Finite Symmetric Trilinear Integral Transform of Distributions. Part Ii

The finite symmetric trilinear integral transform is extended to distributions by using quite different technique than Zemanian (1968) and Dube (1976) and an inversion formula is established using Parseval's identity. The operational calculus generated is applied to find the temperature inside an equilateral prism of semi-infinite length.


Introduction
Sen [6] with the help of trilinear coordinates has solved different types of boundary value problems relating to boundaries in the form of an equilateral triangle.
An equilateral triangular region T is described by the set x = x 1 ,x 2 ,x 3 / 0 < x i < p, x 1 + x 2 + x 3 = p, x i ∈ R, i = 1,2,3 , ( where x 1 , x 2 , and x 3 are the trilinear coordinates of a point and p is the height of an equilateral triangle.
Sen [6] has also expressed two-dimensional Laplace operator in trilinear coordinates as Later Patil [4] has developed the symmetric integral transform of function of trilinear coordinates which is defined on T as where ψ n (x 1 ,x 2 ,x 3 ) = sinλ n x 1 + sinλ n x 2 + sinλ n x 3 are eigenfunctions corresponding to the eigenvalues λ n = 2nπ/ p, n = 1,2,3,..., in an eigenvalue problem subjected to the Dirichlet type of boundary conditions ψ n (x) = 0 at x 1 = 0, x 2 = 0, x 3 = 0. (1.5) If f (x 1 ,x 2 ,x 3 ) is continuous, has piecewise continuous first-and second-order partial derivatives on T, and satisfies the above Dirichlet type of boundary conditions, then inverse transform of (1.6) is given by where 1/c n = p 0 p 0 p 0 ψ 2 n dx 1 dx 2 dx 3 = 3p 3 /2 (see [4, page 129]).In this paper we extend the finite symmetric trilinear integral transform to distributions analogous to the method employed in [1] which is quite different than Zemanian [8] and Dube [2] and establish an inversion theorem by using Parseval's identity as in [3].At the end we find the temperature inside an equilateral prism of semi-infinite length.

The testing function space A
Let A denote the set of all infinitely differentiable complex-valued functions φ defined on T which satisfy the following two conditions: (i) m φ satisfy Dirichlet type of boundary conditions on T for each m = 0,1,2,...; (ii) We note that A is nonempty and for each n ∈ N, eigenfunction ψ n (x) is in A.
A is a linear space.The topology of A is that generated by the countable multinorm The proof of this theorem is similar to the proof of [7,Theorem 3.1].For every φ ∈ A, the finite symmetric trilinear integral transform exists and by (1.6), one has G. L. Waghmare and S. V.More 3 We call the sequence (S(φ)(n)) n∈N as the finite symmetric trilinear integral transform S(φ) of φ.Therefore The expression (2.3) can be seen as an inversion formula for the said transform.
Proposition 2.2.The map φ → S(φ) is a continuous linear transformation from A into l ∞ .
Proof.It can be proved by making use of the property of integrals and Let L 2 (T) denote the set of complex-valued functions φ defined on T such that where ψ(x) denotes the complex conjugate of ψ(x).We prove the following results which we need in subsequent sections.
Proof.Integrating by parts and using boundary conditions, we get Using (2.9), it is quite simple to see that holds for each m = 0,1,2,3,....

Finite symmetric trilinear integral transform
Proof.Expanding the inner product and using the fact that {ψ n } is an orthogonal set, we get and so, for any n ∈ N, It is clear that the series (2.11) converges and taking the limit as n → ∞ we get (2.12).
Proposition 2.5.If φ ∈ A, then the series converges absolutely and uniformly over T.
Proof.We have and the series converges uniformly over T.
Proof.From (1.6), we have (2.19) G. L. Waghmare and S. V.More 5 In view of (2.19) and in view of Proposition 2.5, it is inferred that the series The following is an immediate consequence of Corollary 2.6. (2.20) (2.21) Proof.We have from Proposition 2.4 that (2.24) Taking the limit as n → ∞ and using Corollary 2.7, we get (2.20).By using polarization identity, we get (2.21).

