function f (t), then the Sumudu of the nth derivative, f (n) (t), Gn (u), is given by, Gn (u) =

n−1 G(u) n−1 f (k) (0) −n − = u [G(u) − ∑ ∑ uk f k (0)], n > 1. n−k un u k=0 k=0

(2.4)

Now, we generalize Theorem 2 to fractional order integrals and derivatives, using Proposition 1.

102


Theorem 3. If G(u) is the Sumudu of f (t), then the Sumudu of the fractional integral of f (t) of order v, D−v ( f (t)), is given by S[D−v f (t)] = uv G(u), Re v > 0. (2.5) Proof. Since by equation (1.2) above, D−v ( f (t)) = 1.2 of [7], we have, S[D−v f (t)] =

1 v−1 ∗ Γ(v) t

f (t), then by Theorem 2, and Corollary

u S[t v−1 ]S[ f (t)] = uv S[ f (t)] = uv G(u). Γ(v)

(2.6) u t

This theorem implies that integrating a function, f (t), in the t-domain to a fractional order v, is akin to multiplication of its Sumudu by a fractional power of the variable u, of the same order. For instance, we have um+v−1 S[D−vt m−1 eat ] = Γ(m) , m > 0, v > 0. (2.7) (1 − au)m−1 Applying the theorem to the Mittag-Leffler functions, with, a > 0, yields the following, S[Ev,a (t)] = S[D−v eat ] = uv S(eat ) =

uv , u ε (−1/a, 1/a), v > 0, 1 − au

S[Cv,a (t)] = S[D−v cos(at)] = uv S[cos(at)] =

uv , v > 0, 1 + a2 u2

(2.8) (2.9)

and,

au1+v , v > 0. (2.10) 1 + a2 u2 We can then easily apply Theorem 3 to compute the Sumudu for basic and special functions fractional integrals, as is done for a hundred instances in the Appendix. S[Sv,a (t)] = S[D−v sin(at)] = uv S[sin(at)] =

3 Sumudu transform of fractional derivatives. So far, the Sumudu transform for the Riemann-Liouville fractional integrals was given in Theorem 3. In the same manner, we can extend these formulations to the Riemann-Liouville fractional derivatives (see [12, 15]). Theorem 4. If for a positive integer, n, n − 1 ≤ w < n, and G(u) is the Sumudu of the function f (t), then the Sumudu, Gw (u), of the Riemann-Liouville fractional derivative of f (t) of order w, Dw f (t), is given by n

Gw (u) = S[Dw f (t)] = u−w [G(u) − ∑ uw−k [Dw−k f (t)]t=0 ],

− 1 < n − 1 ≤ w < n.

(3.1)

k=1

Proof. Since the Laplace transform Fw (s) of the Riemann-Liouville fractional derivative, Dw f (t), (see eq. 2.248 in [15]), is given by n−1

n

k=0

k=1

Fw (s) = £[Dw f (t)] = sw F(s) − ∑ sk [Dw−k−1 f (t)]t=0 = sw F(s) − ∑ sk−1 [Dw−k f (t)]t=0 .

(3.2)


103

Hence, by substituting 1/u for the variable s, we get n

Fw (1/u) = (1/u)w F(1/u) − ∑ (1/u)k−1 [Dw−k f (t)]t=0 .

(3.3)

k=1

Now, by using the LSD, we get the claimed result Gw (u) = Fw (1/u)/u =

n [Dw−k f (t)]t=0 G(u) − . ∑ uw uk k=1

(3.4) u t

Note that the terms, [Dw−k f (t)]t=0 , have the look of initial conditions, but they are fractional in nature, and to our knowledge, unless, w, happens to be itself an integer, they have yet to be associated with a physical interpretation! Theorem 5. If for a positive integer n, n − 1 < w ≤ n , and G(u) is the Sumudu of the function f (t), then the Sumudu, GCw (u), of the Caputo fractional derivative of f (t) of order w, C Dw f (t) , is given by n

GCw (u) = S[C Dw f (t)] = u−w [G(u) − ∑ uk−1 [Dk−1 f (t)]t=0 ], − 1 < n − 1 < w ≤ n

(3.5)

k=1

Proof. Since the Laplace transform, FwC (s), of the Caputo fractional derivative, Dw f (t), (see eq. 2.253 in [15]), is given by n−1

n

k=0

k=1

FwC (s) = £[C Dw f (t)] = sw F(s) − ∑ sw−k−1 [Dk f (t)]t=0 = sw F(s) − ∑ sw−k [Dk−1 f (t)]t=0 ,

(3.6)

substituting the variable 1/u for s, we get n

FwC (1/u) = u−w F(1/u) − ∑ u−w+k [Dk−1 f (t)]t=0 .

(3.7)

k=1

Now, using the LSD, and rearranging terms, proves our claim n

GCw (u) = FwC (1/u)/u = u−w [G(u) − ∑ uk−1 [Dk−1 f (t)]t=0 ].

(3.8)

k=1

u t Note that the derivatives at t = 0, in the Riemann-Liouville case are fractional with no yet known physical meaning, while their Caputo counterparts are all integer order, and hence lend themselves to the usual calculus interpretations. This fact, renders the Sumudu transform a valuable tool for solving Caputo fractional differential equations. In the Riemann-Liouville setup this issue becomes more delicate, and may be a subject of study in future works. Now, we extend the previous results to the Miller-Ross sequential fractional derivatives. Theorem 6. Let G(u) be the Sumudu of the function f (t), then the Sumudu for the Miller-Ross sequential fractional derivatives of f (t) of order σm as defined by equations (1.6-7), is given by, S[Dσm f (t)] = u−σm [G(u) −

m−1

∑ uσ

m−k −1

[Dσm−k −1 f (t)]t=0 ]

(3.9)

k=0

where Dσm−k −1 = D−1 Dσm−k = Dαm−k −1 Dαm−1 ...Dα1 , k = 0, 1, ..., m − 1.

(3.10)

104


Proof. To prove the claim, the esteemed reader can easily see that once more, the LSD may be applied to the Laplace transform of the Miller-Ross sequential fractional derivative, (See eq. 2.259 in [15]), given by £[Dσm f (t)] = sσm F(s) −

m−1

∑ sσ

m −σm−k

[Dσm−k −1 f (t)]t=0,

(3.11)

k=0

u t Applying this theorem to the function, et , with σm = 3/2, setting α1 = α2 = α3 = 1/2, and noting that, (see [12] page 315), [D−1/2 et ]t=0 = [E1/2,1 (t)]t=0 = E0 (1/2, 1) = 0,

(3.12)

for yields, S[D3/2 et ] = u−3/2 [

2 1 1 − ∑ u−1/2 [D−1/2 et ]t=0 ] = 3/2 , u ∈ (−1, 1). 1 − u k=0 u (1 − u)

(3.13)

4 Solving fractional differential equations using the Sumudu transform. The Sumudu transform can serve as an alternate tool for solving fractional differential equations to the existing techniques. In this section we first prove the well-known existence and uniqueness theorem for the solutions of initial value fractional differential equations using this transform. Many authors have proved this result using other techniques, [12-15]. Proving it in light of this particular transform is an attempt to lend some autonomy to the Sumudu method. Rt

Theorem 7. If g(t) is continuous function on the interval, I = (0,t), such that, |g(t)| < ∞, then the 0

fractional differential equation,   

Dσm y(t) = g(t)

in I (4.1)

[Dσ j −1 y(t)]t=0 = b j , j = 1, ..., m,

has a unique solution y(t) in L1 (I). Proof. Let G(u), and, H(u), be the Sumudi of g(t) and, y(t), respectively, then by setting, j = m − k, and using Theorem 6, we have, S[Dσm y(t)] =

m H(u) m−1 [Dσm−k −1 y(t)]t=0 −σm = u [H(u) − − [Dσ j −1 y(t)]t=0 ]uσ j −1 = G(u), (4.2) ∑ ∑ σm −σm−k +1 uσm u j=1 k=0

from which we get m

m

j=1

j=1

H(u) = uσm G(u) + ∑ uσ j −1 [Dσ j −1 y(t)]t=0 = uσm G(u) + ∑ b j uσ j −1 .

