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Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables

  • Oksana Malanchuk EMAIL logo and Zinoviy Nytrebych
Published/Copyright: February 17, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

We prove that homogeneous problem for PDE of second order in time variable, and generally infinite order in spatial variables with local two-point conditions with respect to time variable, has only trivial solution in the case when the characteristic determinant of the problem is nonzero. In another, opposite case, we prove the existence of nontrivial solutions of the problem, and we propose a differential-symbol method of constructing them.

Keywords: Null space; PDE; Two-point problem; Differential-symbol method
MSC 2010: 35G15

1 Introduction

The problem of finding the solution T(t) of ordinary differential equation (ODE) of order n ∈ ℕ\{1}, which satisfies conditions T(t1) = c1, …, T(tn) = cn, where t1 < … < tn, t1, …, tn, c1, …, cn ∈ ℝ, in the literature is found as the Vallee-Poussin problem or multipoint (n-point) problem.

For the first time such problems were studied in the articles [1–4], in which the importance to study the problems is indicated from the point of view of generalization of Cauchy problem. In contrast to Cauchy problem, the multipoint problem is ill-posed, because corresponding homogeneous problem can have nontrivial solutions.

The first results on solving the problems with multipoint time conditions for linear partial differential equations (PDE) have been obtained in [5] based on the metric approach. In particular, this paper points out the problem of small denominators, which is typical for multipoint problems. Also, ill-posedness of these problems was proved, morover it was shown that the classes of uniqueness of the multipoint in time problem solutions for PDE were significantly different from the classes of uniqueness of the solutions of the corresponding Cauchy problem for the same equations.

The investigation of the n-point problem for equations and systems of PDE’s in the bounded domains that is based on the metric approach has significantly developed in recent years (see works [6, 7] and bibliography).

Papers [8–10] are devoted to establish the classes of unique solvability of problems with multipoint conditions in time for PDE’s in unbounded domains (strip, layer). The technique of investigation multipoint problems in spaces of functions in which there is no problem of small denominators is proposed in these works.

The solvability of problems with multipoint conditions for differential-operator equations was studied in [11].

This article is devoted to study of the null space of problem for the PDE of the second order with respect to time variable, and arbitrary order with respect to spatial variables with local two-point conditions in time, and it is a continuation of researches [12] for the case of equations with several spatial variables.

2 Problem statement

In the domain ℝ1 + s, s ∈ ℕ\{1}, we investigate the set of solutions U = U(t,x), x =(x1,…, xs), of the PDE

L(∂∂t,∂∂x)U≡∂2U∂t2+2a(∂∂x)∂U∂t+b(∂∂x)U=0, (1)

in which the operator coefficients a(∂∂x),b(∂∂x),∂∂x=(∂∂x1,…,∂∂xs), are considered as arbitrary differential expressions with complex coefficients of the finite or infinite order, and the symbols a(ν), b(ν) of those coefficients are entire functions of complex vector-variable ν ∈ ℂs.

Under condition that a(ν) and b(ν) are polynomials, denote their degrees by the set of variables as pa ∈ ℤ+ and pb ∈ ℤ+. Also we assume that pa = ∞ and pb = ∞ if a(ν) and b(ν) respectively are not polynomials. Denote p = max {pa, pb/2}. Besides, assume p = ∞ if pa = ∞ or pb = ∞.

In the solutions set of equation (1), we will find the solutions that satisfy the homogeneous local two-point conditions:

A1(∂∂x)U(0,x)+A2(∂∂x)∂U∂t(0,x)=0,B1(∂∂x)U(h,x)+B2(∂∂x)∂U∂t(h,x)=0,h>0,x∈Rs, (2)

where A1(∂∂x),A2(∂∂x),B1(∂∂x),B2(∂∂x) are differential polynomials with complex coefficients. Moreover their symbols A1(ν), A2(ν), B1(ν), B2(ν) for ν ∈ ℂs satisfy the conditions:

|A1(ν)|2+|A2(ν)|2≠0,|B1(ν)|2+|B2(ν)|2≠0.