The space of rapidly decreasing sequences
Let B be the set of all complex sequences (a n ) n∈N satisfying B is a linear space and defines a countable multinorm on B. B is complete and therefore a Fréchet space.
Theorem 3.1.For each continuous linear functional h defined on B, there exist a positive constant C and a nonnegative integer r such that for every The proof is similar to the proof of [8, Theorem 1.8.1].
Theorem 3.2.The finite symmetric trilinear integral transform S is a homeomorphism from A onto the space B. Proof.
Hence S −1 : B → A exists and is given by This proves S is continuous.To prove S −1 is continuous, we proceed as follows. Let This completes the proof.
G. L. Waghmare and S. V.More 7

Distribution space
In this section we will introduce the space of distributions and study its basic properties.The set of all distributions is a complex linear space and it will be denoted by A .
Let A 0 denote the set of all functions f (x) which are continuous, have piecewise continuous first-and second-order partial derivatives on T, and satisfy the Dirichlet type of boundary conditions on T.
, for all values of n.Moreover, χ n → f uniformly on T follows from Corollary 2.6.Therefore which proves that (4.2) defines a distribution U f on A.
It is clear that the map f → U f is linear, one-to-one, and continuous.Finally, if as n → ∞ for all φ ∈ A. This implies f n → f as n → ∞.Thus A 0 can be embedded in A .
There are distributions that do not have the form (4.2) with f ∈ A 0 .
Example 4.3.Dirac function δ x centered at x ∈ T is given by It is easy to prove that δ x is linear.
Take χ n (y)= n k=1 c k ψ k (x)ψ k (y), x, y ∈ T, and x is fixed.Then χ n is in A for each n∈N, Proof.Let f : A → C be a continuous linear functional.For each φ ∈ A, by Corollary 2.7, We have Thus the condition in the definition of distribution is satisfied with  Here C and r depend on f but not on φ.
The proof is similar to proof of [8, Theorem 1.8.1].

Generalized finite symmetric trilinear integral transform
The generalized finite symmetric trilinear integral transform S of f ∈ A is defined by , then, by Theorem 3.2, (a n ) n∈N = (S(φ)(n)) n∈N = S(φ) and (5.1) can be written as ( Theorem 5.1.S is a homeomorphism from B onto the space A .
Proof.This result can be seen as a consequence of Theorem 3.2 and [8, Theorem 1.10.2].
Proposition 5.2.The finite symmetric trilinear integral transform S is a special case of the generalized transform S .That is, S f = S f for every f ∈ A.
Proof.By using Proposition 4.6, (5.1), Theorems 3.2, and 2.8, it is simple to prove Motivated by the above result, we define the generalized integral transform S f of f ∈ A as and we set (5.5) We now state and prove an inversion theorem for the elements of A that can be seen as an inversion formula for the S -transformation.
where the limit is taken in the sense of A .
Proof.Let F n (x) = n k=1 c k f ,ψ k ψ k .Since F n ∈ A for every n, by Proposition 4.2, (5.7) By using Theorem 2.8, Parseval's identity, we have (5.8)By Corollary 2.7, E n (φ) → φ for all φ ∈ A. Therefore (5.9) Theorem 5.4 (uniqueness theorem).If f ,g ∈ A are such that S ( f )(n) = S (g)(n) for every n, then f = g in the sense of equality in A .
The following example illustrates the inversion theorem.
Example 5.5.Dirac function δ x centered at x ∈ T is given by (5.10) In Example 4.3, we have shown that δ x is in A .The finite symmetric trilinear integral transform of δ x is given as (5.11)By virtue of Proposition 4.2, for all φ(t) ∈ A, (5.12) G. L. Waghmare and S. V.More 11 A trivial consequence of Theorem 5.3 is the following version of Parseval's identity.

Characterization of distributions in
J is one-one.Moreover, the linear mapping M : is continuous by virtue of (6.3).By Hahn-Banach theorem, M can be extended to l 1 as a member of l 1 .Then there exists ( (6.7) Theorem 6.2.Let (b n ) n∈N be a complex sequence.There exists Proof.The condition is necessary and follows from Proposition 4.7.
Let (b n ) n∈N be a sequence satisfying condition (6.8).
For any φ ∈ A, From (6.8) and Theorem 3.2, it is clear that the series ∞ n=1 c n b n S(φ)(n) is absolutely convergent.
Define f : A → C by the formula Then f is a linear functional on A and where χ n (x) = n k=1 c k b k ψ k (x) ∈ A for every n ∈ N. By using orthogonal relations, we have S ( f Thus the condition is sufficient.

