New Laplace-type integral transform for solving steady heat-transfer problem
aa r X i v : . [ m a t h . G M ] M a y NEW LAPLACE-TYPE INTEGRAL TRANSFORMFOR SOLVING STEADY HEAT-TRANSFERPROBLEM
Shehu Maitama*, Weidong Zhao
School of Mathematics, Shandong University, Jinan, People’s Republic of China
Abstract
The fundamental purpose of this paper is to propose a new Laplace-typeintegral transform (NL-TIT) for solving steady heat-transfer problems. Theproposed integral transform is a generalization of the Sumudu, and theLaplace transforms and its visualization is more comfortable than the Sumudutransform, the natural transform, and the Elzaki transform. The suggestedintegral transform is used to solve the steady heat-transfer problems, andresults are compared with the results of the existing techniques.
Keywords: integral transforms, analytical solutions, heat transferproblems, Laplace-type integral transform.
1. Introduction
For more than 150 years, the motivation behind integral transforms iseasy to understand. The integral transforms have a widely-applicable spiritof converting differential operators into multiplication operators from itsoriginal domain into another domain. Besides, the symbolic manipulatingand solving the equation in the new domain is easier than manipulation andsolution in the original domain of the problem [1]-[8]. The inverse integraltransforms are always used to mapped the manipulated solution back to theoriginal domain to obtain the required result.In the mathematical literature, the famous classical integral transformsused in differential equations, analysis, theory of functions and integraltransforms are the Laplace transform [9] which was first introduced by a *Corresponding author e-mail address: [email protected] (S. Maitama),[email protected] (W. Zhao). Preprint submitted to Thermal Science May 16, 2019 rench mathematician Pierre-Simon Laplace (1747-1827), the Fourier inte-gral transform [10] devised by another French mathematician Joseph Fourier(1768-1830), and the Mellin integral transform [11] which was introduced bya Finnish mathematician Hjalmar Mellin (1854-1933). Besides, the Laplacetransform, the Fourier transform, and the Mellin integral transforms are sim-ilar, except in different coordinates and have many applications in scienceand engineering [12]. Moreover, in mathematics there are many Laplace-types integral transforms such as the Laplace-Carson transform used in therailway engineering [13], the z-transform applied in signal processing [14],the Sumudu transform used in engineering and many real-life problems [15],the Hankel’s and Weierstrass transform applied in heat and diffusion equa-tions [16, 17]. In addition, we have the natural transform [18] and Yangtransform [19, 20] used in many fields of physical science and engineering.This paper aims to further introduce a suitable Laplace-type integraltransform for solving steady heat-transfer problems. We will prove someimportant theorems and properties of the suggested integral transform andillustrated their applications. In the next section, we begin with the defi-nition of the proposed Laplace-type integral transform and introduce someuseful theorems of the integral transform.
Definition and TheoremsDefinition 1.1.
The new Laplace-type integral transform of the function v ( t ) of exponential order is defined over the set of functions, A = (cid:26) v ( t ) : ∃ C, ξ , ξ > , | v ( t ) | < Ce | t | ξi , if t ∈ ( − i × [0 , ∞ ) (cid:27) ,by the following integral: Θ [ v ( t )] ( s, u ) = V ( s, u ) = u Z ∞ e − st v ( ut ) dt = lim β →∞ Z β e − stu v ( t ) dt, s > , u > , (1) where Θ is the NL-TIT operator. It converges if the limit of the integralexists, and diverges if not.The inverse NL-TI transform is given by: Θ − [ V ( s, u )] = v ( t ) , f or t ≥ . (2) Equivalently, the complex inversion formula of the NL-TI transform isgiven by: v ( t ) = Θ − [ V ( s, u )] = 12 πi Z β + i ∞ β − i ∞ u e stu V ( s, u ) ds, t > , (3)2 here s and u are the NL-TI transform variables, and β is a real constantand the integral in eq. (3) is computed along s = β in the complex plane s = x + iy . Theorem 1.1.
