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MATHEMATICAL PHYSICS SOLVING SCHLÖMILCH’S INTEGRAL EQUATION BY THE REGULARIZATION-ADOMIAN METHOD ABDUL-MAJID WAZWAZ Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA E-mail: [email protected] Received September 3, 2014 We present a reliable study for the linear and the nonlinear Schlömilch’s integral equation and its generalized forms. We also introduce Schlömilch-type integral equations and study it as well. The Schlömilch’s integral equation has been used for many ionospheric problems, atmospheric and terrestrial physics. We use the regularization method combined with the Adomian decomposition method to handle all forms of Schlömilch’s integral equations. The combined regularization-Adomian method can be used in many applied physics, applied mathematics and engineering applications. We justify the reliability of the regularization-Adomian method through illustrative examples. Key words: Schlömilch’s integral equation; regularization; terrestrial physics. 1. INTRODUCTION The ionosphere is a shell of electrons and electrically charged atoms that surrounds the earth in a region of the upper atmosphere. This region is ionized by solar radiation and is important because it influences radio propagation to distant places on the earth. The ionosphere shell owes its existence primarily to ultraviolet radiation from the sun. The dynamics of the ionosphere shell is a very important area research that has sustained and is still going strong for terrestrial physics. The ionosphere shell has a unique structure in which electrons and electrically charged atoms coexist. In this work we will study the Schlömilch’s integral equation that was used in order to derive the electron density profile from the ionospheric for oblique incidence for the quasi-transverse approximation [1–4]. The Schlömilch’s integral equation reads Z π 2 2 f (x) = u(x sin t) dt, (1) π 0 where f (x) is a continuous differential coefficient for −π ≤ x ≤ π. It has been proved in [1–4] that this equation has one solution given by Z π 2 u(x) = f (0) + x f 0 (x sin t) dt, (2) 0 f0 where is the derivative of f with respect to the complicated argument ξ = x sin t. The Schlömilch’s integral equation and its unique solution have been used to deRJP 60(Nos. Rom. Journ. Phys., 1-2), Vol. 56–71 60, Nos. (2015) 1-2, P.(c) 56–71, 2015 Bucharest, - v.1.3a*2015.2.7 2015 2 Solving Schlömilch’s integral equation by the regularization-Adomian method 57 termine the electron density profile from the ionospheric for the case of the quasitransverse (QT) approximations [1–4]. The Schlömilch’s integral equation has been used for many ionospheric problems, atmospheric and terrestrial physics. There are four types of Schlömilch’s integral models that will be touched upon in this paper. They are the standard model (1), a generalized Schlömilch’s integral equation, a Schlömilch-type integral equation, where the cosine function replaces the sine function of the standard form, and the nonlinear Schlömilch’s integral equation, where the linear term u(x sin t) in the standard model is replaced by a nonlinear term F (u(x sin t)) or by F (u(x cos t)). The generalized Schlömilch’s integral equation takes the form Z π 2 2 f (x) = u(x sinn t) dt, n ≥ 1. (3) π 0 Moreover, using the sense of the Schlömilch’s integral equation we introduce a Schlömilch-type integral equation given by Z π 2 2 f (x) = u(x cosn t) dt, n ≥ 1, (4) π 0 where the argument x sin t was replaced by x cos t. It is worth noting that the