Adv Syst Sci Appl 2022; 04:92–102
Published online at https://ijassa.ipu.ru.

Approximate Solutions and Estimate of Galerkin Method for
Variable Third-Order Operator-Differential Equation

Abdel Baset I. Ahmed

Engineering Mathematics and Physics Dept, Helwan University, Cairo, Egypt

Abstract: The paper considers a variable third-order operator-differential equation in a separable
Hilbert space. Under certain assumptions, it is proved that this ODE has a unique solution.
The proof is based on a classical Galerkin discretization of the separable Hilbert space in term
of certain eigenfunctions. The approximation quality of the Galerkin approximations can be
controlled in terms of the eigenvalues. We deduce estimates for the convergence rate of the
approximate solutions to the exact one. An example provided as application to the investigated
method.

Keywords: Galerkin method, operator-differential equation, self-adjoint operator, convergence
rate, orthogonal projection

1. INTRODUCTION

An important direction in modern mathematics is the study of operator-differential equations.
Using the operator method, it is possible to study a wide class of differential equations.
The most different types of equations, such as linear and non-linear ordinary differential
equations, partial differential equations, integral and integral differential equations can be
present in operator form. With the help of the methods of the functional analysis and the
theory of operators we can study the question related to the solvability of boundary-values
problems and develop algorithms for finding the approximate solutions. It is known that
the theory and methodology of operator-differential equations widely used in computational
mathematics.

Among all the differential-operator equations, the most comprehensively studied are the
first and the second-order differential-operator equations. In this regard, we can specify the
works of A.G. Zarubin, P.V. Vinogradova [20], A.B.I. Ahmed [1] which investigate strong
solutions of the Cauchy problem for linear and non-linear differential-operator equations of
the first order. Existence, uniqueness, and continuous dependence of strong solutions of the
Cauchy problem for various second-order linear differential equations with variable domains
proved in the works of D.A. Lyakhov [14] and F. E. Lomovtsev [13]. The solvability of the
third and higher-order linear differential-operator equations were studied in works [3], [4],
[8]. From the standpoint of the spectral theory of linear operators in Hilbert space and the
Fourier method for studying the differential-operator equations of the first and higher orders
are represented in [6]. Through some assumptions, the solvability for differential-operator
equations of certain orders with variable coefficients were considered in [1], [2], [3], [7], [8].

Note that some of the equations that arise during studying of the process soil and
groundwater moisture dynamics, distribution of non-stationary external acoustic waves,

∗Corresponding author: abdel2007@yandex.ru



APPROXIMATE SOLUTIONS AND ESTIMATE OF GALERKIN METHOD 93

relaxation processes during heat transfer are possible lead to the differential-operator equation
of the third order in Hilbert space.

As a means of proving the theorems of the existence of solutions to non-stationary
differential equations, Galerkin’s method was used in the works [1], [7], [21] and other works.

L.A. Kantorovich [9] noted several problems that arise in the general theory of
approximate methods for solving operator-differential equations and questions, leading to
them, namely: the question of establishing the convergence of the algorithm, the process
of studying the fast convergence of an approximate solution, and obtaining effective error
estimates for the established approximate solution. The solution of the indicated tasks was
devoted to many works. However, this area of research requires further development.

When studying Galerkin method, special attention is paid to the choice of the basis,
since the properties of the basis functions significantly affect the rate of convergence of
approximate solution of the differential equation to exact solution. P.E. Sobolevsky [19]
proposed to choose the eigenfunctions of the differential operator as a basis, which does not
depend on time and forms with the investigated operator, so-called, acute angle. This idea was
used in [18] for the study of non-stationary differential-operator equations with subordinates
operators.

At the present time, there is many works on the Galerkin method for solving operator-
differential equations of a high order with an arbitrary basis. It should be noted that the
dependence of the estimates for the rate of convergence of the behavior of the approximate
solution on the type of key elements, properties of operators of the equation and its solution

2. PROBLEM STATEMENT AND AUXILIARY ASSERTIONS

Let H− separable Hilbert space over the real scalar field with inner product (., .)H and norm
∥.∥H = ∥.∥ and a separable Hilbert space H1 is compactly embedded into H . We denote the
Hilbert space of all strongly measurable functions f : [0, T ] → H by B2(0, T ;H), where

B2(0, T ;H) =

{
f : ∥f∥B2(0,T ;H) =

(∫ T

0

∥f(t)∥2Hdt
) 1

2

< +∞

}
.

