A nonlocal boundary-value problem for partial differential equations of infinite order

Ільків, В. С.

doi:10.1007/bf01093095

Cited by 5 publications

(7 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The solvability of problem (1), (2) is equivalent to the solvability of the system of equations L kr u kr = f kr , (k, r) ∈ Z p+1 . Let u be a solution of problem (1), (2). Then f = Lu ∈ W −∞ {Aŝ}, and we obtain…”

Section: Theorem 4 Problem (1) (2) Has a Solution In W ∞ {Aŝ} If Anmentioning

confidence: 99%

“…Let us find the correspondence between the infinite-order Sobolev spaces W ∞ {Aŝ}, W −∞ {Aŝ} and the finite-order spaces W q (Ω T ) and use it to justify the solvability of problem (1), (2) in the scale of spaces W q (Ω T ).…”

Section: Theorem 4 Problem (1) (2) Has a Solution In W ∞ {Aŝ} If Anmentioning

confidence: 99%

“…We also assume that the summation in (3) and throughout the following (unless the summation limits are indicated) is performed over all integer vectors (k, r) ∈ Z p+1 . The functions v kr = v kr (t, x) satisfy the nonlocal conditions (2) and form a Riesz basis in the space L 2 (Ω T ) [1,6]. Therefore, in the case of an equation or a system of equations of finite order, we consider finite-order Sobolev spaces W q Ω T , q ∈ R, which are formed by completion of finite vector sums u kr v kr (t, x) in the norm u 2 q = k 2 + r 2 q u * kr u kr , wherek 2 = 1 + k 2 1 + · · · + k 2 p , and the asterisk stands for the Hermitian conjugation of matrices.…”

mentioning

confidence: 99%

“…The scalar (m = 1) problem (1), (2) was considered in [1] in the case of a finite-order equation and in [2] in the case of an infinite-order equation. The investigation of a problem for a partial differential equation of infinite order mainly implies the construction of the corresponding function spaces, which are infinite-order spaces.…”

mentioning

confidence: 99%

“…Sobolev spaces of infinite order were introduced in [3-5] for the investigation of the Dirichlet problem and the problem on periodic solutions for infinite-order equations of elliptic and hyperbolic types. For no-type equations of infinite order, such spaces were introduced and analyzed in [2]. In the scalar case, the Sobolev space of infinite order corresponding to the nonlocal problem (1), (2) is defined as the space [2]…”

mentioning

confidence: 99%

See 4 more Smart Citations

A Nonlocal Boundary Value Problem for Systems of Infinite-Order Partial Differential Equations

Ільків

2005

Diff Equat

View full text Add to dashboard Cite

In the domain Ω T = (0, T ) × Ω, where Ω is a p-dimensional torus, we consider the problemAŝ are square matrices of order m with complex entries, and µ is a nonzero complex number. The desired solution u = u(t, x) and the givenThe scalar (m = 1) problem (1), (2) was considered in [1] in the case of a finite-order equation and in [2] in the case of an infinite-order equation. The investigation of a problem for a partial differential equation of infinite order mainly implies the construction of the corresponding function spaces, which are infinite-order spaces. Sobolev spaces of infinite order were introduced in [3-5] for the investigation of the Dirichlet problem and the problem on periodic solutions for infinite-order equations of elliptic and hyperbolic types. For no-type equations of infinite order, such spaces were introduced and analyzed in [2]. In the scalar case, the Sobolev space of infinite order corresponding to the nonlocal problem (1), (2) is defined as the space [2]where λ kr = |L(τ (r), k)| if |L(τ (r), k)| = 0 and λ kr = 1 otherwise, v kr (t, x) = e τ (r)t+i(k,x) , τ(r) = (− ln µ)/T + i × 2πr/T, (k, x) = k 1 x 1 + · · · + k p x p , and ln µ is treated in the sense of principal value. If the series L(τ (r), k) is divergent, then, by definition, λ kr = ∞, u kr is set to zero, and we assume that the corresponding term in (3) is absent. We also assume that the summation in (3) and throughout the following (unless the summation limits are indicated) is performed over all integer vectors (k, r) ∈ Z p+1 . The functions v kr = v kr (t, x) satisfy the nonlocal conditions (2) and form a Riesz basis in the space L 2 (Ω T ) [1,6]. Therefore, in the case of an equation or a system of equations of finite order, we consider finite-order Sobolev spaces W q Ω T , q ∈ R, which are formed by completion of finite vector sums u kr v kr (t, x) in the norm u 2 q = k 2 + r 2 q u * kr u kr , wherek 2 = 1 + k 2 1 + · · · + k 2 p , and the asterisk stands for the Hermitian conjugation of matrices. Now for problem (1), (2), we construct infinite-order Sobolev spaces of vector functions with using the same functions v kr (t, x) as a basis.For each vector (k, r) ∈ Z p+1 , we consider the matricesAŝ(τ (r)) s0 k s and (L * kr L kr )

show abstract

Section: Theorem 4 Problem (1) (2) Has a Solution In W ∞ {Aŝ} If Anmentioning

confidence: 99%

Section: Theorem 4 Problem (1) (2) Has a Solution In W ∞ {Aŝ} If Anmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations