c Pleiades Publishing, Ltd., 2015. ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2015, Vol. 9, No. 2, pp. 227–240. c A.A. Kosov, E.I. Semenov, A.V. Sinitsyn, 2015, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2015, Vol. XVIII, No. 1, pp. 69–83. Original Russian Text
Constructing Solutions for the Systems of the Nonlinear Equations Modeling Magnetic Insulation A. A. Kosov1* , E. I. Semenov1** , and A. V. Sinitsyn2*** 1
Institute for System Dynamics and Control Theory, ul. Lermontova 134, Irkutsk, 664033 Russia 2 Universidad Nacional de Colombia, Carrera 45, Bogota, Colombia Received August 11, 2014
Abstract—Under consideration is some model of magnetic insulation of a vacuum diode. It is represented by a system of two nonlinear ordinary differential equations of the second order. The integrability of the system under study is substantiated, and a method for solving a singular boundary value problem is developed. A generalized model of magnetic insulation with multidimensional Laplace operator is suggested, and this is the main study object in the paper. The conditions are obtained under which the exact solutions are found of the boundary value problem for a spherical layer. DOI: 10.1134/S1990478915020088 Keywords: solution method for a singular boundary value problem, integrability, equation of elliptic type, exact solution
The limit model of magnetic insulation is proposed in [1] for a planar vacuum diode which is represented by a system of two nonlinear ordinary differential equations of the second order. The problem for this system is to find the solution of a singular boundary value problem and investigate its properties. Some results on this model were obtained earlier using both analytical and numerical methods [1–5]. In the present article, we continue to study the model of [1]. Our main goal here is to obtain a complete system of first integrals, develop a method of solution of the corresponding singular boundary value problem, as well as to obtain the exact solutions for a generalized model of magnetic insulation with multidimensional Laplace operator. Note that an important role of construction of exact solutions of the nonlinear systems of partial differential equations was indicated in [6–9]. 1. DESCRIPTION OF THE MODEL AND STATEMENT OF THE PROBLEM The limit model of a planar vacuum diode was obtained in [1]. This model is represented by a system of two nonlinear ordinary differential equations of the second order of the following form: (1 + ϕ) d2 ϕ , = j 2 dx (1 + ϕ)2 − a2 − 1
d2 a a . = j 2 dx (1 + ϕ)2 − a2 − 1
(1.1)
Here the independent variable x ∈ [0, 1] expresses the relative distance from the cathode, and x = 1 corresponds to the anode. The function ϕ(x) describes the change of the electric field potential when moving from the cathode to the anode; the function a(x) is the magnetic field potential, and j is the density of the current through the diode. System (1.1) describes the electric and magnetic fields inside the diode, and its solution satisfies the boundary conditions dϕ (0) = 0, (1.2) ϕ(0) = 0, a(0) = 0, ϕ (0) = dx a(1) = a1 . (1.3) ϕ(1) = ϕ1 , *
E-mail:
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Note that the boundary value problem (1.1)–(1.3) is singular since conditions (1.2) inserting into (1.1) lead to zero denominator. Thus, the classical definition of solution as a pair of functions (ϕ(x), a(x)) satisfying (1.2), (1.3) and making (1.1) an identity for x ∈ [0, 1] (while understanding the derivatives at the endpoint of the segment as one-sided) is inapplicable to this problem; and it is necessary to define what we mean by a solution of (1.1)–(1.3). Moreover, in solving (1.1)–(1.3) it is assumed [1] that the parameter j is free and should be found together with the solution of the boundary value problem. We consider system (1.1) in the domain Ω = {(ϕ, a) : (1 + ϕ)2 − a2 − 1 > 0} such that, on each compact subsets of Ω, the right-hand sides of (1.1) have the bounded partial derivatives and, thus, the conditions hold of the existence and uniqueness theorem for the solution of the Cauchy problem for (1.1). Moreover, by an obvious symmetry, we can study only the solutions with positive 1 + ϕ(x) and a(x); i.e., we can carry out our entire reasoning in the domain Ω+ = Ω ∩ {(ϕ, a) : 1 + ϕ > 0, a > 0}. Let the conditions of the existence and uniqueness theorem for the Cauchy problem hold at the right endpoint of the segment: Θ1 = (1 + ϕ1 )2 − 1 − a21 > 0. Definition 1. Let a solution of the problem (1.1)–(1.3) be understood to be a twice differentiable function (ϕ(x), a(x)) defined on the interval x ∈]0, 1] and with values in Ω+ and such that (1) it satisfies the boundary conditions at the right endpoint ϕ(1) = ϕ1 and a(1) = a1 ; (2) on each segment x ∈ [ε, 1], 0 < ε < 1, it turns the original equations (1.1) into identities when inserted there (the derivatives at the endpoints of the segment are considered to be one-sided); (3) the limits exist lim ϕ(ε) = 0,
ε→+0
lim a(ε) = 0,
ε→+0
lim ϕ (ε) = 0.