(4.3)

We can now find the solution to the fractional differential equation (4.1) by Sumudu inversion in view of the previous results,


1 y(t) = Γ(σm )

Zt

105

m

b j σ j −1 t . j=1 Γ(σ j )

(t − τ)σm−1 g(τ)dτ + ∑ 0

(4.4)

Clearly, from equation (4.4) we see that, y(t) ∈ L1 (I). To prove the uniqueness part, we assume that equation (4.1) has two solution y1 (t) and y2 (t), then h(t) = y1 (t) − y2 (t) is a solution for the homogenous fractional differential equation, Dσm h(t) = 0, with zero valued initial conditions. Then, the Sumudu, H(u), is identically zero. Therefore, h(t) = 0, and hence, y1 (t) = y2 (t), on the given interval, I . So, the solution must be unique. u t As was done with the Laplace transform (see, [13]), this result may be extended to linear fractional differential equations such as, Dσm y(t) +

m−1

∑ ak (t)Dσ

m−k

y(t) + am (t)y(t) = f (t), [Dσ j−1 y(t)]t=0 = b j , for, j = 1, 2..., m.

(4.5)

k=1

In particular, for, 0 < b < a < n, the following fractional differential equation, Da f (t) + Db f (t) = h(t),

(4.6)

was studied by using many techniques[12,15]. Without loss of generality, we Sumudu-solve equation (4.6), only for the choice, 0 < b < a < 1, as other instances can be solved in a similar manner. Letting, G(u), and, H(u) be the respective Sumudi of f (t), and h(t), upon Sumudu transforming (4.6) we get the expression, G(u) [Da−1 ( f (t))]t=0 G(u) [Db−1 ( f (t))]t=0 − + b − = H(u), (4.7) ua u u u which upon rearranging terms becomes, G(u) =

([Da−1 ( f (t))]t=0 + [Db−1 ( f (t))]t=0 + uH(u)) . (u1−a + u1−b )

(4.8)

After simplifications, and using the relation, " (k) t αk+β−1 Eα,β (±ct α )

=S

−1

# k! , uαk+β−1 ( u1α ∓ c)k+1

(4.9)

where, Eα,β , is the two-parameter Mittag-Leffler exponential function in series form, ∞

Eα,β (t) = Et (α, β) =

tk

∑ Γ(αk + β) , (α > 0, β > 0).

(4.10)

k=0

Setting, α = a − b, β = a, and using Sumudu convolution, we obtain the solution of the fractional equation (4.6), Zt b−1

f (t) = [D

f (t) + D

a−1

f (t)]t=0t

a−1

Ea−b,a (−t

a−b

)+

g(t − τ)h(τ)dτ.

(4.11)

0

Now, for, n−1 < α < n, consider the nonhomogeneous differential equation under the non-zero initial conditions,

106


Dα y(t) − λy(t) = h(t), t > 0, subject to, [Dα−k−1 y(t)]t=0 = bk , for k = 0, 1, 2, ..., n − 1.

(4.12)

By Sumudu transforming both sides of equation (4.12), we get the expression, G(u) n−1 bk − ∑ k+1 − λG(u) = H(u), uα k=0 u

(4.13)

which yields, G(u) =

n−1 H(u) bk + . ∑ −α k+1 (u − λ) k=0 u (u−α − λ)

(4.14)

Once more, using equation (4.9), we obtain, Zt

n−1

(t − τ)α−1 Eα,α (λ(t − τ)α )h(τ)dτ + ∑ bk t α−k−1 Eα,α−k (λt α ).

y(t) = +

(4.15)

k=0

0

In [15], Podlubny used the Laplace transform to solve a generalized version of the simple typical initial value problem, 1 1 (4.16) D 2 f (t) + f (t) = 0, (t > 0); [D− 2 f (t)]t=0 = 2. Aside from fitting into either one of the previous two solved general examples, we can equally directly Sumudu-solve, equation (4.16), to get, G(u) 1

u2

−1

[D 2 ( f (t))]t=0 + G(u) − = 0, u

(4.17)

which, after rearranging, yields, G(u) =

2 1 2

1 2

u (u + 1)

=

2 1

u(u− 2 + 1)

.