In the space H , we will study the following variable third-order operator-differential equation
with zero initial conditions:

Φ(t)
d3u(t)

dt3
− S(t)u(t) = f(t), t ∈ [0, T ], (2.1)

u(0) = u(T ) = u
′
(0) = 0, (2.2)

where u(t) is the required solution and f(t) is a given scalar function.

In this work, we introduce some assumptions which concerning the operators Φ(t) and
S(t) as follows:
(i) Φ(t) is self-adjoint operator defined only on H1, (Φ(t) = Φ∗(t) ≥ σ0E, σ0 > 0), E− is
the unit operator and σ0− is the spectrum lower bound (σ0 ∈ σ(Φ(t)) [1], [8]. Φ(t) is a
positive definite operator where, ∥u∥H1 = ∥Φ(t)u∥H .
(ii) Φ(t) and S(t) are three-times strongly continuously differentiable operators on the
interval [0, T ] [11].
(iii) There exist some positive constants C00, C01, C03 ≥ 0 for all v ∈ H1 such that

(Φ(t)v, v)H ≥ C00

∥∥Φ1/2(0)v
∥∥2 , (Φ

′
(t)v, v)H ≤ −C01

∥∥Φ1/2(0)v
∥∥2 ,

(Φ
′′′
(t)v, v)H ≥ C03

∥∥Φ1/2(0)v
∥∥2 .

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94 A.B.I. AHMED

(iv) S(t) is α−subordinate (with order α : 0 ≤ α < 1) to Φ(0) [5], [11] i.e., For any v ∈ H1

there exists a constant k2 ≥ 0 independent on t, v, the S(t) satisfies the following inequality:

∥S(t)v∥ ≤ k2
∥Φ(0)v∥α

∥v∥α−1
. (2.3)

(v) There exists a linear operator Ψ is similar to Φ(0) [16] i.e., Ψ is a self-adjoint positive
definite operator. Moreover, the domain of (Ψ) is equal to the domain of (Φ(0)) and
Φ−1(t),Ψ−1 : H → H1 are compact in H . Both that operators are forming an acute angle
in H [19], that is there exists a positive constant k3 independent of the choice v ∈ H1 and t
such that (

Φ(t)v, Ψv
)
≥ k3∥Φ(0)v∥∥Ψv∥. (2.4)

Suppose that u(t) and Φ(t) have continuous derivatives
diu(t)

dti
,
diΦ(t)

dti
, i = 1, 2, 3

respectively in H . Moreover,
(
Φ(i)(t)u, v

)
H
=
(
u,Φ(i)(t)v

)
H
, i = 1, 2, 3 for all u, v ∈ H1

and for almost t ∈ (0, T ) [4] such that

W 3
2 (H,H1) =

{
u(t) ∈ B2(0, T ;H1) :

d3

dt3

(
Φ(t)u(t)

)
∈ B2(0, T ;H)

}
,

with norm

∥u(t)∥2W 3
2 (H,H1)

=

∫ T

0

(
∥u(t)∥2H1

+

∥∥∥∥ d3

dt3

(
Φ(t)u(t)

)∥∥∥∥2
H

)
dt.

Define the subspace which contains the strong solution u(t) of problem (2.1)-(2.2) by

◦
W 3

2 (H,H1) =
{
u(t) ∈ W 3

2 (H,H1) : u(0) = u(T ) = u
′
(0) = 0

}
.

According to [11], for all t under assumptions (i) and (ii) in the space H and for some
constants c1, c2 > 0 that do not depend on t, we have

∥Φ(t)Φ−1(0)∥H→H ≤ c1, ∥Φ(0)Φ−1(t)∥H→H ≤ c2. (2.5)

3. GALERKIN METHOD

Let e1, . . . , en, . . . be eigenvectors of the spectral problem Ψer = λrer, r = 1, . . . , n, . . . ,
which form complete orthonormal system and λ1, . . . λn, . . . be the corresponding
eigenvalues, in which 0 < λ1 ≤ . . . ≤ λn ≤ . . . , λn approaches infinity as n approaches
infinity.
The Galerkin solution (approximate) of (2.1) - (2.2) is considered to be in the discrete form
un(t) =

∑n
i=1 Gi(t)ei, i = 1, 2, . . . , n where the functions Gi(t) assumed to be unknown and

represent the exact solutions to the following Cauchy problem:

Φ(t)
d3Gj(t)

dt3
−

n∑
i=1

Gj(t)
(
S(t)ei, ej

)
=
(
f(t), ej

)
H
, (3.6)

Gj(0) = Gj(T ) = G ′

j(0) = 0, j = 1, 2, . . . , n. (3.7)

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



APPROXIMATE SOLUTIONS AND ESTIMATE OF GALERKIN METHOD 95

Next we denote the orthogonal projection in H onto Hn by Pn, where Hn is the linear span
of {e1, e2, . . . , en}. Then (3.6) - (3.7) is equivalent to the following problem:

Φ(t)
d3un(t)

dt3
− PnS(t)un(t) = Pnf(t), (3.8)

un(0) = un(T ) = u
′

n(0) = 0. (3.9)
Let us establish the unique solvability of problem (2.1) - (2.2) and obtain the necessary

estimates for the rate of convergence of the approximate solutions by using the Galerkin
method. Now and then, c is used to denote different positive constants independent of t and
n.
Theorem 3.1:
Under the preceding assumptions, there exists a unique solution un(t) of (3.8)-(3.9) in
◦

W 3
2 (H,H1) for each n. Moreover, the sequence {un(t)}n=1,2,... is convergent in W 3

2 (H,H1)
and the solution of (2.1) - (2.2) is unique.

Proof
Look into the Cauchy problem:

Φ(t)
d3On(t)

dt3
= Png(t), (3.10)

On(0) = On(T ) = O
′

n(0) = 0. (3.11)
If g(t) ∈ B2(0, T ;H), then problem (3.10) - (3.11) has a unique solution On(t) ∈
◦

W 3
2 (H,H1) and hence ∫ T

0

∥∥∥∥Φ(t)d3On(t)

dt3

∥∥∥∥2 dt = ∫ T

0

∥Png(t)∥2dt.

From assumption (i) it follows that the operator Φ(t) has an inverse Φ−1(t) : H → H1. Let

the operator
(
Φ(t) d3

dt3

)−1

: B2(0, T ;H) →
◦

W 3
2 (H,H1) be a homeomorphism, then holds the

following inequality: ∥∥∥∥∥
(
Φ(t)

d3

dt3

)−1
∥∥∥∥∥

B2(0,T ;H)→B2(0,T ;H)

≤ c. (3.12)

Multiply equation (3.10) by On(t) in H and then integrate with respect to t, where t varies
from 0 to T , we get∫ T

0

(
Φ(t)

d3On(t)

dt3
,On(t)

)
dt = −3

2

∫ T

0

(
dΦ(t)

dt

dOn(t)

dt
,
dOn(t)

dt

)
dt

+
1

2

∫ T

0

(
d3Φ(t)

dt3
On(t),On(t)

)
dt =

∫ T

0

(Png(t),On(t))dt.

(3.13)

As the operator Φ(0)Φ−1(t) is uniformly bounded. From the Heinz inequality [11] yields that
the operator Φ

1
2 (0)Φ− 1

2 (t) is also uniformly bounded. From assumption (iii), it follows that

c

2

∥∥∥Φ 1
2 (0)On(t)

∥∥∥2 ≤ ∥Png(t)∥B2(0,T ;H)

(∫ T

0

∥On(t)∥2
)1/2

.

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96 A.B.I. AHMED

Furthermore, ∥∥∥Φ 1
2 (0)On(t)

∥∥∥ ≤ c∥Png(t)∥B2(0,T ;H).

Hence, ∥∥∥∥∥Φ 1
2 (0)

(
Φ(t)

d3

dt3

)−1

Png(t)

∥∥∥∥∥ ≤ c ∥Png(t)∥B2(0,T ;H) . (3.14)

Then using the following substitution Φ(t)
d3On(t)

dt3
≡ ωn(t) in (3.8) - (3.9), we get

ωn(t)− PnS(t)
(
Φ(t)

d3

dt3

)−1

ωn(t) = Pnf(t). (3.15)

In the coming step we need to prove that the operator PnS(t)
(
Φ(t) d3

dt3

)−1

is compact in
B2(0, T ;H).
According (2.3) and Hölder inequality for all ν ∈ H1, we get∥∥∥∥PnS(t)

(
Φ(t)

d3

dt3

)−1

ν

∥∥∥∥
B2(0,T ;H)

≤ c

∥∥∥∥PnΦ(0)
(
Φ(t)

d3

dt3

)−1

ν

∥∥∥∥α
B2(0,T ;H)

∥∥∥∥Pn

(
Φ(t)

d3

dt3

)−1

ν

∥∥∥∥1−α

B2(0,T ;H)

.