The sufficient condition for the existence of the new Laplace-type integral transform. If the function v ( t ) is piecewise continues on everyfinite interval [0 , t ] and satisfies | v ( t ) | ≤ Ce βt , (4) for all t ∈ [ t , ∞ ) , and a constant β , then Θ [ v ( t )] ( s, u ) exists for all su > β . Proof.To prove the theorem, we must first show that the improper integralconverges for su > β . Without loss of generality, we first split the improperintegral into two parts namely: Z ∞ e − stu v ( t ) dt = Z t e − stu v ( t ) dt + Z ∞ t e − stu v ( t ) dt. (5)The first integral on the right hand side of eq. (5) exists by the firsthypothesis, hence the existence of the Laplace-type integral transform com-pletely depends on the second integral. Then by the second hypothesis wededuce: (cid:12)(cid:12)(cid:12) e − stu v ( t ) (cid:12)(cid:12)(cid:12) ≤ Ce − stu e βt = Ce − ( s − βu ) tu . (6)Thus Z ∞ t Ce − ( s − βu ) tu dt = us − βu Ce − ( s − βu ) t u . (7)Hence, eq. (7) converges for β < su . This implies by the comparison testfor improper integrals theorem, Θ [ v ( t )] ( s, u ) exists for β < su .This completethe proof. (cid:3) In the next theorem, we prove the uniqueness of the NL-TI transform.
Theorem 1.2.
Uniqueness of the new Laplace-type integral transform.Let v ( t ) and w ( t ) be continuous functions defined for t ≥ and havingNL-TI transforms of V ( s, u ) and W ( s, u ) respectively. If V ( s, u ) = W ( s, u ) ,then v ( t ) = w ( t ) . v ( t ) = 12 πi Z β + i ∞ β − i ∞ u e stu V ( s, u ) ds. (8)Since V ( s, u ) = W ( s, u ) by the second hypothesis, then replacing this ineq. (8), we obtain v ( t ) = 12 πi Z β + i ∞ β − i ∞ u e stu W ( s, u ) ds. (9)This implies v ( t ) = 12 πi Z β + i ∞ β − i ∞ u e stu V ( s, u ) ds = w ( t ) . (10)Hence, eq. (10) proves the uniqueness of the NL-TI transform. (cid:3) Theorem 1.3.
Convolution theorem of the NL-TI transform. Let the func-tions v ( t ) and w ( t ) be in set A. If V ( s, u ) and W ( s, u ) are the NL-TI trans-forms of the functions v ( t ) and w ( t ) respectively, then Θ [( v ∗ w )( t )] = V ( s, u ) W ( s, u ) . (11) Where v ∗ w is the convolution of two functions v ( t ) and w ( t ) which is definedas: Z t v ( τ ) w ( t − τ ) dτ = Z t v ( t − τ ) w ( τ ) dτ. (12)Proof.Based on eq. (1) and eq. (12), we get:Θ (cid:20)Z t v ( τ ) w ( t − τ ) (cid:21) = Z ∞ e − stu (cid:18)Z t v ( τ ) w ( t − τ ) (cid:19) dτ. Changing the limit of integration yields:Θ (cid:20)Z t v ( τ ) w ( t − τ ) (cid:21) = Z ∞ (cid:18) v ( τ ) Z ∞ τ e − stu w ( t − τ ) dt (cid:19) dτ. Substituting ϑ = t − τ in the inner integral, we deduce: Z ∞ τ e − stu w ( t − τ ) dt = Z ∞ e − ( ϑ + τ ) u s w ( ϑ ) dϑ = e − τsu Z ∞ e − ϑsu w ( ϑ ) dϑ = e − τsu W ( s, u ) . (cid:20)Z t v ( τ ) w ( t − τ ) (cid:21) = Z ∞ v ( τ ) e − sτu W ( s, u ) dτ = W ( s, u ) Z ∞ v ( τ ) e − sτu dτ = V ( s, u ) W ( s, u ) . (13)This complete the proof. (cid:3) Theorem 1.4.