In this article, we establish the conditions of existence of nontrivial solutions (nontrivial null space) of this problem. In the case of existence of such solutions, we propose the way of their construction by using the differential-symbol method [13].

3 Main results

In equation (1), we replace ∂∂x with vector-parameter ν and the symbol ∂∂twithddt. We obtain the ODE

L(ddt,ν)T(t,ν)≡(d2dt2+2a(ν)ddt+b(ν))T(t,ν)=0, (3)

in which, from now on, we consider ν ∈ ℂs.

The normal fundamental system of solutions of equation (3) at the point t = 0 has the form:

T0(t,ν)=e−a(ν)t{a(ν)sinh⁡[tD(ν)]D(ν)+cosh[tD(ν)]},T1(t,ν)=e−a(ν)tsinh⁡[tD(ν)]D(ν),

where D(ν) = a2(ν)−b(ν).

Since the coefficients a(ν), b(ν) of equation (3) are entire functions by assumption, so by the Poincare Theorem [14, p. 59] the functions T0(t, ν), T1(t, ν) are also entire functions of vector-parameter ν with order p.

We write down the determinant of the form:

Δ(ν)=A1(ν)A2(ν)B1(ν)T0(h,ν)+B2(ν)dT0dt(h,ν)B1(ν)T1(h,ν)+B2(ν)dT1dt(h,ν).

Such determinant will be called the characteristic determinant of problem (1), (2). It can also be represented in a matrix form:

Δ(ν)=(B1(ν)B2(ν))T0(h,ν)T1(h,ν)dT0dt(h,ν)dT1dt(h,ν)−A2(ν)A1(ν).

Note that function Δ(ν) as a superposition of entire functions is an entire function.

3.1 The case when the set of zeroes of the characteristic determinant is empty

Theorem 3.1

If Δ(ν)≠ 0 ∀ ν ∈ ℂs, then problem (1), (2) in the class of entire functions has only trivial solution.

Proof

Let there exists a nontrivial integer solution U(t, x) of equation (1), i.e. entire function of the form

U(t,x)=∑k¯∈Z+s+1uk¯tk0xk,k¯=(k0,k1,…,ks),uk¯∈C,

of variables t and x = (x1, … xs), where xk=x1k1…xsks, that satisfies conditions (2).

Let’s denote U(0,x)=φ(x),∂U∂t(0,x)=ψ(x). Then φ(x) and ψ(x) are also entire functions. Write down the solution of problem (1), (2) according to differential-symbol method [13] as the solution of Cauchy problem for equation (1) with initial data φ and ψ in the form:

U(t,x)=φ(∂∂ν){T0(t,ν)eν⋅x}|ν=O+ψ(∂∂ν){T1(t,ν)ev⋅x}|ν=O, (4)

where ν · x = ν1x1 + … + νsxs, O = (0,…,0).

Since T0(t, ν)eν·x and T1(t, ν)eν · x are entire in ν functions of order p¯=max{p,1}≥1, so both entire functions φ(x) and ψ(x) must have the adjoint [15, p. 316] with p¯orderq,i.e.q=p¯/(p¯−1)for1<p¯<∞, and q=∞forp¯=1andq=1forp¯=∞.

Since two-point conditions (2) for function φ and ψ are satisfied, we obtain in ℝs the system of identities

φ(∂∂ν){A1(ν)eν⋅x}|ν=O+ψ(∂∂ν){A2(ν)eν⋅x}|ν=O≡0,φ(∂∂ν){(B1(ν)T0(h,ν)+B2(ν)T0′(h,ν))eν⋅x}|ν=O++ψ(∂∂ν){(B1(ν)T1(h,ν)+B2(ν)T1′(h,ν))eν⋅x}|ν=O≡0, (5)

where T0′(h,ν)=d T 0 dt(h,ν),T1′(h,ν)= d T 1 d t(h,ν).