ε→+0
At the point x = 0 we extend the definition of this function by continuity, while taking into account the first two equalities from its third property. This definition (a) does not require insertion of the boundary conditions at the left endpoint of the segment into the system, which allows escaping the necessity of division by zero; (b) imposes no restrictions on the behavior of the first derivative a (x) and the second derivatives at the left endpoint of the segment; (c) can be modified in an obvious way also for the case Θ1 = (1 + ϕ1 )2 − 1 − a21 = 0, when there is a singularity at the right endpoint of the segment. In addition, within the framework of this definition, a situation is allowed when the limit of the derivative lim a (ε)
ε→+0
does not exist or equals infinity. One of our goals is to develop a method for constructing solutions of the singular boundary value problem (1.1)–(1.3) in the sense of the above definition. To do this, we show that the system (1.1) can be integrated in quadratures and construct a complete system of the first integrals. JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS
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2. THE HAMILTONIAN REPRESENTATION AND INTEGRABILITY We introduce the new notations for the independent and dependent variables: t = x,
q1 = ϕ(x),
q2 = a(x),
p1 = −ϕ (x),
p2 = a (x),
q = col(q1 , q2 ) ∈ R2 ,
p = col(p1 , p2 ) ∈ R2 , and consider the Hamiltonian function H(q, p) = − p21 + p22 /2 + j (1 + q1 )2 − q22 − 1. Then (1.1) can be written in the canonical form: q˙ =
∂H , ∂p
p˙ = −
∂H . ∂q
(2.1)
Correspondingly, the boundary conditions in the new variables are rewritten as (2.2) (2.3)
q2 (0) = 0, p1 (0) = 0, q1 (0) = 0, q2 (1) = Q2 ≡ a1 . q1 (1) = Q1 ≡ ϕ1 ,
The new boundary value problem (2.1)–(2.3) is completely equivalent to the original (1.1)–(1.3) and differs from it only by the Hamiltonian form of presenting the system of differential equations, which allows us to use the integration methods developed for the problems of analytical mechanics [10]. The Hamiltonian system (2.1) has the energy integral 1 2 2 − p1 + p2 + j (1 + q1 )2 − q22 − 1 = c1 = const. J1 ≡ H(q, p) = 2
(2.4)
It can be easily checked that another first integral of system (2.1) has the form (2.5)
J2 = (1 + q1 )p2 + q2 p1 = c2 = const. The first integrals J1 and J2 do not depend explicitly on time and are in involution. In the domain D = {(q1 , q2 , p1 , p2 ) : (q1 , q2 ) ∈ Ω+ , (p1 , p2 ) ∈ R2 }, the rank of the Jacobian matrix for the first integrals J1 and J2 becomes less than 2 only on M = (q1 , q2 , p1 , p2 ) : (q1 , q2 ) ∈ Ω+ , √ √ j q2 j (1 + q1 ) p1 = ± 1/4 , p2 = ± 1/4 . (1 + q1 )2 − q22 − 1 (1 + q1 )2 − q22 − 1
Note that M is two-dimensional and does not divide the four-dimensional set D into subdomains. In what follows, we restrict consideration to the domain D \ M where J1 and J2 are functionally independent. So, by the Liouville theorem [10] the model (2.1) of the diode is integrable in this domain. In order to construct two more first integrals which we need, we express from (2.4) and (2.5) the momenta in terms of coordinates √ c22 − 2z 2 (c1 − j z 2 − 1) c2 q2 (1 + q1 ), (2.6) p1 = − 2 ∓ z z2 √ c22 − 2z 2 (c1 − j z 2 − 1) c2 (1 + q1 ) ± q2 , (2.7) p2 = z2 z2 where z 2 = (1 + q1 )2 − q22 . These formulas can be presented as p1 =
∂Φ , ∂q1
p2 =
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where Φ(q1 , q2 , c1 , c2 ) is given by the formula
√ √ (1+Q1 )2 −Q22 c22 − 2z 2 (c1 − j z 2 − 1) c2 1 + q1 + q2 Φ(q1 , q2 , c1 , c2 ) = ln dz. ± 2 1 + q1 − q2 √ z
(2.8)
(1+q1 )2 −q22
The integral in (2.8) is reduced to the elementary and elliptic functions. By the Liouville theorem, the two additional first integrals are expressed through the function Φ(q1 , q2 , c1 , c2 ): √ (1+Q1 )2 −Q22 ∂Φ 1 1 + q1 + q2 c2 dz = ln (2.9) ± J3 ≡ √ = c3 , ∂c2 2 1 + q1 − q2 2 − 2z 2 c − j z 2 − 1 √ z c 1 2 2 2 √ ∂Φ =∓ J4 ≡ ∂c1 √
(1+q1 ) −q2
(1+Q1 )2 −Q22
(1+q1 )2 −q22
zdz
c22
− 2z 2 (c1
−j
√
z2
− 1)
= t + c4 ,
(2.10)
where c3 and c4 are some constants. Note that, naturally, we use (2.9) and (2.10) only for nonzero radicals in the denominator. Since (1 + q1 )p1 + q2 p2 ≡ 0 on M, whereas the formulas (2.6) and (2.7) yield (1 + q1 )p1 + q2 p2 = ∓ c22 − 2z 2 c1 − j z 2 − 1 ; therefore, the nonzero radicals in the denominators of (2.9) and (2.10) provide also the independence of the first integrals J3 and J4 . 3. SOLUTION OF THE SINGULAR BOUNDARY VALUE PROBLEM Putting in (2.10) t = 1 and using (2.3) for the right endpoint of the segment, we obtain c4 = −1. By analogy, we find from (2.9) 1 1 + Q1 + Q2 . c3 = ln 2 1 + Q1 − Q2 It follows from (1.2) or (2.2) at the left endpoint of the segment and integrals (2.4) and (2.5) that p1 (0) = 0, p2 (0) = c2 , and c22 = 2c1 . Let us consider the functions √ (1+Q1 )2 −Q22 s ds , F (u, v, Q1 , Q2 ) = √ 2 (1 − s2 ) + 2vs2 s2 − 1 u 1 √ (1+Q1 )2 −Q22 u ds . G(u, v, Q1 , Q2 ) = √ 2 (1 − s2 ) + 2vs2 s2 − 1 s u 1 These functions can be represented as combinations of some elementary and elliptic functions of their arguments. Now, from (2.9) and (2.10) we can derive the following theorem taking into account the relations between the arbitrary constants obtained from the boundary conditions: Theorem 1. If c2 = u∗ and j = v∗ form a solution of the system of two nonlinear equations 1 1 + Q1 + Q2 G(u, v, Q1 , Q2 ) = ln (3.1) F (u, v, Q1 , Q2 ) = 1, 2 1 + Q1 − Q2 JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS
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then a solution of the boundary value problem (2.1)–(2.3) exists and is also a solution of the Cauchy problem for (2.1) with the initial conditions at the right end of the segment which are determined by the equalities following from (2.6) and (2.7): √ u2∗ (1 − Z 2 ) + 2v∗ Z 2 Z 2 − 1 u∗ Q2 q1 (1) = Q1 , p1 (1) = − 2 − (1 + Q1 ), (3.2) Z Z2 √ u2∗ (1 − Z 2 ) + 2v∗ Z 2 Z 2 − 1 u∗ (1 + Q1 ) p2 (1) = + Q2 , (3.3) q2 (1) = Q2 , Z2 Z2 where Z 2 = (1 + Q1 )2 − Q22 . Note that if (3.1) is incompatible then it does not mean that the boundary value problem (2.1)–(2.3) has no solutions. There are situations when, on a solution of the boundary value problem, a sign change occurs in the expressions (2.6) and (2.7) for the momenta in terms of coordinates, and in this case (3.1) do not hold longer. In [1] and [5], a solution of the original singular boundary value problem (1.1)–(1.3) was obtained as a solution of the initial value problem for (1.1) at the left endpoint of the segment: dϕ da (0) = 0, a (0) ≡ (0) = C > 0. (3.4) dx dx Note that the conditions of the standard existence theorems for the Cauchy problem do not hold for (1.1), (3.4); but it is not explained in [1] and [5] how we could understand a solution of this initial value problem. Let us show by two examples that this approach may result in losing the solutions of the singular boundary value problem. ϕ(0) = 0,
a(0) = 0,
ϕ (0) ≡
Example 1. Consider the following singular boundary value problem: 12x2 − 1 d2 ϕ = ϕ(x) + 6a(x), dx2 x4 ϕ(0) = 0,
d2 a a(x) , = − 2 2 dx ϕ (x) + x6 a2 (x) dϕ (0) = 0, dx a(1) = a1 = sin 1.
(3.6)
a(0) = 0,
ϕ(1) = ϕ1 = cos 1,
(3.5)
(3.7)
The right-hand sides of (3.5), the same as for (1.1), have a singularity only at the left endpoint x = 0 of the segment. An exact solution of problem (3.5)–(3.7) in the sense of Definition 1 is given by the pair of functions 1 1 a(x) = x sin . (3.8) ϕ(x) = x4 cos , x x Here we have 1 1 1 a (x) = sin − cos , x x x therefore, lim a (x) as x → +0 does not exist. Thus, there does not exist C ∈ R for which (3.8) would be a solution of the initial value problem (3.5), (3.4) according to [1, 5]. Example 2. Consider the singular boundary value problem 3 d2 ϕ , = 2 dx 4a(x)
d2 a 1 , =− 2 dx 4ϕ(x) dϕ (0) = 0, ϕ(0) = 0, a(0) = 0, dx a(1) = a1 = 1. ϕ(1) = ϕ1 = 1, JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS
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A solution of (3.9)–(3.11) in the sense of Definition 1 is given by the pair of function ϕ(x) = x3/2 ,
a(x) = x1/2 .
(3.12) √ Here we have a (x) = 1/(2 x); therefore, lim a (x) = +∞. Thus, there does not exist C ∈ R for which x→+0
(3.12) would be, by [1, 5], a solution of the initial value problem (3.9), (3.4). Let us note that, surely, the above examples neither disprove nor question the above-mentioned results from [1] and [5]. These examples clearly demonstrate that, in the general case, the transition from the boundary value problem with a singularity at the left endpoint of the segment to the initial value problem at the left endpoint of the segment is not correct and may lead to losing a solution of the singular boundary value problem. Our approach to solving the singular boundary value problem (1.1)–(1.3) differs principally from the approaches of [1, 5]. First, we use Definition 1 which allows us to avoid the substitution of the initial conditions (1.2) at the left endpoint of the segment for x = 0 and so avoid division by zero. Second, we use the initial value problem only at the right endpoint of the segment for x = 1, where the conditions of the existence and uniqueness theorem are satisfied. The initial conditions themselves for the right endpoint of the segment can be obtained from solving the system of two nonlinear equations (3.1) according to Theorem 1. This allows us to calculate a solution of the singular boundary value problem (1.1)–(1.3) only by numerical integration of (1.1) from right to left, and the standard numerical methods can be used. 4. A GENERALIZED MODEL OF MAGNETIC INSULATION Consider now the system of two partial differential equations of the following form: Δψ ≡