Schlömilch’s integral equation may arise in a nonlinear form as Z π 2 2 f (x) = F (u(x sin t)) dt, (5) π 0 where f (x) is a continuous differential coefficient for −π ≤ x ≤ π, and F is a nonlinear form of u(x sin t) such as u2 (x sin t) and u3 (x sin t). It is obvious that the Schlömilch’s integral equation is a special case of Fredholm integral equation of the first kind. Recall that the linear Fredholm integral equations of the first kind are of the form [5–16] Z b f (x) = λ K(x, t)u(t) dt, x ∈ Ω, (6) a and the nonlinear Fredholm integral equation of the first kind is given by Z b f (x) = λ K(x, t)F (u(t)) dt, x ∈ Ω, (7) a where Ω is a closed and bounded region. Fredholm integral equations of the first kind (6) and (7) are characterized by the occurrence of the unknown function u(x) only inside the integral sign. Some special difficulties arise from the existence of u(x) only inside the integral sign. The Fredholm integral equations of the first kind appear in many physical models such as radiography, spectroscopy, cosmic radiation, image processing and in the theory of signal processing. RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7 58 Abdul-Majid Wazwaz 3 The complete study of ionospheric problems, atmospheric and terrestrial physics should cover both theoretical analysis, which exists reasonably in the literature, and computational, which we believe needs more studies, and this is one of our aims of this work. For this reason, we will combine the method of regularization [9–13] and the Adomian decomposition method [14–16] to handle the linear and the nonlinear Schlömilch’s integral equation and the other related equations (1) and (3)–(5). The combined regularization-Adomian method proved to be reliable, effective [14–16], and hence can be extremely useful in carrying out further works especially in illposed problems as Fredholm integral equations of the first kind. In what follows, we give brief outlines of each method. The Adomian decomposition method (ADM) and the method of regularization are given thoroughly in the literature. In what follows we will present a brief summary for each method. 2. THE ADOMIAN DECOMPOSITION METHOD The details of the ADM are now well known and widely applied in the literature; see, for example [14–16]. The ADM admits the use of the infinite decomposition series u(x) = ∞ X un (x), (8) n=0 for the solution u(x). The components ui (x), i ≥ 1 are determined by using a recursive relation, and the zero-th component u0 (x) is given all terms that are not included inside the integral sign. Substituting (8) into both sides of the integral equation, and assigning u0 (x) as indicated earlier, the classic ADM admits the use of the recursive relation u0 (x) = all terms not included inside the integral sign, Z x (9) uj+1 (x) = (uj (x sin t)) dx, j ≥ 0, 0 that will lead to the complete determination of the components un (x), for n ≥ 0, of u(x). The series solution of u(x) follows immediately and converges to the closed form solution if such a solution exists. For concrete problems, the obtained series can be used for numerical purposes. 3. THE METHOD OF REGULARIZATION In what follows we will present a brief summary of the method of regularization. Details can be found in [9–13]. RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7 4 Solving Schlömilch’s integral equation by the regularization-Adomian method 59 The method of regularization was established independently by Tikhonov [11, 12] and Phillips [13]. The method transforms a first kind Fredholm integral equation to a second kind equation. The regularization method transforms the linear Fredholm integral equation of the first kind Z b f (x) = K(x, t)u(t) dt, (10) a to the approximation Fredholm integral equation Z b K(x, t)uα (t) dt, αuα (x) = f (x) − (11) a where α is a small positive parameter called the regularization parameter. It is clear that (11) is a Fredholm integral equation of the second kind that can be rewritten as Z 1 1 b uα (x) = f (x) − K(x, t)uα (t) dt. (12) α α a Moreover, it was proved by Tikhonov [11, 12] and Phillips [13] that the solution uα of equation (12) converges to the solution u(x) of (10) or (12) as α → 0. This means that u(x) = lim uα (x). (13) α→0 It was also found that the obtained series mostly includes infinite geometric series. We will apply the aforementioned methods to the Schlömilch’s integral equation and its other related forms. Two examples will be given for each case to illustrate the power of the proposed approach. 4. THE LINEAR SCHLÖMILCH’S INTEGRAL EQUATION The method of regularization converts the linear Schlömilch’s integral equation Z π 2 2 f (x) = u(x sin t) dt, x ∈ D, π 0 to the Schlömilch’s integral equation of the second kind in the form Z π 2 2 αuα (x) = f (x) − uα (x sin t) dt, π 0 or equivalently ! Z π 1 1 2 2 uα (x sin t) dt , uα (x) = f (x) − α α π 0 (14) (15) (16) where α is a small positive parameter. It is obvious that the solution uα of equation (16) converges to the solution u(x) of (14) as α → 0. Consequently, we can apply RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7 60 Abdul-Majid Wazwaz 5 any method we studied before for solving the Schlömilch’s integral equations of the second kind. However, we will use the Adomian decomposition method, where we set the recurrence relation 1 uα0 (x) = f (x), α ! Z π (17) 1 2 2 uαk+1 (x) = − uαk (x sin t) dt , k ≥ 0. α π 0 In what follows we will present two illustrative examples. Our focus will be on transforming the first kind equation to a second kind equation by using the method of regularization, and hence we can use any appropriate method such as the Adomian decomposition method. Example 1: Solve the linear Schlömilch’s integral equation Z π 2 2 2 1 + πx = u(x sin t) dt, −π ≤ x ≤ π. π 0 Using the method of regularization, Eq. (18) becomes ! Z π 2 1 1 2 uα (x) = (1 + πx2 ) − uα (x sin t) dt . α α π 0 (18) (19) We select the Adomian method for solving this equation. The Adomian method admits the use of the recurrence relation 1 uα0 (x) = (1 + πx2 ), α ! Z π (20) 1 2 2 uαk+1 (x) = − uαk (x sin t) dt , k ≥ 0. α π 0 This in turn gives the first few components 1 (1 + πx2 ), α 2 + π2 uα1 (x) = − , 2α2 4 + πx2 uα2 (x) = , 4α3 8 + πx2 , uα3 (x) = − 8α4 .. .. uα0 (x) = RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7 (21) 6 Solving Schlömilch’s integral equation by the regularization-Adomian method 61 We have uα (x) = 1 1 1 πx2 1 1 1 1 1− + 2 − 3 +··· + 1− + 2 − 3 + · · · , (22) α α α α α 2α 4α 8α which gives 1 2π + x2 , (23) 1 + α 1 + 2α obtained upon finding the summation of each infinite geometric series. The exact solution is given by u(x) = lim uα (x) = 1 + 2πx2 . (24) uα (x) = α→0 Example 2: Solve the linear Schlömilch’s integral equation Z π 2 2 u(x sin(3t) dt, −π ≤ x ≤ π. 2x = π 0 Using the method of regularization, Eq. (25) becomes ! Z π 2 1 2 2 uα (x) = x − uα (x sin(3t)) dt . α α π 0 (25) (26) We select the Adomian method for solving this equation. The Adomian method admits the use of the recurrence relation 2 uα0 (x) = x, α ! Z π (27) 1 