Derivative of the NL-TI transform. Suppose that
Θ [ v ( t )] ( s, u ) exists and that v ( t ) is differentiable n-times on the interval (0 , ∞ ) with n th derivative v ( n ) ( t ) , then Θ (cid:2) v ′ ( t ) (cid:3) ( s, u ) = su V ( s, u ) − v (0) , (14)Θ (cid:2) v ′′ ( t ) (cid:3) ( s, u ) = s u V ( s, u ) − su v (0) − v ′ (0) , (15)Θ (cid:2) v ′′′ ( t ) (cid:3) ( s, u ) = s u V ( s, u ) − s u v (0) − su v ′ (0) − v ′′ (0) , (16) ... (17)Θ h v ( n ) ( t ) i ( s, u ) = s n u n V ( s, u ) − n − X k =0 (cid:16) su (cid:17) n − ( k +1) v ( k ) (0) . (18) Proof.Using Definition 1.1 of the NL-TI transform and integration by parts,we deduce: Θ (cid:2) v ′ ( t ) (cid:3) ( s, u ) = Z ∞ e − stu v ′ ( t ) dt = − v (0) + su Z ∞ e − stu v ( t ) dt = − v (0) + su V ( s, u ) . (19)Θ (cid:2) v ′′ ( t ) (cid:3) ( s, u ) = su Θ (cid:2) v ′ ( t ) (cid:3) ( s, u ) − v ′ (0) = su h − v (0) + su V ( s, u ) i − v ′ (0)= − v ′ (0) − su v (0) + s u V ( s, u ) . (20)Θ (cid:2) v ′′′ ( t ) (cid:3) ( s, u ) = su Θ (cid:2) v ′′ ( t ) (cid:3) ( s, u ) − v ′′ (0) = su (cid:20) − v ′ (0) − su v (0) + s u V ( s, u ) (cid:21) − v ′′ (0)= − v ′′ (0) − su v ′ (0) − s u v (0) + s u V ( s, u ) . (21)5inally, eq. (18) follows using mathematical induction. (cid:3) In the next theorems, we prove the NL-TI transform of Riemann-Liouvillefractional derivative RL D αt [6], and the Caputo fractional derivative C D αt [6]. Theorem 1.5.
New Laplace-type integral transform of Riemann-Liouvillefractional derivative. If α > , n = 1+[ α ] and v ( t ) , I n − α v ( t ) , ddt I n − α v ( t ) , · · · , d n dt n I n − α v ( t ) , RL D α v ( t ) ∈ A , then Θ (cid:2) RL D αt v ( t ) (cid:3) = (cid:16) su (cid:17) α Θ [ v ( t )] − n − X k =0 (cid:16) su (cid:17) n − k − d k − dt k − I n − α v (0+) , (22) where I α is the Riemann-Liouville fractional integral. Proof.Since RL D αt v ( t ) = d n dt n I n − α v ( t ). Let g ( t ) = I n − α v ( t ), then RL D αt v ( t ) = d n dt n v ( t ). Applying the hypothesis of theorem (1.4), we getΘ (cid:2) RL D αt v ( t ) (cid:3) = (cid:16) su (cid:17) n Θ [ v ( t )] − n − X k =0 (cid:16) su (cid:17) n − k − v ( k ) (0+)= (cid:16) su (cid:17) α Θ [ v ( t )] − n − X k =0 (cid:16) su (cid:17) n − k − d k − dt k − I n − α v (0+) . The proof ends. (cid:3)
Theorem 1.6.