(3.16)

Taking into account the uniform boundedness of Φ(0)Φ−1(t) and inequalities (3.12) (3.16),
yields that∥∥∥∥PnS(t)

(
Φ(t)

d3

dt3

)−1

ν

∥∥∥∥
B2(0,T ;H)

≤ c

∥∥∥∥Pn

(
Φ(t)

d3

dt3

)−1

ν

∥∥∥∥1−α

B2(0,T ;H)

∥ν∥αB2(0,T ;H) . (3.17)

As a result of the compact embedding of H1 into H and based on the lemma of compactness
[15], yields that the space W 3

2 (H,H1) is compactly embedded in B2(0, T ;H). Consequently,(
Φ(t) d3

dt3

)−1

is actually compact in B2(0, T ;H). Hence, from inequality (3.17) yields that

the operator PnS(t)
(
Φ(t) d3

dt3

)−1

is compact in B2(0, T ;H).
The second step, we shall prove that problem (3.15) is resolvable.

In (3.15), let the right-hand side be zero, then (3.15) will be equivalent to the following
Cauchy problem:

Φ(t)
d3un(t)

dt3
− PnS(t)un(t) = 0, (3.18)

un(0) = un(T ) = u
′

n(0) = 0, (3.19)

with a solution un(t) ∈
◦

W 3
2 (H,H1). Using assumption (2.3), yields∫ T

0

∥∥∥∥Φ(t)d3un(t)

dt3

∥∥∥∥2 dt ≤ k2
2

∫ T

0

∥PnΦ(0)un(t)∥2α ∥Pnun(t)∥2(1−α)dt

≤ k2
2∥Φ(0)Φ−1(t)∥2αH→H

∫ T

0

∥PnΦ(t)un(t)∥2α∥Pnu(t)n∥2(1−α)dt.

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



APPROXIMATE SOLUTIONS AND ESTIMATE OF GALERKIN METHOD 97

Thus, ∫ T

0

∥∥∥∥Φ(t)d3un(t)

dt3

∥∥∥∥2 dt ≤ c

∫ T

0

∥PnΦ(t)un(t)∥2α∥Pnun(t)∥2(1−α)dt.

Apply Hölder inequality, yields that∫ T

0

∥∥∥∥Φ(t)d3un(t)

dt3

∥∥∥∥2 dt ≤ c

(∫ T

0

∥PnΦ(t)un(t)∥2
)α(∫ T

0

∥Pnun(t)∥2dt
)(1−α)

.

Consequently, apply the Young inequality :

ab ≤ εa1/α +
(α
ε

)α/1−α

(1− α)b1/1−α,

yields that∫ T

0

∥∥∥∥Φ(t)d3un(t)

dt3

∥∥∥∥2 dt
≤ c

(
ε

∫ T

0

∥PnΦ(t)un(t)∥2dt+
(α
ε

) α
1−α

(1− α)

∫ T

0

∥Pnun(t)∥2dt
)
.

Take ε = 1
2c

, then ∫ T

0

∥∥∥∥Φ(t)d3un(t)

dt3

∥∥∥∥2 dt ≤ c

∫ T

0

∥Pnun(t)∥2dt. (3.20)

Whenever we multiply the above-mentioned equation (3.18) by un(t) and integrate the
outcome, we get

c03
2
∥Φ

1
2 (0)un(t)∥2 ≤

∫ T

0

∥un(t)∥∥PnS(t)un(t)∥dt ≤ c

∫ s

0

∥PnΦ(0)un(t)∥α∥un(t)∥2−αdt

≤ c

(∫ T

0

∥PnΦ(0)un(t)∥2dt
)α

2
(∫ T

0

∥un(t)∥2dt
) 2−α

2

.

Using (3.20), yields that

∥un(t)∥2 ≤ c

∫ T

0

∥un(t)∥2dt. (3.21)

From the Bellman–Gronwall theorem [5] it follows that this inequality is possible only in the
case when un(t) ≡ 0. Thus, by using the the Fredholm alternative [17] yields that problem

(3.8)-(3.9) has a unique solution un(t) ∈
◦

W 3
2 (H,H1).

Let δn(t) = Φ(t)
d3un(t)

dt3
− S(t)un(t)− f(t), follows that

∥δn(t)∥B2(0,T ;H) ≤ ∥(I − Pn)f(t)∥B2(0,T ;H) + ∥(I − Pn)S(t)un(t)∥B2(0,T ;H).

From (3.21), yields that {un(t)} is bounded in W 3
2 (H,H1). Since the subspace H1

is compactly embedded in the space H , yields that S(t) : W 3
2 (H,H1) → B2(H, H1) is

compact. Consequently, {S(t)un(t)} is also compact in B2(H, H1). As in [10], the sequence
{I − Pn} uniformly converges to zero. Therefore, ∥δn(t)∥B2(0,T ;H) → 0 as n → ∞.
Now, we need to prove that {un(t)} in W 3

2 (H,H1) is a Cauchy sequence.

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98 A.B.I. AHMED

From (3.13) we have∥∥∥Φ′′′
(0) (un(t)− un+p(t))

∥∥∥
B2(0,T ;H)

≤ c

∥∥∥∥Φ(t)(d3un(t)

dt3
− d3un+p(t)

dt3

)∥∥∥∥
B2(0,T ;H)

≤ c
(
∥δn(t)∥B2(0,T ;H) + ∥δn+p(t)∥B2(0,T ;H) + ∥S(t)(un(t)− un+p)∥B2(0,T ;H)

)
.

Applying Young inequality, we get∥∥∥Φ′′′
(0) (un(t)− un+p(t))

∥∥∥
B2(0,T ;H)

≤ c
(
∥δn(t)∥B2(0,T ;H) + ∥δn+p(t)∥B2(0,T ;H) + ∥(un(t)− un+p)∥B2(0,T ;H)

)
.

(3.22)

In equation (3.8), replace un(t) by un+p(t) and then subtract the resulting equation from
equation (3.8), we get the following relation

Φ(t)

(
d3un+p(t)

dt3
− d3un(t)

dt3

)
− S(t)(un+p − un)

= (I − Pn)S(t)un(t)− (I − Pn)S(t)un+p(t) + (Pn+p − Pn)f(t).

Multiply the preceding relation by (un+p(t)− un(t)) and then integrate the outcome. Further,
employ Hölder inequality, yields that∥∥Φ1/2(0)(un+p(t)− un(t))

∥∥2
B2(0,T ;H)

≤ (∥(I − Pn)S(t)un(t)∥B2(0,T ;H) + ∥(I − Pn)S(t)un+p(t)∥B2(0,T ;H)

+ ∥(Pn+p − Pn)f(t)∥B2(0,T ;H))× ∥Φ−1/2(0)∥H→H

∥∥Φ1/2(0)(un+p(t)− un(t))
∥∥

B2
.

Consequently,∥∥Φ1/2(0)(un+p(t)− un(t))
∥∥

B2(0,T ;H)

≤ (∥(I − Pn)S(t)un(t)∥B2(0,T ;H) + ∥(I − Pn)S(t)un+p(t)∥B2(0,T ;H)

+ ∥(Pn+p − Pn)f(t)∥B2(0,T ;H))× ∥Φ−1/2(0)∥H→H .

From the compactness of {S(t)un(t)} in B2(0, T ;H), yields that the right side of the
preceding inequality approaches to zero. Therefore, according to (3.22), the {un(t)} in
W 3

2 (H,H1) is a Cauchy sequence.
Let {un(t)} → u(t) as n → ∞. Then∥∥∥∥−f(t) + Φ(t)

d3u(t)

dt3
− S(t)u(t)

∥∥∥∥
B2(0,T ;H)

≤
∥∥∥∥Φ(t)(d3u(t)

dt3
− d3un(t)

dt3

)
− S(t) (u(t)− un(t))

∥∥∥∥
B2(0,T ;H)

+ ∥δn∥B2(0,T ;H)

≤ c
(
Φ(0)(∥un(t)− u(t))∥B2(0,T ;H) + δn∥B2(0,T ;H)

)
.

Therefore, there exists a strong solution u(t) ∈ W 3
2 (H,H1) of problem (2.1)-(2.2).

Now we will show that the solution of problem (2.1)-(2.2) is unique.
Assume that problem (2.1), (2.2) has two solutions u and û. Then

Φ(t)
d3

dt3
(u− û)− S(t) (u− û) = 0.