2 2 uαk (x sin(3t)) dt , k ≥ 0. uαk+1 (x) = − α π 0 This in turn gives the first few components 2 x, α 4 uα1 (x) = − x, 3απ 8 uα2 (x) = 3 2 x, 9α π 16 uα3 (x) = − x, 27α4 π 3 .. .. uα0 (x) = (28) We have 2 2 4 8 uα (x) = x 1 − + − +··· , α 3απ 9α2 π 2 27α3 π 3 RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7 (29) 62 Abdul-Majid Wazwaz 7 which gives 6π x, (30) 2 + 3απ obtained upon finding the summation of the infinite geometric series. The exact solution is given by u(x) = lim uα (x) = 3πx. (31) uα (x) = α→0 5. THE GENERALIZED SCHLÖMILCH’S INTEGRAL EQUATION We follow the discussion presented before to handle generalized Schlömilch’s integral equation. The method of regularization converts this equation of the first kind Z π 2 2 u(x sinn t) dt, n ≥ 1, x ∈ D, (32) f (x) = π 0 to the Schlömilch’s integral equation of the second kind in the form Z π 2 2 αuα (x) = f (x) − uα (x sinn t) dt, (33) π 0 or equivalently ! Z π 1 1 2 2 uα (x) = f (x) − (34) uα (x sinn t) dt , α α π 0 where α is a small positive parameter. As stated before, the solution uα of equation (34) converges to the solution u(x) of (32) as α → 0. Proceeding as before, we use the Adomian decomposition method that admits the use of the recurrence relation 1 uα0 (x) = f (x), α ! Z π (35) 1 2 2 uαk+1 (x) = − uαk (x sinn t) dt , k ≥ 0. α π 0 In what follows we will present two illustrative examples. Example 1: Solve the Schlömilch’s integral equation Z π 2 2 2 u(x sin2 t) dt, −π ≤ x ≤ π. x + 3x = π 0 Using the method of regularization, Eq. (36) becomes ! Z π 1 1 2 2 2 2 uα (x) = (x + 3x ) − uα (x sin t) dt . α α π 0 RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7 (36) (37) 8 Solving Schlömilch’s integral equation by the regularization-Adomian method 63 We select the Adomian method for solving this equation. The Adomian method admits the use of the recurrence relation 1 (x + 3x2 ), α ! Z π 1 2 2 uαk (x sin2 t) dt , k ≥ 0. uαk+1 (x) = − α π 0 uα0 (x) = (38) This in turn gives the first few components 1 (x + 3x2 ), α 1 uα1 (x) = − 2 x(4 + 9x), 8α 1 uα2 (x) = x(16 + 27x), 64α3 1 uα3 (x) = − x(64 + 81x), 512α4 .. .. uα0 (x) = (39) We have x 1 1 1 3x2 3 9 27 1− + 2 − 3 +··· + 1− + − + · · · , α 2α 4α 8α α 8α 64α2 512α3 (40) which gives uα (x) = 2 24 x+ x2 . 1 + 2α 3 + 8α (41) u(x) = lim uα (x) = 2x + 8x2 . (42) uα (x) = The exact solution is given by α→0 Example 2: Solve the Schlömilch’s integral equation Z π 5 2 2 2 4x − x = u(x sin3 (t) dt, −π ≤ x ≤ π. 16 π 0 (43) Using the method of regularization, Eq. (43) becomes 1 5 1 uα (x) = (4x − x2 ) − α 16 α 2 π Z π 2 ! 3 uα (x sin (t)) dt . 0 RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7 (44) 64 Abdul-Majid Wazwaz 9 The Adomian method admits the use of the recurrence relation 1 5 (4x − x2 ), α 16 ! Z π 1 2 2 3 uαk+1 (x) = − uαk (x sin (t)) dt , k ≥ 0. α π 0 uα0 (x) = (45) This in turn gives the first few components 1 5 (4x − x2 ), α 16 16 25 2 x , uα1 (x) = − 2 x + 3α π 256α2 64 125 2 uα2 (x) = 3 2 x + x , 9α π 256α2 625 2 256 x+ x , uα3 (x) = − 4 3 27α π 4096α3 .. .. uα0 (x) = (46) Proceeding as before, and using the infinite geometric series gives uα (x) = 5 12π x− x2 , 4 + 3απ 5 + 16α (47) obtained upon finding the summation of the infinite geometric series. The exact solution is given by u(x) = lim uα (x) = 3πx − x2 . α→0 (48) 6. THE LINEAR SCHLÖMILCH-TYPE INTEGRAL EQUATION In this section we will examine a Schlömilch-type integral equation. We will follow the analysis presented in the preceding sections, hence we skip details. The method of regularization converts the Schlömilch-type integral equation Z π 2 2 f (x) = u(x cos t) dt, x ∈ D, (49) π 0 to the Schlömilch-type integral equation of the second kind in the form ! Z π 1 2 2 1 uα (x) = f (x) − uα (x cos t) dt . α α π 0 RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7 (50) 10 Solving Schlömilch’s integral equation by the regularization-Adomian method 65 It is obvious that the solution uα of equation (50) converges to the solution u(x) of (49) as α → 0. The Adomian decomposition method gives the recurrence relation 1 f (x), α ! Z π 1 2 2 uαk+1 (x) = − uαk (x cos t) dt , k ≥ 0. α π 0 uα0 (x) = In what follows we will examine two illustrative examples. Example 1: Solve the Schlömilch-type integral equation Z π 2 2 1 + 2x = u(x cos t) dt, −π ≤ x ≤ π. π 0 Using the method of regularization, Eq. (52) becomes ! Z π 1 1 2 2 uα (x) = (1 + 2x) − uα (x cos t) dt . α α π 0 (51) (52) (53) We select the Adomian method for solving this equation. The Adomian method admits the use of the recurrence relation 1 uα0 (x) = (1 + 2x), α ! Z π (54) 1 2 2 uαk+1 (x) = − uαk (x cos t) dt , k ≥ 0. α π 0 This in turn gives the first few components 1 (1 + 2x), α π + 4x uα1 (x) = − , πα2 π 2 + 8x uα2 (x) = 2 3 , π α π 3 + 16x uα3 (x) = − 3 4 , π α .. .. uα0 (x) = (55) This in turn gives 1 1 1 1 2 2 4 8 uα (x) = 1− + 2 − 3 +··· + x 1− + − +··· , α α α α α πα π 2 α2 π 3 α3 (56) RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7 66 Abdul-Majid Wazwaz 11 which gives 1 2π + x, (57) 1 + α 2 + πα obtained upon finding the summation of each infinite geometric series. The exact solution is given by uα (x) = u(x) = lim uα (x) = 1 + πx. α→0 Example 2: Solve the Schlömilch-type integral equation Z π 1 2 2 2 x = u(x cos(2t)) dt, −π ≤ x ≤ π. 2 π 0 (58) (59) Proceeding as before, we obtain the recurrence relation 1 2 x , 2α ! Z π 1 2 2 uαk+1 (x) = − uαk (x cos(2t)) dt , k ≥ 0. α π 0 uα0 (x) = (60) This in turn gives the first few components 1 2 x , 2α 1 uα1 (x) = − 2 x2 , 4α 1 uα2 (x) = 3 x2 , 8α 1 2 x , uα3 (x) = − 16α4 .. .. (61) 1 2 1 1 1 x 1− + − +··· , uα (x) = 2α 2α 4α2 8α3 (62) uα0 (x) = We have which gives 1 x2 , (63) 1 + 2α obtained upon finding the summation of the infinite geometric series. The exact solution is given by uα (x) = u(x) = lim uα (x) = x2 . α→0 RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7 (64) 12 Solving Schlömilch’s integral equation by the regularization-Adomian method 67 7. THE NONLINEAR SCHLÖMILCH’S INTEGRAL EQUATION The nonlinear Schlömilch’s integral equation takes the form Z π 2 2 f (x) = F (u(x sin t)) dt, π 0 (65) where F (u(x sin t)) is a nonlinear function of u(x sin t), such as ui (x sin t), ui (x cos t), i ≥ 2, and f (x) is a continuous differential coefficient for −π ≤ x ≤ π. To handle this nonlinear equation, we will follow the same analysis presented earlier for linear equations. To achieve this goal, we should first transform this nonlinear equation to a linear form. To transform (65) to a linear form of the first kind, we first use the transformation F (u(x sin t)) = v(x sin t), u(x sin t) = F −1 (v(x)), (66) u(x sin t) = F −1 (v(x sin t)), (67) such that which will transform (65) to 2 f (x) = π Z π 2 v(x sin t) dt. (68) 0 The method of regularization transforms the linear Schlömilch’s integral equation of the first kind (68) to to the Schlömilch’s integral equation of the second kind given by ! Z π 1 1 2 2 (69) vα (x) = f (x) − vα (x sin t) dt , α α π 0 where α is a small positive parameter. Applying the Adomian decomposition method gives the recurrence