New Laplace-type integral transform of Caputo fractionalderivative. Assume α > , n = 1 + [ α ] and v ( t ) , ddt v ( t ) , d dt v ( t ) , · · · , d n dt n v ( t ) , C D α v ( t ) ∈ A , then Θ (cid:2) C D αt v ( t ) (cid:3) = (cid:16) su (cid:17) α Θ [ v ( t )] − n − X k =0 (cid:16) su (cid:17) α − k − v ( k ) (0+) , (23) where C D αt is the Caputo fractional derivative. Proof.Applying the Caputo fractional derivative [6] and theorem (1.3), we de-duce: 6 D αt v ( t ) = 1Γ( n − α ) Z t ( t − τ ) n − α − v ( n ) ( τ ) dτ = 1( n − α ) t n − α − ∗ v ( n ) ( τ ) . Finally, using the hypothesis of theorem (1.4) yields:Θ (cid:2) C D αt v ( t ) (cid:3) = 1Γ( n − α ) Θ (cid:2) t n − α − (cid:3) Θ h v ( n ) ( τ ) i = (cid:16) su (cid:17) α Θ [ v ( t )] − n − X k =0 (cid:16) su (cid:17) α − k − v ( k ) (0+) . This complete the proof. (cid:3)
Some Properties of the NL-TI transformProperty 1.
Linearity property of the NL-TI transform. Let
Θ [ v ( t )] ( s, u ) = V ( s, u ) and Θ [ w ( t )] ( s, u ) = W ( s, u ) , then Θ [ αv ( t ) + βw ( t )] ( s, u ) = α Θ [ v ( t )] ( s, u ) + β Θ [ w ( t )] ( s, u ) , (24) where α and β are constants.Proof.Linearity property follows directly from Definition 1.1. (cid:3) Property 2.
Exponential Shifting Property of the NL-TI transform. Letthe function v ( t ) ∈ A and α is an arbitrary constant, then Θ (cid:2) e αt v ( t ) (cid:3) ( s, u ) = V ( s − αu ) . (25)Proof.Using Definition 1.1 of the NL-TI transform, we get:Θ [ v ( t )] ( s, u ) = u Z ∞ e − st v ( ut ) dt. (26)Then 7 (cid:2) e αt v ( t ) (cid:3) ( s, u ) = Z ∞ e ( αt ) e − stu v ( t ) dt = Z ∞ e − ( s − αu ) u v ( t ) dt = u Z ∞ e − ( s − αu ) t v ( ut ) dt = Θ [ v ( t )] ( s − αu ) = V ( s − αu ) . (27)In particular,Θ (cid:2) sin(3 t ) e − t (cid:3) ( s, u ) = Θ [sin(3 t )] ( s − ( − u )) = Θ [sin(3 t )] ( s + 4 u ) . (28)Based on Definition 1.1, the NL-TI transform of sin(3 t ) is given by:Θ [sin(3 t )] ( s, u ) = Z ∞ e − stu sin(3 t ) dt = 12 i (cid:18) us − iu − us + 3 iu (cid:19) = 3 u s + 9 u . (29)So, replacing the variable s with ( s + 4 u ) in eq. (29), we obtain:Θ [sin(3 t )] ( s + 4 u ) = 3 u ( s + 4 u ) + 9 u = 3 u s + 8 us + 25 u . (30)Alternatively, Z ∞ sin(3 t ) e − t e − stu dt = 3 u s + 8 us + 25 u . (31)Moreover,Θ (cid:2) te αt (cid:3) ( s, u ) = u ( s − αu ) = s − α ) , u = 1 , Laplace transform [6] u (1 − αu ) , s = 1 , Yang transform [19] . (32)This complete the proof. (cid:3) Property 3.
New Laplace-type transform of integral. Let
Θ [ v ( t )] = V ( s, u ) and v ( t ) ∈ A , then Θ (cid:20)Z t v ( ζ ) dζ (cid:21) = us V ( s, u ) . (33)8roof.Let w ( t ) = R t v ( ζ ) dζ , then w ′ ( t ) = v ( t ) and w (0) = 0. Computing theNL-TI transform of both sides, we get:Θ (cid:2) w ′ ( t ) (cid:3) ( s, u ) = Θ [ v ( t )] ( s, u ) = su Θ [ w ( t )] ( s, u ) − w (0) = V ( s, u ) . (34)This implies Θ (cid:20)Z t v ( ζ ) dζ (cid:21) = us V ( s, u ) . (35)This complete the proof. (cid:3) Property 4.