∂2ψ ψ ∂2ψ , + · · · + = j(x) 2 2 2 ∂xn ∂x1 ψ − a2 − 1
(4.1)
∂2a a ∂2a . + · · · + = j(x) Δa ≡ 2 2 ∂xn ∂x1 ψ 2 − a2 − 1
This is a generalization of the magnetic insulation model of [1], which we obtain by introducing the new variable ψ = 1 + ϕ, replacing the second derivative on the left-hand sides by the multidimensional Laplace operator, and using on the right-hand sides the current density j(x) that depends on the spatial variables, x ∈ Rn , n ∈ N, n ≥ 2. In the case of n = 2, if the current density j(x1 , x2 ) is the squared gradient of an arbitrary harmonic function then it can be shown [3] that, by some change of variables, the system (4.1) in partial derivatives is reducible to a system of ordinary differential equations of the form (1.1). Thus, the set of solutions of (1.1) lies certainly in the solution set of (4.1). Therefore, we will call henceforth (4.1) the generalized model of magnetic insulation. ¯ Let ψ = ψ(x) and a = a ¯(x) be solutions of (4.1). Then we obtain from (4.1) that Δa ψ 2 − a2 − 1 Δψ ψ 2 − a2 − 1 , j(x) ≡ . j(x) ≡ ψ a Moreover, since the radical ψ 2 − a2 − 1 must be positive, by cancelling it out and taking it into ¯ account that j(x) ≥ 0, we obtain from the two previous identities that ψ = ψ(x) and a = a ¯(x) must satisfy the identities Δa Δψ ≡ ≡ λ(x) ≥ 0. (4.2) ψ a Here on the right-hand side of (4.2) λ(x) is some nonnegative function. Thus, we have proved ¯ Theorem 2. The solutions ψ = ψ(x) and a = a ¯(x) of (4.1) there only be the solutions of the second order linear equation of the following form: Δu = λ(x)u, (4.3) JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS
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¯ where λ(x) ≥ 0 is a certain nonnegative function specific for each solution (ψ(x), a ¯(x)). It follows from this theorem that the solutions of (4.1) inherit some properties of the solutions of the Helmholtz equation (4.3). In particular, the maximum value principle [13] holds for (4.1) in the following form: Proposition 1. If the density function j(x) is positive inside some domain D ⊂ Rn with the boundary ∂D then, for every solution (ψ(x), a(x)) of (4.1) inside D, both components cannot attain the positive maximum and negative minimum values at the interior points of D. Let us show that, using the connection between solutions of (4.1) and (4.3), we can construct the exact solutions of (4.1) by selection some appropriate current density function; i.e., using the current density essentially as a control that provides the exact solutions of a certain class. Fix a certain nonnegative function λ(x) ≥ 0 (for example, some constant) and take two arbitrary ¯ solutions ψ = ψ(x) and a = a ¯(x) of (4.3). Then in the domain ¯ = 0, a ¯(x) = 0, ψ¯2 (x) − a ¯2 (x) − 1 > 0} D = {x ∈ Rn : ψ(x) these functions will be exact solutions of nonlinear system (4.1) for the current density given by the formula: ¯2 (x) − 1 ≥ 0. (4.4) j = λ(x) ψ¯2 (x) − a Thus, we proved the statement that is converse to Theorem 2: ¯ Theorem 3. Let the functions ψ = ψ(x) and a = a ¯(x) be solutions of (4.3) in D ⊂ Rn for some function λ(x) ≥ 0 nonnegative in this domain, while everywhere in D the following hold: ¯ ¯2 (x) − 1 > 0. ψ(x) = 0, a ¯(x) = 0, ψ¯2 (x) − a ¯ Then ψ = ψ(x) and a = a ¯(x) are solutions in D of (4.1) in which the current density is given by (4.4). Let us look for a solution of (4.1) in the form √ ¯ ψ(x) = 2 cosh(ξ(x)),
a ¯(x) =
√
2 sinh(ξ(x)),
(4.5)
where ξ(x) = 0 is an arbitrary, for the time being, function which is twice differentiable with respect to ¯2 − 1 = 1, and we obtain from (4.4) that the variables (x1 , . . . , xn ). It is easy to check that ψ¯2 − a ¯ we have j = λ(x). Let us insert (4.5) into (4.3) and express the function λ(x). For ψ, λ(x) =
sinh(ξ) Δξ + |∇ξ|2 cosh(ξ)
and, correspondingly, for a ¯ we obtain cosh(ξ) Δξ + |∇ξ|2 . sinh(ξ) Since λ(x) should be the same for both functions ψ¯ and a ¯; therefore, comparing the last two relations, we demand the fulfilment of the equality Δξ(x) = 0. In this case, λ(x) =
j = λ(x) = |∇ξ|2 ≥ 0,
(4.6)
where ξ(x) is an arbitrary nonconstant harmonic function. Thus, taking Theorems 2 and 3 into account, we arrive at Theorem 4. The generalized model of magnetic insulation Δψ ≡
∂2ψ ψ ∂2ψ , + · · · + = |∇ξ|2 2 2 2 ∂xn ∂x1 ψ − a2 − 1
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where ξ(x) is an arbitrary nonconstant harmonic function, has the exact solution (4.5). Corollary 1. In the case of n = 2, system (4.7) of the form k−1 ∂2ψ ∂2ψ ψ + = k2 x21 + x22 , 2 2 2 ∂x1 ∂x2 ψ − a2 − 1
(4.8)
k−1 a ∂2a ∂2a + 2 = k2 x21 + x22 2 2 ∂x1 ∂x2 ψ − a2 − 1