Let’s act by the expression B1(∂∂x)T1(h,∂∂x)+B2(∂∂x)T1′(h,∂∂x) onto the first identity of system (5) and act by A2(∂∂x) onto the second identity of the system, after that subtract the second identity from the first one. We obtain

φ(∂∂ν){eν⋅xΔ(ν)}|ν=O≡0,x∈Rs.

Since the function Δ−1 (ν) is entire, so, acting by expression Δ−1(∂∂x) onto the last identity, we obtain φ(∂∂ν){eν⋅x1Δ(ν)Δ(ν)}|ν=O≡0, whence we have φ(∂∂v){ev⋅x}|v=O≡0orφ(x)≡0.

Similarly, if we act by expression −B1(∂∂x)T0(h,∂∂x)−B2(∂∂x)T0′(h,∂∂x) onto the first identity of system (5) and by expression A1(∂∂x) onto the second identity, and then add the obtained identities, we get

ψ(∂∂ν){ev⋅xΔ(ν)}|v=O≡0,x∈Rs,

that implies ψ (x) ≡ 0.

We have the identity U(t, x) ≡ 0, which contradicts to the assumption on nontriviality of the solution of problem (1), (2).

3.2 The case when the set of zeroes of the characteristic determinant coincides with ℂs

Let’s investigate the solvability of problem (1), (2) in case Δ(ν) ≡ 0.

Consider the function

Φ(t,x,ν)={A2(ν)T0(t,ν)−A1(ν)T1(t,ν)}eν⋅x, (6)

which is an entire function with respect to the vector-parameter ν ∈ ℂs. Let’s denote the order of this function in the set of parameters ν1, . . ., νs by p~. . Note that 1≤p~≤p¯.

Theorem 3.2

Let Δ(ν) ≡ 0 for problem (1), (2), then entire nontrivial solutions of the problem exist, and they can be represented in form

U(t,x)=φ(∂∂v){Φ(t,x,v)}|ν=O, (7)

where φ(x) is an arbitrary entire function, that has order, adjoint with the order p~.

Proof

Since Δ(ν) ≡ 0 then there holds the equality

A1(ν){B1(ν)T1(h,ν)+B2(ν)T1′(h,ν)}≡A2(ν){B1(ν)T0(h,ν)+B2(ν)T0′(h,ν)}. (8)

Let’s show firstly that function (7) satisfies equation (1). Taking into account the commutativity of the operations ∂∂t,∂∂xand∂∂ν, we obtain

L(∂∂t,∂∂x){φ(∂∂ν){Φ(t,x,ν)}|ν=O}=φ(∂∂ν){L(∂∂t,∂∂x){Φ(t,x,ν)}|ν=O}==φ(∂∂ν){eν⋅xL(ddt,ν){A2(ν)T0(t,ν)−A1(ν)T1(t,ν)}}|ν=O==φ(∂∂ν){eν⋅x[A2(ν)L(ddt,ν)T0(t,ν)−A1(ν)L(ddt,ν)T1(t,ν)]}|ν=O=0.

Let’s prove that the first condition (2) is satisfied:

A1(∂∂x)U(O,x)+A2(∂∂x)∂U∂t(0,x)==A1(∂∂x)φ(∂∂ν){Φ(0,x,ν)}|ν=O+A2(∂∂x)φ(∂∂ν){∂Φ∂t(0,x,ν)}|ν=O==φ(∂∂ν){A1(∂∂x)[A2(ν)eν⋅x]}|ν=O−φ(∂∂ν){A2(∂∂x)[A1(ν)eν⋅x]}|ν=O==φ(∂∂ν){eν⋅x[A1(ν)A2(ν)−A1(ν)A2(ν)]}|ν=O=0.

Due to equality (8), function (7) also satisfies the second condition (2):

B1∂∂xU(h,x)+B2∂∂x∂U∂t(h,x)==B1∂∂xφ∂∂νΦ(h,x,ν)|ν=O+B2∂∂xφ∂∂ν∂Φ∂t(h,x,ν)|ν=O==φ∂∂νB1(ν)Φ(h,x,ν)|ν=O+φ∂∂νB2(ν)∂Φ∂t(h,x,ν)|ν=O==φ∂∂νeν⋅xΔ(ν)|ν=O=0.