Multiple Shift Property of the NL-TI transform. Let
Θ [ v ( t )] ( s, u ) = V ( s, u ) and v ( t ) ∈ A , then Θ [ t n v ( t )] ( s, u ) = ( − u ) n d n ds n V ( s, u ) . (36)Proof.By Definition 1.1 of the NL-TI transform and Leibnizs rule, we obtain: dds V ( s, u ) = dds Z ∞ e − stu v ( t ) dt = Z ∞ ∂∂s (cid:16) e − stu (cid:17) v ( t ) dt = − u Z ∞ e − stu tv ( t ) dt = Θ [ tv ( t )] ( s, u ) = − u dds V ( s, u ) . (37)Thus, eq. (37) above proves the theorem for n = 1. To generalized thetheorem, we apply the induction hypothesis. Let assume the theorem holdsfor n = k that is Z ∞ e − stu t k v ( t ) dt = ( − u ) k d k ds k V ( s, u ) . (38)Then dds Z ∞ e − stu t k v ( t ) dt = ( − u ) k d k +1 ds k +1 V ( s, u ) . (39)Alternatively, using Leibnizs rule, we deduce: dds Z ∞ e − stu t k v ( t ) dt = Z ∞ ∂∂s (cid:16) e − stu (cid:17) t k v ( t ) dt = − u Z ∞ e − stu t k +1 v ( t ) dt = ( − u ) k d k +1 ds k +1 V ( s, u ) . (40)9his implies Z ∞ e − stu t k +1 v ( t ) dt = ( − u ) k +1 d k +1 ds k +1 V ( s, u ) . (41)Since, eq. (41) holds for n = k + 1, then by induction hypothesis theprove is complete. (cid:3) Applications
In this section, we illustrate the applicability of the proposed Laplace-type integral transform on steady heat-transfer problems to proves its effi-ciency and high accuracy.
Example 1.
Consider the following steady heat-transfer problem: − hM v ( t ) = ρ Λ c p v ′ ( t ) , (42) subject to the initial condition v (0) = β, (43) where h − is the convection heat transfer coefficient, M − is the surfacearea of the body, ρ − is the density of the body, Λ − is the volume, c p − is thespecific heat of the material, and v ( t ) is the temperature.Applying the NL-TI transform on both sides of eq. (42), we get: − hM V ( s, u ) = ρ Λ c p (cid:16) su V ( s, u ) − v (0) (cid:17) . (44)Substituting the given initial condition and simplifying, we get: V ( s, u ) = βus + hMρ Λ cp u (45)Taking the inverse NL-TI transform of eq. (45), we get: v ( t ) = βe − hMρ Λ cp t . (46)The exact solution is in excellent agreement with the result obtained in[5, 20]. 10 xample 2. Consider the following steady heat-transfer problem: v t ( x, t ) = 2 v xx ( x, t ) , < x < , t > . (47) Subject to the boundary and initial conditions v (0 , t ) = 0 , v (5 , t ) = 0 , v ( x,
0) = 10 sin(4 πx ) − πx ) . (48) Applying the NL-TI transform on both sides of eq. (47), we deduce: su V ( x, s, u ) − v ( x,
0) = 2 d V ( x, s, u ) dx . (49)Substituting the given initial condition and simplifying, we get:2 d V ( x, s, u ) dx − su V ( x, s, u ) = −
10 sin(4 πx ) + 5 sin(6 πx ) . (50)The general solution of eq. (50) can be written as: V ( x, s, u ) = V h ( x, s, u ) + V p ( x, s, u ) , (51)where V h ( x, s, u ) is the solution of the homogeneous part which is given by: V h ( x, s, u ) = α e √ su x + α e − √ su x , (52)and V p ( x, s, u ) is the solution of the inhomogeneous part which is given by: V p ( x, s, u ) = α sin(4 πx ) + β sin(6 πx ) . (53)Applying the boundary conditions on eq. (52), yields α + α = 0 ⇒ α e √ su + α e − √ su = 0 ⇒ V h ( x, s, u ) = 0 , (54)since, α = α = 0 . Using the method of undetermined coefficients on the inhomogeneous