(4.18)

Using the simplified version of (4.9), (see also [14-15]), S[t

k−1 2

√ E 1 , 1 (± t)] = 2 2

k! √ , −1 u( u ∓ 1)k+1

(4.19)

with k = 0, we find the same solution given in (4.7), is in the expression, f (t) = 2 t

−1 2

√ 1 1 E 1 , 1 (−t 2 ) = 2[ √ − et erfc( t)]. 2 2 πt

(4.20)

5 Conclusion. Prior to this work, various approaches have been used to solve fractional differential equations. In this paper, fractional differential equations have been solved using the Sumudu transform after deriving the related formulae for fractional integrals, and derivatives. Now, by using the results in the previous sections, and enlisting the help of the sumudi table in the Appendix, along with the sumudi tables previously published in the works of Belgacem et al. [6-7], the Sumudu technique can be used to solve many types initial value problems in applied and engineering fields. Applications in quantum mechanics and fractional diffusion seem to be highly suitable for such methods, despite the intricacies. We plan to look in this direction in future attempts.


107

Acknowledgments. The authors wish to thank their respective institutions for their research support, Professor Stephan Samko, and anonymous referees for helpful comments that improved the paper.

Appendix: Basic and Special Functions Fractional Integrals Sumudu. n 1 2 3 4 5 6 7 8 9 10

f (t) 1 t t n−1 , n = 1, 2, 3, ... n!p t ,p>0 Γ[p] eat t n−1 eat , n = 1, 2, 3, ... n! t p−1 eat ,p>0 Γ[p] sin(at) a cos(at) eat sin(at) a

11 ebt cos(at) 12 sinh(at) 13 cos h(at) 14 15 16 17 18 19

ebt − eat , a 6= b (a − b) bebt − aeat , a 6= b (b − a) sin(at) − at cos(at) 2a3 t sin(at) 2a sin(at) + at cos(at) 2a 1 cos(at) − at sin(at) 2

20 t cos(at)

S[D−v f (t)] uv uv+1 un+v−1 u p+v−1 uv 1 − au un+v−1 (1 − au)n u p+v−1 (1 − au) p uv+1 1 + a2 u2 v u (1 − bu)2 + a2 u2 uv+1 1 + a2 u2 uv − buv+1 (1 − bu)2 + a2 u2 auv+1 1 − av2 u2 u 1 − a2 u2 uv+1 (1 − au)(1 − bu) uv (1 − au)(1 − bu) uv+3 (1 + a2 u2 )2 uv+2 (1 + a2 u2 )2 uv+1 (1 + a2 u2 )2 uv (1 + a2 u2 )2 uv+1 (1 − a2 u2 ) (1 + a2 u2 )2

108


n 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

41

S[D−v f (t)] uv+3 (1 − a2 u2 )2 uv+2 (1 − a2 u2 ) (1 + a2 u2 )2 uv+1 (1 − a2 u2 )2 uv (1 − a2 u2 )2 uv+1 (1 + a2 u2 ) t cosh(at) (1 − a2 u2 )2 2 2 (1 + a t ) sin(at) − at cos(at) uv+3 8a3 (1 + a2 u2 )3 3t sin(at) + at 2 cos(at) uv+2 8a (1 + a2 u2 )3 2 2 (3 − a t ) sin(at) + 5at cos(at) uv+1 8a (1 + a2 u2 )3 2 2 (8 − a t ) cos(at) − 7at sin(at) uv 8 (1 + a2 u2 )3 2 t sin(at) uv+3 (3 − a2 u2 ) 2a (1 + a2 u2 )3 v+2 1 2 u (1 − 3a2 u2 ) t cos(at) 2 (1 + a2 u2 )3 v+3 1 3 u (1 − 6a2 u2 + a4 u4 ) t cos(at) 6 (1 + a2 u2 )4 3 v+4 t sin(at) u (1 − a2 u2 ) 24a (1 + a2 u2 )4 2 2 (3 + a t ) sinh(at) − 3at cosh(at) uv+5 8a5 (1 − a2 u2 )3 2 2 at cosh(at) + (a t − 1)t sinh(at) uv+3 8a3 (1 − a2 u2 )3 2 3t sinh(at) + at cosh(at) uv+2 8a (1 − a2 u2 )3 2 t sinh(at) uv+3 (1 + 3a2 u2 ) 2a (1 − a2 u2 )3 t 2 cosh(at) uv+2 (3 + a2 u2 ) 2 (1 + a2 u2 )3 v+3 1 3 cosh(at) u (1 + 6a2 u2 + a4 u4 ) t 6  (1 − a2 u2 )4  √ √ −3at  eat √ 3at 3at uv+2 2 3 sin( ) − cos( ) + e  (1 + a3 u3 ) 3a2  2 2   √ √ −3at  √ eat  3at 3at uv+1 cos( ) + 3 sin( )−e 2  (1 + a3 u3 ) 3a  2 2 f (t) at cosh(at) − sinh(at) 2a3 t sinh(at) 2a at cosh(at) + sinh(at) 2a 1 cosh(at) + a sinh(at) 2