Operational calculus
Integrating by parts and using boundary conditions, one can easily prove that if f ∈ A 0 , then It allows us to define that for any f ∈ A , It can also be seen inductively that for any integer m, and G. L. Waghmare and S. V.More 13 That is, which gives an operation transform formula.Now consider the partial differential equation of the form where given h and unknown f are required to be in A , and G is a polynomial such that G(−λ 2 n ) = 0, n = 1,2,3,....By applying the operation transform formula (7.5) to (7.6), we obtain (7.7) By applying the inversion theorem (Theorem 5.3), we get

Application
In this section we apply the present theory to find the temperature inside an equilateral prism of semi-infinite length.The formulation of the problem is given below.Find the conventional function v(x,z) on the domain that satisfies Laplace equation in R and the following boundary conditions: (i) as z → 0+, v(x,z) → f (x) ∈ A in the sense of convergence in A ; (ii) as x i → 0+, i = 1,2,3, v(x,z), converges to zero uniformly on Z ≤ z < ∞ for each Z > 0; (iii) as z → ∞, v(x,z) converges uniformly to zero uniformly on 0 < x i < p, i = 1,2,3.Every section of the prism by a plane perpendicular to z-axis is an equilateral triangle with its centroid on the z-axis, By applying the finite symmetric trilinear integral transform S to (8.2), we arrive at whose general solution is In view of boundary condition (iii), it is reasonable to choose A(n) = 0 and B(n) = F(n) because of boundary condition (i).Therefore, Applying the inversion theorem (Theorem 5.3) to the above equation, we get We now verify that (8.7) is truly a solution of (8.2).By Theorem 6.2, there exist C > 0 and k ∈ N such that |F(n)| ≤ Cλ 2k n .For Z ≤ z < ∞ where Z > 0, the nth term of the series (8.7) satisfies the condition Using c n = 2/3p 3 , λ n = 2nπ/ p, and e −λnz < (2k + 2)!/λ 2k+2 n Z 2k+2 , we get By Weierstrass M test, the series on the right-hand side of (8.7) converges absolutely and uniformly over R. The factor e −λnz ensures the uniform convergence of any series obtained by term-by-term differentiation of (8.7) with respect to x i , i = 1,2,3, or z.We may apply the operator I + D 2 z under the summation sign in (8.7).Since e −λnz ψ n (x 1 ,x 2 ,x 3 ) satisfies Laplace equation, so does v.Thus the differential equation (8.2) is satisfied in the conventional sense.
To verify the boundary condition (i), we have to show that for each  The series in (8.11) converges uniformly for all z > 0. By taking the limit as z → 0+, one has lim = f ,φ by virtue of Theorem 5.3. (8.12) The series in (8.13) converges absolutely and uniformly on T. So we may take limit as x i → 0+ under the summation sign in (8.13), which verifies boundary condition (ii).
In the same way we have The series in (8.14) converges uniformly on 0 < z < ∞.By taking the limit as z → ∞ under the summation sign in (8.14), one verifies boundary condition (iii).

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Definition 4 . 1 .
A linear functional U on a Fréchet space A, U : A → C, is called a distribution if there exists a sequence (χ n ) n∈N in A such that U,φ = lim x)φ(x)dx 1 dx 2 dx 3 exists for each φ ∈ A.(4.1)

Theorem 4 . 4 .
Then d is a compatible translation invariant metric on A [5, page 27].Furthermore (A,d) is a complete metric space.Every distribution is a continuous linear functional on A. The proof is similar to the proof of [3, Theorem 3.143].Proposition 4.5.A is the dual of A, that is, A is precisely the collection of all continuous linear functionals from A into C.

. 6 )
Taking b n = d n λ 2k n /c n , where c n = 2/3p 3 [4, page 129], n ∈ N. Since every sequence in l ∞ is bounded sequence, we have H, a n n∈N = ∞ n=1 c n a n b n .

First
Round of ReviewsMarch 1, 2009 Proposition 4.7.For each f ∈ A , there exist a nonnegative integer r and a positive constant C such that .11) implies the second statement.G. L. Waghmare and S. V.More 9

A and their generalized finite symmetric trilinear integral transform S
Proposition 6.1.Let H : B → C. Then H ∈ B if and only if there exists a complex sequence (b n ) n∈N such that Proof.Assume that H takes the form (6.2) where (b n ) n∈N satisfies (6.1).By using (3.2) it is simple to prove H ∈ B .Conversely, let H ∈ B .Then by Theorem 3.1, there exists k ∈ N such that