has two one-parametric families of exact anisotropic solutions of the following form: √ √ a1 (x1 , x2 ) = 2 sinh r k cos(kθ1 ) + c01 , (4.9) ψ1 (x1 , x2 ) = 2 cosh r k cos(kθ1 ) + c01 , √ √ k k a2 (x1 , x2 ) = 2 sinh r sin(kθ2 ) + c02 , (4.10) ψ2 (x1 , x2 ) = 2 cosh r sin(kθ2 ) + c02 , where r = x21 + x22 , θ1 = arccos(x1 /r), θ2 = arcsin(x2 /r), k ≥ 2, k ∈ N, and c01 , c02 are arbitrary constants. Example 3. Let n = 3, and let us take a particular exact solution of the Laplace equation of the form f (x1 , x2 , x3 ) =
1 (x1 − x10
)2
+ (x2 − x20 )2 + (x3 − x30 )2
which has a simple physical interpretation, namely, it is the potential of the unit charge at the point (x10 , x20 , x30 ). Then it follows from Theorem 4 that the functions
√ 1 + C0 , ψ(x1 , x2 , x3 ) = 2 cosh (x1 − x10 )2 + (x2 − x20 )2 + (x3 − x30 )2
√ 1 + C0 , a(x1 , x2 , x3 ) = 2 sinh (x1 − x10 )2 + (x2 − x20 )2 + (x3 − x30 )2 where x10 , x20 , x30 , and C0 are some arbitrary constants, are the exact solutions of the following system: 1 ψ , 2 ((x1 − x10 )2 + (x2 − x20 )2 + (x3 − x30 )2 )2 ψ − a2 − 1 1 a . Δx1 x2 x3 a = 2 ((x1 − x10 )2 + (x2 − x20 )2 + (x3 − x30 )2 )2 ψ − a2 − 1
Δx1 x2 x3 ψ =
Note that in this example √ the current density j(x1 , x2 , x3 ) → 0 as (x1 , x2 , x3 ) → ∞, whereas the functions ψ(x1 , x2 , x3 ) → 2 cosh(C0 ) and a(x1 , x2 , x3 ) → sinh(C0 ); i.e., the density vanishes at infinity, while the solution stabilizes towards a certain constant value. However, in the general case, the vanishing of the current density at infinity does not yet guarantee stabilization of this kind for all solutions. For instance, for n = 3, the system (4.1) with the density 3r 5 + 11r 4 + 8r 3 + 16r + 16 √ > 0, r = x21 + x22 + x23 > 0, j(r) = 4r 4 (2 + r)2 1 + r has the exact solution √ 2+r cosh (2 + r)8 exp(r 2 − 4r) , ψ(r) = √ 2
√ a(r) =
2+r √ sinh (2 + r)8 exp(r 2 − 4r) 2
which grows unboundedly at infinity, while, at the same time, the density tends to zero. Example 4. The second order linear partial differential equation (4.3) for 2 2 |x| = x21 + x22 + · · · + x2n = 0, λ(x1 , . . . , xn ) = m /|x| > 0,
n ≥ 2,
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has the exact radially symmetric solutions: u(x) ≡ ψ(x) = A|x|− 2 n+1+ 2 1
1
− 12 n+1+ 12
u(x) ≡ a(x) = A|x|
√ √
4m2 +n2 −4n+4
+ B|x|− 2 n+1− 2
4m2 +n2 −4n+4
− 12 n+1− 12
1
− B|x|
1
√
235
4m2 +n2 −4n+4
√ 4m2 +n2 −4n+4
,
(4.11)
,
where A > 0 and B > 0 are arbitrary constants. For these functions, the current density defined by (4.4) has the form m2 4AB|x|2−n − 1 . j(x) = |x|2 We consider this current density in the domain D = {x ∈ Rn : 4AB|x|2−n − 1 ≥ 0}, where the functions (4.11) satisfy the conditions of Theorem 3 and, because of that, are the solutions in D of the generalized model of magnetic insulation of the following form: m2 4AB|x|2−n − 1 ψ(x) , Δψ(x) = 2 2 |x| ψ (x) − a2 (x) − 1 m2 4AB|x|2−n − 1 a(x) . Δa(x) = 2 |x|2 ψ (x) − a2 (x) − 1 Note that, in the case of n = 2 and for j(x1 , x2 ) ≡ j0 = const, system (4.1) is a particular case of the system in [12] of the form Δx1 x2 a = aF (ψ 2 − a2 ) + ψG(ψ 2 − a2 ), Δx1 x2 ψ = ψF (ψ 2 − a2 ) + aG(ψ 2 − a2 ), where F (ψ 2 − a2 ) = j0 / ψ 2 − a2 − 1 and G(ψ 2 − a2 ) ≡ 0 in our case.
(4.12)
In [12], it is suggested to look for a solution of (4.12) in the form a(x1 , x2 ) = r(z) sinh θ(z) + C1 x2 + C2 . ψ(x1 , x2 ) = r(z) cosh θ(z) + C1 x2 + C2 , Here the following notations are used: z = k1 x1 + k2 x2 ; C1 , C2 , k1 , and k2 are arbitrary constants, while the functions r(z) and θ(z) are to be determined from a system of two second order nonlinear ordinary differential equations. Following the solution structure proposed in [12], we continue considering the multidimensional system (4.1) in the general case of a variable current density. We look for a solution of (4.1) in the form ψ(x) = f cosh(ω),
(4.13)
a(x) = f sinh(ω),
where f = f (x), ω = ω(x) are yet arbitrary functions, twice differentiable with respect to the variables (x1 , . . . , xn ). After inserting (4.13) into (4.1) we obtain f cosh(ω) , [Δf + f |∇ω|2 ] cosh(ω) + [f Δω + 2∇f · ∇ω] sinh(ω) = j(x) f2 − 1 f sinh(ω) . [Δf + f |∇ω|2 ] sinh(ω) + [f Δω + 2∇f · ∇ω] cosh(ω) = j(x) f2 − 1 From now on the symbol (·) means the inner product. Divide both parts of the last equalities, respectively, by cosh(ω) and sinh(ω): Δf + f |∇ω|2 +
f sinh(ω) , [f Δω + 2∇f · ∇ω] = j(x) cosh(ω) f2 − 1
(4.14)
Δf + f |∇ω|2 +
f cosh(ω) . [f Δω + 2∇f · ∇ω] = j(x) sinh(ω) f2 − 1
(4.15)
Since the right-hand sides of (4.14) and (4.15) coincide, the left-hand sides must coincide as well, which is possible only when the expression in the brackets equals zero. Thus, formulas (4.14) and (4.15) are JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS
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reduced to joint fulfillment of two equalities: Δf + f |∇ω|2 = j(x)
f
f2 − 1 f Δω + 2∇f · ∇ω = 0.