Thus, we have proved that function (7) is a solution of problem (1), (2), in which φ(x) is an arbitrary entire function with the order adjoint to the order p~.

Let’s show that function (7) is nontrivial. In fact,

U(O,x)=φ(∂∂ν){Φ(0,x,ν)}|ν=O=φ(∂∂ν){[A2(ν)eν⋅x]}|ν=O,∂U∂t(0,x)=φ(∂∂ν){∂Φ∂t(0,x,ν)}|ν=O=−φ(∂∂ν){[A1(ν)eν⋅x]}|ν=O.

Since |A1(ν)|2 + |A2(ν)|2 > 0 for arbitrary ν ∈ ℂs, thus either U(0,x)or∂U∂t(0,x) are nontrivial. Hence, function (7) is also nontrivial as a solution of Cauchy problem with nonzero initial data.

3.3 The case when the set of zeroes of the characteristic determinant is not empty and does not coinside with ℂs

Consider the set

M={ν∈Cs:Δ(ν)=0}, (9)

moreover we assume M ≠ ∅ and M ≠ ℂs.

Let α ∈ M. For complex vector α consider the set of multi-indexes:

Ω1(α)=ω=(ω1,...,ωs)∈Z+s:∂∂νωΔ(ν)|ν=α≡Δ(ω)(α)≠0, (10)

where ∂∂νωΔ(ν)=∂|ω|.Δ(ν)∂ν1ω1…∂νsωs,|ω|=ω1+…+ωs.

For two vectors ω = (ω1,…, ωs) and σ=(σ1,…,σs)∈Z+s we assume ω ≥ σ, when ω1 ≥ σ1,…, ωs ≥ σs. Also let’s denote Cωσ=ω!σ!(ω−σ)!, where ω!=∏k=1sωk!,ωk!=1⋅2⋅…⋅ωk,0!=1.

Besides the set (10) let us consider such sets:

Ω(α)={ω~∈Z+s:ω~>_ω,ω∈Ω1(α)},Ω¯(α)=Z+s∖Ω(α).

Note the following fact: if ω∈Ω¯(α),thenΔ(σ)(α)=0 for all σ∈Z+s for which σ ≤ ω.

Theorem 3.3

If ω∈Ω¯(α), then

U(t,x)=∂∂νω{Φ(t,x,ν)}|ν=α (11)

is a nontrivial solution of problem (1), (2), where Φ(t, x, ν) is the function (6).

Proof

The function U(t, x) is the solution of equation (1). It follows from equality (7) for entire function φ(x) = xω. The first condition from two-point conditions (2) is fulfiled:

A1∂∂xU(0,x)+A2∂∂x∂U∂t(0,x)=∂∂νω[{A1(ν)A2(ν)−A1(ν)A2(ν)}eν⋅x]|ν=α≡0.

Let’s show that the second condition in (2) is satisfied:

B1∂∂x∂∂νωA2(ν)T0(h,ν)−A1(ν)T1(h,ν)eν⋅x|ν=α++B2∂∂x∂∂νωA2(ν)T0′(h,ν)−A1(ν)T1′(h,ν)eν⋅x|ν=α==∂∂νωB1(ν)A2(ν)T0(h,ν)−A1(ν)T1(h,ν)eν⋅x|ν=α++∂∂νωB2(ν)A2(ν)T0′(h,ν)−A1(ν)T1′(h,ν)eν⋅x|ν=α==∂∂νωΔ(ν)eν⋅x|ν=α=eα⋅x∂∂α+xωΔ(α)=eα⋅x∑O≤q≤ωCωqxω−qΔ(q)(α).

Since ω∈Ω¯(α), , then according to the remark for arbitrary q∈Z+s such that q ≤ ω, equalities Δ(q) (α) = 0 are fulfiled. Hence, the function (11) satisfies the second condition (2).

Let’s prove that the formula (11) defines nontrivial solution of problem (1), (2). Let’s calculate the value of the function (11) and its derivative at the point t = 0:

U(0,x)=∂∂νωΦ(0,x,ν)|ν=α=∂∂νωA2(ν)eν⋅x|ν=α==eα⋅x∂∂α+xωA2(α)=eα⋅x∑O≤q≤ωCωqxω−qA2(q)(α);∂U∂t(0,x)=∂∂νωA2(ν)T0′(0,ν)−A1(ν)T1′(0,ν)eν⋅x|ν=α==−∂∂νωA1(ν)eν⋅x|ν=α=−eα⋅x∂∂α+xωA1(α)=−eα⋅x∑O≤q≤ωCωqxω−qA1(q)(α).

In the last sums, the coefficients of xω are A2(α) and A1(α). From the condition |A1(α)|2 + |A2(α)|2 ≠ 0, it follows that at least one of the expressions U(0,x)or∂U∂t(0,x) is a nonzero quasipolynomial. This implies that function (11) is a nontrivial solution of problem (1), (2).

4 Examples

Example 4.1

Find in domain (t, x1, x2) ∈ ℝ3 the solutions of equation

[∂2∂t2+2(∂∂x1+∂∂x2)∂∂t+((∂∂x1+∂∂x2)2+1)]U(t,x1,x2)=0, (12)

that satisfy local two-point conditions

∂∂x1+∂∂x2U(0,x)+∂U∂t(0,x)=0,∂∂x1+∂∂x2U(h,x)+∂U∂t(h,x)=0. (13)

For this problem, we obtain a(ν) = ν1 + ν2, b(ν) = (ν1+ν2)2 + 1, D(ν) = − 1, A1(ν) = B1(ν) = ν1+ν2, A2(ν) = B2(ν) = 1, ν = (ν1, ν2), x = (x1, x2), s = 2.

The normal at the point t = 0 fundamental system of solutions of the corresponding to (12) ODE

d2dt2+2ν1+ν2ddt+ν1+ν22+1T(t,ν)=0

has the form

T0(t,ν)=e−(ν1+ν2)t{(ν1+ν2)sin⁡t+cos⁡t},T1(t,ν)=e−(ν1+ν2)tsin⁡t.

Let’s write down the characteristic determinant of problem (12), (13):

Δ(ν)=ν1+ν21(ν1+ν2)T0(h,ν)+T0′(h,ν)(ν1+ν2)T1(h,ν)+T1′(h,ν)=e−(ν1+ν2)hsin⁡h.

In the case h ≠ πk, k ∈ ℕ, the condition Δ(ν) ≠ 0 is satisfied for arbitrary ν ∈ ℂ2, so by Theorem 3.1, problem (12), (13) has only trivial solution.

If h = πk, where k ∈ ℕ, then there holds the equality Δ(ν) ≡ 0. So, by Theorem 3.2, there exist nontrivial solutions of problem (12), (13). Function (6) gets the form:

Φ(t,x,ν)={T0(t,ν)−(ν1+ν2)T1(t,ν)}eν⋅x=e−(ν1+ν2)t+ν⋅xcos⁡t.

We search the nontrivial solutions of problem (12), (13) in the case h = πk, where k ∈ ℕ, by formula (7):

U(t,x)=φ∂∂ν{e−(ν1+ν2)t+ν⋅xcos⁡t}|ν=O==cos⁡tφ∂∂ν{e−(ν1+ν2)t+ν1x1+ν2x2}|ν=O=φ(x1−t,x2−t)cos⁡t.

The null space of the problem is infinite-dimensional. Note that the condition of entirety of the function φ can be weakened, considering classical solutions of problem (12), (13): φ can be arbitrary twice continuously differentiable function on ℝ2.

Example 4.2

In the domain (t, x1, x2) ∈ ℝ3, find the solutions of the two-point problem

∂2∂t2U(t,x1,x2)+2∂2∂t∂x1U(t,x1,x2)+∂2∂x12U(t,x1,x2+1)=0,∂U∂x1(0,x1,x2)+∂U∂t(0,x1,x2)=0,U(1,x1,x2)=0. (14)

For this problem, we have a(ν)=ν1,b(ν)=ν12eν2,D(ν)=ν12(1−eν2), A1(ν) = ν1, B2(ν) = 0, A2(ν) = B1(ν) = 1, s = 2, h = 1. The functions

T0(t,ν)=e−ν1tsinh⁡[tν11−eν2]1−eν2+cosh⁡[tν11−eν2],T1(t,ν)=e−ν1tsinh⁡[tν11−eν2]ν11−eν2

form the normal at the point t = 0 fundamental system of solutions of ODE

d2dt2+2ν1ddt+ν12eν2T(t,ν)=0.

The characteristic determinant of problem (14) and the corresponding set M will have the form:

Δ(ν)=e−ν1coshν11−eν2,M=ν∈C2:ν11−eν2=π2+πki,k∈Z,i2=−1.

The set M consists of the following vectors:

αk(μ)=(π2+πk)i1−eμ,μ,μ∈C∖{2πmi,m∈Z},k∈Z.

Let’s find the first order partial derivatives of the function Δν at the point v = αk(μ) :

Δ(1,0)(αk(μ))=e−ν11−eν2sinhν11−eν2−coshν11−eν2|ν=αk(μ)==(−1)k1−eμe−π2+πki1−eμ≠0,Δ(0,1)(αk(μ))=−ν121−eν2eν2−ν1sinhν11−eν2|ν=αk(μ)==π2+πki2(1−eμ)(−1)keμ−π2+πki1−eμ≠0.

Hence, Ω¯(αk(μ))={(0,0)}. Function (6) gets the form

Φ(t,x,ν)=coshtν11−eν2eν1(x1−t)+ν2x2.

By Theorem 3.3, for arbitrary μ∈ ℂ\ {2πmi, m ∈ ℤ} and k ∈ ℤ, we find such nontrivial solutions of problem (14):

Uk(t,x1,x2,μ)=Φ(t,x,νk(μ))=cosπ2+πkte(π2+πk)i1−eμ(x1−t)+μx2.

Note that the obtained solutions of problem (14) are linearly independent.

Example 4.3

In the domain (t, x) ∈ ℝ4 find the solutions of two-point problem

∂∂t+a(∂∂x)2U(t,x)=0,a∂∂x+c∂∂xU(0,x)+∂U∂t(0,x)=0,U(1,x)=0, (15)

in which a∂∂x,c∂∂x are differential polynomials with complex coefficients.

Problem (15) is problem (1), (2), in which b(ν) = a2(ν), A1(ν) = a(ν) + c(ν), A2(ν) = B1(ν) = 1, B2(ν) = 0, h = 1, s = 3.

The functions

T0(t,ν)=e−a(ν)t{a(ν)t+1},T1(t,ν)=te−a(ν)t

form the normal at the point t = 0 fundamental system of solutions of ODE

ddt+a(ν)2T(t,ν)=0.

We calculate the characteristic determinant of problem (15):

Δ(ν)=a(ν)+c(ν)1T0(1,ν)T1(1,ν)=e−a(ν){c(ν)−1}.

Let’s consider the cases.

Case 1. Let c(ν) = c ∈ ℂ\{1}. Then for arbitrary ν ∈ ℂ3 condition

Δ(ν)=e−a(ν){c−1}≠0

is satisfied. Therefore by Theorem 3.1 problem (15) has only trivial solution.

Case 2. If c(ν) = 1, then Δ(ν)≡ 0. Function (6) has the form

Φ(t,x,ν)={T0(t,ν)−[a(ν)+1]T1(t,ν)}eν⋅x=(1−t)e−a(ν)t+ν⋅x.