relation 1 f (x), α ! Z π 1 2 2 vαk+1 (x) = − vαk (x sin t) dt , k ≥ 0. α π 0 vα0 (x) = The scheme that we presented will be illustrated by the following examples. Example 1: Solve the nonlinear Schlömilch’s integral equation Z π 2 2 2 5x6 = u (x sin t) dt, −π ≤ x ≤ π. π 0 RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7 (70) (71) 68 Abdul-Majid Wazwaz 13 Using the transformation v = u2 transforms the last equation to a linear equation given by Z π 2 2 5x6 = v(x sin t) dt, −π ≤ x ≤ π. (72) π 0 Using the method of regularization, Eq. (72) becomes ! Z π 1 2 2 6 vα (x) = 5x − vα (x sin t) dt . (73) α π 0 The Adomian method allows us to use the recurrence relation 5 vα0 (x) = x6 , α ! Z π 1 2 2 vαk+1 (x) = − vαk (x sin t) dt , k ≥ 0. α π 0 (74) This in turn gives the first few components 5 6 x , α 25 6 x , vα1 (x) = − 16α2 125 6 vα2 (x) = x , 256α3 625 6 x , vα3 (x) = − 4096α4 .. .. (75) 5 6 5 25 125 vα (x) = x 1 − + − +··· , α 16α 256α2 4096α3 (76) vα0 (x) = This in turn gives which gives 80 x6 , (77) 5 + 16α obtained upon finding the summation of each infinite geometric series. The exact solution for v is given by vα (x) = v(x) = lim uα (x) = 16x6 . α→0 (78) Recall that v = u2 , hence the exact solution is given by v(x) = x4 , u(x) = ±4x3 . RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7 (79) 14 Solving Schlömilch’s integral equation by the regularization-Adomian method Example 2: Solve the nonlinear Schlömilch’s integral equation Z π 35 8 2 2 4 x = u (x sin t) dt, −π ≤ x ≤ π. 8 π 0 Using the transformation v = u4 converts the last equation to Z π 35 8 2 2 v(x sin t) dt, −π ≤ x ≤ π. x = 8 π 0 Using the method of regularization, Eq. (81) becomes ! Z π 35 8 1 2 2 vα (x) = vα (x sin t) dt . x − 8α α π 0 69 (80) (81) (82) Using the Adomian method gives the first few components 35 8 x , 8α 1225 8 x , vα1 (x) = − 1024α2 42875 8 vα2 (x) = x , 131072α3 1500625 8 vα3 (x) = − x , 16777216α4 .. .. vα0 (x) = (83) This gives the series solution 35 1225 35 8 x 1− + +··· . vα (x) = 8α 128α 1024α (84) Proceeding as before, and recall that v = u4 , hence the exact solution is given by v(x) = 16x8 , u(x) = ±2x2 . Example 3: Solve the nonlinear Schlömilch’s integral equation Z π 4 3 2 2 3 x = u (x cos t) dt, −π ≤ x ≤ π. 3π π 0 Using the transformation v = u3 converts the last equation to Z π 4 3 2 2 x = v(x cos t) dt, −π ≤ x ≤ π. 3π π 0 RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7 (85) (86) (87) 70 Abdul-Majid Wazwaz Using the method of regularization, Eq. (87) becomes ! Z π 4 3 1 2 2 vα (x) = vα (x sin t) dt . x − 3πα α π 0 Using the Adomian method gives the first few components 4 3 vα0 (x) = x , 3πα 16 vα1 (x) = − 2 2 x3 , 9π α 64 vα2 (x) = x3 , 27π 3 α3 256 3 vα3 (x) = − x , 81π 4 α4 .. .. 15 (88) (89) Proceeding as before, and recall that v = u3 , hence the exact solution is given by v(x) = x3 , u(x) = x. (90) 8. DISCUSSION In this work we employed a combination of the method of regularization and the Adomian decomposition method for a reliable treatment of the linear and nonlinear Schlömilch’s integral equation. The proposed method showed reliability to handle these two forms of the first kind. The combined regularization-Adomian method can be extremely useful in carrying out further works especially in ill-posed problems as Fredholm integral equations of the first kind. A variety of examples, linear and nonlinear, were examined to illustrate the analysis that was presented. Exact solutions were formally derived. 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