n 42 43

44

45 46 47 48 49 50 51 52

53

f (t) √ at 1 −at 3at (e + 2e 2 cos( )) 3  2  √ √ 3at  −at √ e 3at 3at  2 e − cos( ) − 3 sin( ) 3a2  2 2    √ √ 3at   −at √ e 3at 3at 3 sin( ) − cos( )+e 2  3a  2 2 √ −at 1 at 3at 2 (e + 2e )) cos( 3 2 1 (sin(at) cosh(at) − cos(at) sinh(at)) 4a3 1 sin(at) sinh(at) 2a3 1 (sinh(at) − sin(at)) 2a3 1 (cosh(at) − cos(at)) 2a2 1 (sinh(at) + sin(at)) 2a 1 (cosh(at) + cos(at)) 2 (e−bt − e−at ) √ 2(b − a)( πt 3 )) √ erf( at) √ a

√ eat erf( at) √ 54 ½ a ¾ √ 1 b2t at √ − be erfc(b t) 55 e πt 56 J0 (at) 57 I0 (at) 58 an Jn (at); n > −1 59 an In (at); n > −1 a 60 erf( √ ) 2 t

S[D−v f (t)] uv+2 (1 + a3 u3 ) uv (1 − a3 u3 ) uv+1 (1 − a3 u3 ) uv (1 − a3 u3 ) uv+3 (1 + 4a4 u4 ) uv+2 (1 + 4a4 u4 ) uv+2 (1 − a4 u4 ) uv+1 (1 − a4 u4 ) uv (1 − a4 u4 ) uv (1 − a4 u4 ) uv √ √ √ u( 1 + au + 1 + bu) 1 v+ u 2 p (1 + au) 1 v+ u 2 (1 − au) uv √ √ u( 1 − au + b) uv √ 1 +va2 u2 u √ 1√ − a2 u2 v u ( 1 + a2 u2 − 1)n √ un √1 − au2 uv (1 − 1 − a2 u2 )n √ un ( 1 + a2 u2 ) −a √ uv (1 − e u )