(4.16)
,
(4.17)
Hence, solvability of (4.1) in the form of ansatz (4.13) is reduced to solvability of the system of two partial differential equations of the second order (4.16) and (4.17). Thus, we arrive at Theorem 5. If f (x) and ω(x) satisfy equations (4.16) and (4.17) then system (4.1) has the exact solution (4.13). √ Remark 1. Theorem 4 is a particular case of Theorem 5 if we put in the latter f (x) ≡ 2. The problem of constructing exact solutions of the nonlinear system of partial differential equations (4.16) and (4.17) for an arbitrary function j(x) is rather nontrivial. However, for certain types of the density j(x), equations (4.16) and (4.17) possess solutions f (x) and ω(x) from the class of harmonic functions. Thus, the following two statements are true: Theorem 6. If the density of the current in (4.1) is representable as j(x) = |∇ω|2 f 2 − 1,
(4.18)
where f (x) and ω(x) are arbitrary harmonic functions and their gradients satisfy the orthogonality condition ∇f · ∇ω = 0, then this system has a solution of the form (4.13) in the domain D = {x ∈ Rn : f (x) > 1}. Proof. By assumptions, f (x) and ω(x) are harmonic and their gradients are orthogonal. Then (4.17) holds identically. In turn, if the current density j(x) has the form (4.18) then equation (4.16) becomes an identity too. The proof is over.
Remark 2. In the case of n = 2, the functions f (x1 , x2 ) and ω(x1 , x2 ) in Theorem 7 can be chosen to be conjugate harmonic, so for them the orthogonality condition for their gradients ∇f · ∇ω = 0 is automatically satisfied. Theorem 7. Let the function f (x) be harmonic in some domain D = {x ∈ Rn : f (x) > 1}, whereas the current density is representable as (4.19) j(x) = f −4 f 2 − 1|∇f |2 . Then (4.1) possesses an exact solution of the form ψ(x) = f (x) cosh 1/f (x) + const , a(x) = f (x) sinh 1/f (x) + const .
(4.20)
Proof. Let us verify that (4.1) with the current density (4.19) has the exact solution (4.20). By Theorem 5, it suffices to show that f (x) and ω(x) = 1/f (x) + const, where f (x) is harmonic, satisfy (4.16) and (4.17). It is not difficult to check that |∇ω|2 =
|∇f |2 , f4
Δω = 2
|∇f |2 , f3
∇f · ∇ω = −
|∇f |2 . f2
Taking these into account, we see that (4.17) is an identity, whereas (4.16) turns into the identity provided the current density has the form (4.19). This proves the theorem. JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS
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Example 5. System (4.1) in the two-dimensional case of n = 2 with the current density 2 2 2 2 j(x1 , x2 ) = 4 x1 + x2 x1 − x22 − 1
has the exact solution in the domain D = (x1 , x2 ) : x21 − x22 > 1 : a(x1 , x2 ) = x21 − x22 sinh(2x1 x2 ). ψ(x1 , x2 ) = x21 − x22 cosh(2x1 x2 ), Let us turn to studying the system of partial differential equations (4.16), (4.17). For the linear function ω(x) = d · x, (4.21) where d = (d1 , d2 , . . . , dn ) ∈ Rn is an arbitrary constant vector, |d| = 0, the system (4.16), (4.17) is reduced to the equation for f (x) of the form Δf + |d|2 f = j(x)
f
(4.22)
f2 − 1
with the additional orthogonality condition ∇f · d ≡ d1
∂f ∂f ∂f + d2 + · · · + dn = 0. ∂x1 ∂x2 ∂xn
(4.23)
Relation (4.23) is a linear partial differential equation of the first order for f which can be easily integrated by choosing an appropriate integral basis. For example, in the three-dimensional case n = 3, it has the general solution of the form f (x1 , x2 , x3 ) = f (ui , vi ),
(4.24)
i = 1, 2, 3,
where u1 = d2 x1 − d1 x2 , u2 = d3 x2 − d2 x3 , u3 = d1 x3 − d3 x1 ,
v1 = d3 x1 − d1 x3 , v2 = d1 x2 − d2 x1 , v3 = d2 x3 − d3 x2 ,
d1 = 0, d2 = 0, d3 = 0.
(4.25)
In what follows, for the sake of briefness, we consider only the solutions with the variables (u1 = u, v1 = v) because the functions with the variables (u2 , v2 ) and (u3 , v3 ) can be obtained from f (u1 , v1 ) by a cyclic permutation of (x1 x2 x3 ). In this case, the three-dimensional Laplace operator in (4.22) in terms of u and v is written as (4.26) Δf ≡ fx1 x1 + fx2 x2 + fx3 x3 = d21 + d22 fuu + 2d2 d3 fuv + d21 + d23 fvv . It is known that the operator on the right-hand side of (4.26) can be reduced by a nondegenerate transformation to canonical form. So, if we put ξ = a1 u + b1 v, where
(4.27)
η = a2 u + b2 v,
d1 |d|2 d2 d3 a2 − 2 a1 , b1 = ∓ 2 2 d1 + d3 d1 + d23
d1 |d|2 d2 d3 b2 = ± 2 a1 − 2 a2 2 d1 + d3 d1 + d23
and a1 , a2 are arbitrary constants not equal simultaneously to zero then from (4.26) we have 2 d1 + d22 fuu + 2d2 d3 fuv + d21 + d23 fvv = μ(fξξ + fηη ). Here
2 2 μ = a1 + a2 1 +
d22 d21 + d23
|d|2 2 d , d21 = a21 + a22 2 d1 + d23 1
while μ > 0; this condition guarantees that (4.27) is nondegenerate. JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS
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Now, let us assume that j(x1 , x2 , x3 ) = j(ξ, η) for the transformation of variables (4.25) and (4.27). Then, instead of (4.22) with the additional condition (4.23), everything is reduced to solvability of the sole equation in the two-dimensional coordinate space of the variables (ξ, η): μ(fξξ + fηη ) + |d|2 f = j(ξ, η)
f f2 − 1
,
(4.29)
f = f (ξ, η).