part,we get: V p ( x, s, u ) = 10 sin(4 πx ) us + 32 π u − πx ) us + 72 π u , (55)since, α = 10 us +32 π u , and β = − us +72 π u .Then eq. (51) will become: V ( x, s, u ) = 10 sin(4 πx ) us + 32 π u − πx ) us + 72 π u . (56)Taking the inverse NL-TI transform of eq. (56), we obtain: v ( x, t ) = 10 e − π t sin(4 πx ) − e − π t sin(6 πx ) . (57)The exact solution is the same with the result obtained in [9].11 xample 3. Consider the following fractional porous medium equation: D αt v ( x, t ) = D x ( v ( x, t ) D x v ( x, t )) , < α ≤ , (58) subject to the initial condition v ( x,
0) = x. (59) Applying theorem (1.6) on eq. (58) subject to the initial condition, we obtain:
Θ [ v ( x, t )] = us x + u α s α Θ [ D x ( v ( x, t ) D x v ( x, t ))] . (60) Computing the inverse NL-TI transform on both sides of eq. (60), we deduce: v ( x, t ) = x + Θ − (cid:20) u α s α Θ [ D x ( v ( x, t ) D x v ( x, t ))] (cid:21) . (61) Based on the basic idea of the homotopy analysis method (see [6] and refer-ences therein), we have: v ( x, t ) = ∞ X n =0 p n v n ( x, t ) . (62) Then eq. (61) will become: ∞ X n =0 p n v n ( x, t ) = x + p Θ − " u α s α Θ " D x ∞ X n =0 p n H n ! , (63) where H n , is the He’s polynomials [6] which represent the nonlinear terms v ( x, t ) D x v ( x, t ) .Some few components of the nonlinear terms H n are computed below: H = v v x , H = v v x + v v x , H = v x v + v x v + v x v , · · · (64) On comparing the coefficients of same powers of p in eq. (63), the wedetermine the following approximations: p : v ( x, t ) = x,p : v ( x, t ) = Θ − (cid:20) u α s α Θ [ D x ( H )] (cid:21) = t α Γ(1 + α ) , : v ( x, t ) = Θ − (cid:20) u α s α Θ [ D x ( H )] (cid:21) = 0 , ... p n : v n ( x, t ) = Θ − (cid:20) u α s α Θ [ D x ( H n − )] (cid:21) = 0 , for n ≥ . Then the solution of eq. (58)-(59) is given by: v ( x, t ) = x + t α Γ(1 + α ) . (65) The result obtained in eq. (65) is in excellent agreement with the resultobtained in [6]. The special case of eq. (65) when α = 1 is given by: v ( x, t ) = x + t. (66) The result of eq. (66) is in closed agreement with the result obtained in [6, 7].
Conclusion
In this paper, we introduced a powerful Laplace-type integral trans-form for finding a solution of steady heat-transfer problems. The proposedLaplace-type integral transform converges to both Yang transform, and theLaplace transforms just by changing variables. Many interesting propertiesof the suggested integral transform are discussed and successfully applied tosteady heat-transfer problems. Finally, based on the efficiency and simplic-ity of the Laplace-type integral transform, we conclude that it is a powerfulmathematical tool for solving many problems in science and engineering.
Acknowledgment
The authors would like to thank the anonymous reviewers, managing ed-itor, and editor in chief for their valuable help in improving the manuscript.This work is supported by the Natural Science Foundation of China (GrandNo. 11571206). The first author acknowledges the financial support of ChinaScholarship Council (CSC) in Shandong University (Grand No. 2017GXZ025381).13 omenclature
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