By Theorem 3.2 we find such nontrivial solutions of problem (15):

U(t,x)=(1−t)φ∂∂ve−a(v)t+ν⋅x|ν=O,

where φ is an arbitrary entire function of three variables. The order of this function is adjoint with number p = max{1, deg a(ν)}(deg a(ν) defines degree of polynomial a(ν) by the set of variables ν1, ν2, ν3).

Case 3. Let c(ν) = ν3, a(ν) = ν3 − ν1ν2. Then problem (15) gets the form

∂∂t+∂∂x3−∂2∂x1∂x22U(t,x)=0,2∂∂x3−∂2∂x1∂x2+1U(0,x)+∂U∂t(0,x)=0,U(1,x)=0. (16)

For problem (16), we obtain

Φ(t,x,ν)={1−(ν3+1)t}e(−ν3+ν1ν2)t+ν⋅x,Δ(ν)=ν3e−ν3+ν1ν2,M=ν∈C3:ν3=0.

The set M consists of the following vectors: α ≡ α(μ1, μ2) = (μ1, μ2, 0), where μ1, μ2 ∈ ℂ. We calculate

Δ(j,0,0)(α)=ν2jν3e−ν3+ν1ν2|ν=α=0,Δ(0,j,0)(α)=ν1jν3e−ν3+ν1ν2|ν=α=0,Δ(0,0,1)(α)=(1−ν3)e−ν3+ν1ν2|ν=α=eμ1μ2≠0,j∈N.

The set Ω(α) contains all multi-indexes ω∈Z+3 such that ω ≥ (0,0,1). Since for arbitrary m, n ∈ ℤ+ equalities hold Δ((m, n, 0))(α)= 0, then

Ω¯(α)={(m,n,0),m,n∈Z+}.

Using the Theorem 3.3 for arbitrary m, n ∈ ℤ+ and μ1, μ2∈ ℂ we find such nontrivial solutions of problem (16):

Umn(t,x,μ1,μ2)=∂m+n∂ν1m∂ν2n{Φ(t,x,ν)}|ν=α==(1−t)∂m+n∂ν1m∂ν2neν1ν2t+ν1x1+ν2x2+ν3x3|ν=α=(1−t)∂m+n∂μ1m∂μ2neμ1μ2t+μ1x1+μ2x2==m!n!(1−t)eμ1μ2t+μ1x1+μ2x2∑j=0min{m,n}tjj!(μ2t+x1)m−j(m−j)!(μ1t+x2)n−j(n−j)!.

It is easy to see that nontrivial solutions of problem (16) are also functions of the form

U(t,x,μ1,μ2)=(1−t)φ∂∂μ1,∂∂μ2eμ1μ2t+μ1x1+μ2x2,

where φ is an arbitrary nontrivial entire function of two variables, whose order in the set of variables is not greater than second, and μ1, μ2 are arbitrary complex parameters.

5 Conclusions

In this work we proved that the homogeneous problem for PDE of the second order in time variable, in which local two-point conditions are imposed, and generally infinite order in spatial variables, has only trivial solution in the case when the characteristic determinant of the problem is not equal to zero. In the other case, when the characteristic determinant possesses a nonzero value, we proved the existence of nontrivial solutions of the problem and proposed the differential-symbol method for their construction. We also gave some examples of using this method.

The investigation of the null space, of the problem given herein, provide an opportunity for future to specify the classes of unique solvability of the corresponding nonhomogeneous problem.

References

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Received: 2016-10-24
Accepted: 2017-1-17
Published Online: 2017-2-17

© 2017 Malanchuk and Nytrebych

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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Keywords for this article
Null space; PDE; Two-point problem; Differential-symbol method
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  231. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  232. The algebraic size of the family of injective operators
  233. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  234. The history of a general criterium on spaceability
  235. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  236. On sequences not enjoying Schur’s property
  237. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  238. A hierarchy in the family of real surjective functions
  239. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  240. Dynamics of multivalued linear operators
  241. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
  244. Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces
  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
  250. A nonlinear plate control without linearization
  251. Special Issue on Recent Developments in Differential Equations
  252. Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
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