Thus, we arrive at Theorem 8. If, by transformation (4.25), (4.27), the current density j given in the threedimensional coordinate space of the variables (x1 , x2 , x3 ) is reduced to some function j defined in the two-dimensional coordinate space of the variables (ξ, η) then (4.1) in the three-dimensional case has the exact solution: ψ(x1 , x2 , x3 ) = f (x1 , x2 , x3 ) cosh(d1 x1 + d2 x2 + d3 x3 ), a(x1 , x2 , x3 ) = f (x1 , x2 , x3 ) sinh(d1 x1 + d2 x2 + d3 x3 ), while f (x1 , x2 , x3 ) is determined from (4.29) ξ = a1 (d2 x1 − d1 x2 ) + b1 (d3 x1 − d1 x3 ),
η = a2 (d2 x1 − d1 x2 ) + b2 (d3 x1 − d1 x3 ).
Obviously, the theorem remains valid also in the case of constant current density; i.e., for j(x1 , x2 , x3 ) ≡ j0 = const. Note that if in (4.29) the current density can be represented as j(ξ, η) = μ|∇z|2 + |d|2 sinh(z), where z = z(ξ, η) is an arbitrary harmonic function, whereas the parameter μ is given by (4.28), then equation (4.29) has an exact solution f (ξ, η) = cosh(z(ξ, η)). 5. ON EXACT SOLUTIONS OF THE BOUNDARY VALUE PROBLEM IN A SPHERICAL LAYER Consider (4.1) in the case of the three-dimensional Laplace operator: Δψ ≡
ψ ∂2ψ ∂2ψ ∂2ψ , + + = j(x1 , x2 , x3 ) 2 2 2 ∂x1 ∂x2 ∂x3 ψ 2 − a2 − 1
(5.1)
a ∂2a ∂2ψ ∂2a . + + 2 = j(x1 , x2 , x3 ) Δa ≡ 2 ∂x21 ∂x22 ∂x3 ψ − a2 − 1
Suppose that we need to find a pair of functions ψ = ψ(x1 , x2 , x3 ), a = a(x1 , x2 , x3 ) which are defined in the spherical layer
D = (x1 , x2 , x3 ) : 0 < r¯12 < x21 + x22 + x23 < r¯22 < +∞ , turn (5.1) into identity after substitution, and satisfy the boundary conditions ψ(x1 , x2 , x3 ) = ψ¯1 , ψ(x1 , x2 , x3 ) = ψ¯2 ,
a(x1 , x2 , x3 ) = a ¯1 ,
x21 + x22 + x23 = r¯12 ,
(5.2)
a(x1 , x2 , x3 ) = a ¯2 ,
x21 + x22 + x23 = r¯22 .
(5.3)
¯i , and r¯i with i = 1, 2 are some given positive numbers, whereas a nonnegative function Here ψ¯i , a j(x1 , x2 , x3 ) ≥ 0 is to be determined together with the solution ψ(x1 , x2 , x3 ), a(x1 , x2 , x3 ). Let us demonstrate that, in some cases, a solution of the boundary value problem for the spherical layer can be obtained in explicit form. JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS
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Theorem 9. If the boundary values satisfy the relations ψ¯2 − a ¯2 > 1, i
arccosh
ψ¯i
(5.4)
i
= arcsinh
a ¯i
(5.5)
, i = 1, 2, ψ¯i2 − a ψ¯i2 − a ¯2i ¯2i ¯21 − 1 = r¯22 ψ¯22 − a ¯22 − 1 r¯12 ψ¯12 − a
(5.6)
then the boundary value problem (5.1)–(5.3) has an exact solution given by the formulas √ r B A2 + r 2 cosh arctan +C , ψ(x1 , x2 , x3 ) = r A A √ r B A2 + r 2 sinh arctan +C , a(x1 , x2 , x3 ) = r A A for the following values of the parameters:
A = r¯i arccosh
C=−
(5.8)
¯2i − 1, ψ¯i2 − a
ψ¯2 ¯22 ψ¯22 − a
− arccosh
(5.9) ψ¯1 ¯21 ψ¯12 − a
, W (¯ r2 ) − W (¯ r1 ) ψ¯1 ψ¯2 − W (¯ r1 ) arccosh W (¯ r2 ) arccosh ¯21 ¯22 ψ¯12 − a ψ¯22 − a B=
(5.7)
W (¯ r2 ) − W (¯ r1 )
(5.10)
,
(5.11)
where W (r) = B/A arctan(r/A). In this case, the corresponding density function is given by j(x1 , x2 , x3 ) =
A(A2 + B 2 ) . r(A2 + r 2 )2
(5.12)
Proof. It can be verified directly that, with the density (5.12), system (4.1) has in the domain r > 0 an exact solution given by (5.7) and (5.8). It remains to show that, under the conditions of the theorem, the three parameters should be taken precisely in accordance with (5.9)–(5.11) in order to guarantee (5.2) and (5.3) on the boundary. Multiplying the difference of formulas (5.7) and (5.8) by their sum and taking (5.6) into account, we obtain (5.9) for A. Moreover, (5.4) provides the positivity of A calculated by (5.9). Now, taking (5.5) into account, we obtain the system of two linear equations for determining the parameters B and C: ψ¯1 ψ¯2 , W (¯ r2 )B + C = arccosh . W (¯ r1 )B + C = arccosh ¯21 ¯22 ψ¯12 − a ψ¯22 − a Since W (r) is monotonous and r¯2 > r¯1 , the determinant of this system is nonzero and the solution is given by (5.10) and (5.11). The theorem is proved. We now consider a boundary value problem in the exterior of the ball; i.e., we assume that only (5.2) is present, whereas (5.3) is absent. Take the function f (r) = 1 + Ar 2 , where A > 0 is a parameter. We find from ω = r 1−n B/f 2 that √ √ Ar 2 B + − 3 A arctan(r A) + ln C, ω(r) = − 2 1 + Ar 2 r where B > 0 and C > 0. For the density √ A(6A5 r 12 + 30A4 r 10 + 54A3 r 8 + 42A2 r 6 + 12Ar 4 + B 2 Ar 2 + 2B 2 ) √ > 0, j(r) = r 3 2 + Ar 2 (1 + Ar 2 )4 JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS
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(5.1) has the exact solution:
√ √ B 2 Ar + A arctan(r A) , ψ(r) = C(1 + Ar 2 ) cosh − − 3 2 1 + Ar 2 r √ √ Ar B 2 2 a(r) = C(1 + Ar ) sinh − + − 3 A arctan(r A) . 2 1 + Ar 2 r
Owing to the boundary condition (5.2), we find the value of the parameter 1 ¯21 , ψ¯12 − a C= 1 + A¯ r12
(5.14) (5.15)
(5.16)
while A > 0 and B remain free. Thus, under a variable density (5.13), formulas (5.14)–(5.16) yield a two-parametric family of the exact solutions of the boundary value problem (5.1), (5.2) in the exterior of the ball. Note that j(r) → 6A = const as r → +∞, and this convergence is very rapid (for r > 5 the density is practically constant already). Therefore, in the domain r > r¯1 ≥ 5 (i.e., in the exterior of the ball) formulas (5.14) and (5.15) with the parameter values (5.16) and A = j0 /6 can be considered as approximation by an oneparameter family of functions of the solution of (5.1), (5.2) for the system with a given constant density j = j0 = const > 0 and given constant values ψ(x1 , x2 , x3 ) = ψ¯1 = const, a(x1 , x2 , x3 ) = a ¯1 = const on the sphere r = r¯1 ≥ 5. ACKNOWLEDGMENTS The authors were supported by the Russian Foundation for Basic Research (project no. 15-0806680), the Council of the President of the Russian Federation for the State Maintenance of Leading Scientific Schools (project no. NSh–5007.2014.9), and the Siberian Division of the Russian Academy of Sciences (Integration project no. 80). REFERENCES 1. N. Ben Abdallah, P. Degond, and F. Mehats, “Mathematical Model of Magnetic Insulation,” Physics of Plasmas 5, 1522–1534 (1998). 2. A. V. Sinitsyn, “Positive Solutions of a Nonlinear Singular Boundary Value Problem of Magnetic Insulation,” Mat. Modelir. 13 (5), 37–52 (2001). 3. E. I. Semenov and A. V. Sinitsyn, “Mathematical Model of Magnetic Insulation of a Vacuum Diode and Its Exact Solutions,” Izv. Irkutsk. Gos. Univ. Ser. Mat. No. 1, 78–91 (2010). 4. A. A. Kosov and A. V. Sinitsyn, “On Construction of the First Integrals for a Class of Nonlinear Syatems,” Izv. Irkutsk. Gos. Univ. Ser. Mat. No. 1, 57–69 (2012). 5. V. P. Varin,” “An Analysis of a Vacuum Diode Model,” Comput. Math. Math. Phys. 53 (2), 194–204 (2013). 6. A. D. Polyanin and V. F. Zaitsev, Handbook on Nonlinear Equations of Mathematical Physics: Exact Solutions (Fizmatlit, Moscow, 2002) [in Russian]. 7. V. V. Pukhnachev, “Exact Solutions of the Equations of Motion of a Maxwell Incompressible Viscoelastic Medium,” Prikl. Mekh. i Tekhn. Fiz. 50 (2), 16–23 (2009). 8. E. A. Vyazmina and A. D. Polyanin, “New Classes of Exact Solutions to General Nonlinear Diffusion-Kinetic Equations,” Teor. Osn. Khim. Tekhnol. 40 (6), 1–10 (2006) [Theor. Found. Chem. Eng. 40 (6), 555–563 (2006)]. 9. V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachev, and A. A. Rodionov, Application of Group-Theoretical Method in Hydrodynamics (Nauka, Novosibirsk, 1994) [in Russian]. 10. E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (University Press, Cambridge, 1917; Izd. Udmurtskogo Univ, Izhevsk, 1999). 11. V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. Linear Equations of Mathematical Physics (Nauka, Moscow, 1964) [in Russian]. 12. A. D. Polyanin, “Nonlinear Systems of Two Partial Differential Equations,” URL: http://eqworld.ipmnet. ru/ru/solutions/syspde/spde-toc3.htm 13. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1977; Dover, New York, 1990).
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