109

110


n f (t) a 61 erfc( √ ) 2 t

S[D−v f (t)] −a √ v ue u −a √ u √ 1+b u uv e

√ a 62 eb(bt+a) erfc(b t + √ ) 2 t Z ∞ √ √ 1 n −(u2 /4a2 t) 63 √ 2n+1 u e J2n (2 u)du; n > −1 uv+n e−a u πta 0 (e−bt − e−at ) 1 + au 64 uv−1 Ln( ) t 1 + bu 1 v 1 + a2 u2 65 Ci(at) u Ln( 2 2 ) 2 a u 1 + au 66 Ei(at) uv Ln( ) au 67 Ln(t) (−γ + Ln(u))uv 2(cos(at) − cos(bt)) 1 + a2 u2 ) 68 uv−1 Ln( t 1 + b2 u2 π2 69 Ln(t)2 uv ( + (γ − Ln(u))2 ) 6 70 Ln(t) + γ; γ = Euler0 s constant = 0.577216... −uv Ln(u) π2 71 (Ln(t) + γ)2 − ; γ = Euler0 s constant uv Ln2 (u) 6 72 t n Ln(t); n > −1 un+v (Γ0 (n + 1) + Γ(n + 1)Ln(u)) sin(at) 73 uv−1 tan−1 (au) t 1 v− √ 1 −2√at 74 √ e u 2 eau erfc( au) πt 2a −a2t 2 2 2 75 √ e uv−1 e1/4a u erfc(1/2au) πt 1 2 2 a u v 76 erf(at) u e4 erfc(1/2au) 1 a v− 1 a 77 p u 2 e u erfc( √ ) u π(t + a) a 1 a 78 uv−1 e u Ei ( ) t +a u 1 uv−1 a π a a a 79 2 (cos( ){ − Si( )} − sin( )Ci( )) 2 t +a a u 2 u u u t v−1 (sin( a ){ π − Si( a )} + cos( a )Ci( a )) 80 2 u t + a2 u 2 u u u t a π a a a v 81 arctan( ) u (cos( ){ − Si( )} − sin( )Ci( )) a u 2 u u u 2 2 1 t +a a π a a a v 82 Ln( ) u (sin( ){ − Si( )} + cos( )Ci( )) 2 2 a u 2 u u u 1 t 2 + a2 π a 2 2 a v−1 83 Ln( ) u ( − Si( )) + Ci ( ) t a2 2 u u 84 N(t) 0 85 δ(t) uv−1 −a v−1 86 δ(t − a) u e u


S[D−v f (t)] −a v 87 u(t − a) ue u x uv sinh( ) 4 ∞ (−1)n (2n − 1)πx (2n − 1)πt u 88 ∑ 2n − 1 sin( 2a ) sin( 2a ) a π n=1 cosh( ) u v cosh( x ) u ∞ n (−1) t 2 nπt nπx u 89 + ∑ sin( ) cos( ) a a π n=1 n a a sinh( ) ux cosh( ) ∞ 4 (−1)n (2n − 1)πx (2n − 1)πt u 90 1 + ∑ cos( ) sin( ) uv a π n=1 2n − 1 2a 2a sinh( ) u v+1 sinh( x ) u ∞ n (−1) xt 2a nπx nπt u sin( 91 + 2 ∑ ) sin( ) a 2 a π n=1 n a a sinh( ) u v+1 sinh( x ) u ∞ n (−1) 8a (2n − 1)πx (2n − 1)πt u 92 x + 2 ∑ sin( ) cos( ) a 2 π n=1 (2n − 1) a a cosh( ) u x v+1 u cosh( ) t 2 2a ∞ (−1)n nπt nπx u 93 + 2 ∑ )(1 − cos( )) cos( a 2a π n=1 n2 a a sinh( ) u v+1 cosh( x ) u ∞ n 8a (−1) (2n − 1)πt (2n − 1)πx u 94 t + 2 ∑ ) sin( ) cos( a 2 π n=1 (2n − 1) 2a 2a cosh( ) u x v−1 sinh( √ ) u 2π ∞ nπx u n −n2 π2 t/a2 95 2 ∑ (−1) ne sin( ) a a n=1 a sinh( √ ) u a v−1 u cosh( √ ) ∞ 2 2 2 π (2n − 1)πx u 96 2 ∑ (−1)n (2n − 1)e−(2n−1) π t/4a cos( ) a a n=1 2a cosh( √ ) u 1 v− a u 2 sinh( √ ) ∞ 2 2 2 2 (2n − 1)πx u √ 97 ∑ (−1)n−1 e−(2n−1) π t/4a sin( 2a ) a n=1 cosh(a/ u) 1 v− a u 2 cosh( √ ) 2 2 2 1 2 ∞ nπx u √ 98 + ∑ (−1)n e−n π t/a cos( ) a a n=1 a sinh(a/ u) 2 −λnt ix uv J0 ( √ ) ∞ e a2 J0 (λn x/a) u 99 1 − 2 ∑ , where, J0 (λn ) = 0 ia λ J (λ ) n=1 n 1 n J0 ( √ ) u −λ2nt ix uv J0 ( √ ) ∞ 1 2 e a2 J0 (λn x/a) u 2 2 100 (x − a ) + t + 2a ∑ , 3 J (λ ) ia 4 λ n 1 n n=1 J0 ( √ ) u n

f (t)

111

112


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