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



APPROXIMATE SOLUTIONS AND ESTIMATE OF GALERKIN METHOD 99

Multiply the preceding equation by (u− û) and integrate with respect to t, then we can
simply show that ∥u− û∥W 3

2 (H,H1)
≤ 0, which means that the solution of problem (2.1), (2.2)

is unique. The theorem is proved.

After the question of the solvabilty of problem (2.1) - (2.2) is proven by Galerkin method,
we shall use also the same method to estimate the rate of convergence of the approximate
solutions of problem (2.1) - (2.2).
Theorem 3.2:
Suppose that all of assumptions of Theorem 1 be fulfilled. Then∥∥Φ1/2(0) (un(t)− u(t))

∥∥
B2(0,T ;H)

≤ cλ
−1/2
n+1 . (3.23)

Proof
Let Un(t) = u(t)− un(t), where u(t) is the exact solution of problem (2.1) - (2.2) and un(t)
is the approximate solutions of problem (3.8) - (3.9). Then

Φ(t)
d3Un

dt3
− S(t)Un(t) = (I − Pn)(f(t)− S(t)u(t)).

Multiply the preceding equation by Un(t) and hence integrate the outcome taking into account
that for each n, we have that Un(t) belongs to domain Φ(t), we get

3c

2

∥∥∥Φ1/2(0)U ′

n(t)
∥∥∥

B2(0,T ;H)
+

3c

2

∫ T

0

∥∥Φ1/2(0)Un(t)
∥∥

≤
∫ T

o

| f(t)((I − Pn)Un(t))(S(t)un(t), (I − Pn)Un(t)) | dt.

From assumption (iv), we get∥∥Φ1/2(0)Un(t)
∥∥2

B2(0,T ;H)
≤ c

(
∥f(t)∥B2(0,T ;H) + ∥S(t)un(t)∥B2(0,T ;H)

)
×
∥∥Φ−1/2(0)Ψ1/2

∥∥
H→H

∥∥Ψ−1/2(I − Pn)Φ
1/2(0)Un(t)

∥∥ .
Consequently, ∥∥Φ1/2(0)Un(t)

∥∥
B2(0,T ;H)

≤ cλ
−1/2
n+1 .

The proof is complete.

4. APPLICATION

As an applied example on Galerkin method we will investigate an initial-boundary value
problem for non-classical higher order differential equations of composite-mixed type with
smooth coefficients. The theory of equations of mixed type is one of the most important
branches of the theory of non-classical differential equations of mathematical physics.These
types of problems have many applications in gas dynamics [12] and in other branches of
physics
In the rectangle domain Q = [0, 1]× [0, T ], consider the initial-boundary value problem

χ(x, t)
∂5u(x, t)

∂x2∂t3
+ χx(x, t)

∂4u(x, t)

∂x∂t3
− ζ1(x, t)

∂u(x, t)

∂x
− ζ0(x, t)u(x, t) = f(x, t), (x, t) ∈ Q

(4.24)

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



100 A.B.I. AHMED

u(x, 0) = ut(x, 0) = u(x, T ) = 0, 0 ≤ x ≤ 1, (4.25)

u(0, t) = u0(t), 0 ≤ t ≤ T, (4.26)

u(1, t) = u1(t), 0 ≤ t ≤ T. (4.27)

where f(x, t) ∈ B2(Q), ζi(x, t),
∂ζi(x, t)

∂t
∈ C(Q), ui(t) ∈ C3[0, T ], i = 0, 1.

Assume that

χ(x, t) ≥ χ0 > 0,
∂s+1

∂x∂ts
χ(x, t) ∈ C(Q), s = 1, 2, 3,

and the functions u0(t), u1(t) fulfill the following matching conditions:

u0(0) = u0(T ) = u1(0) = u1(T ) = u
′

0(0) = u
′

1(0) = 0.

By using the substitution V(x, t) = u(x, t)− (1− x)u0(t)− xu1(t), problem (4.24) - (4.27)
will be equivalent to the following problem:

χ(x, t)
∂5V(x, t)
∂x2∂t3

+ χx(x, t)
∂4V(x, t)
∂x∂t3

− ζ1(x, t)
∂V(x, t)

∂x
− ζ0(x, t)V(x, t) = h(x, t), (x, t) ∈ Q

(4.28)

V(x, 0) = Vt(x, 0) = V(x, T ) = 0, 0 ≤ x ≤ 1, (4.29)

V(0, t) = V(1, t) = 0, 0 ≤ x ≤ 1, (4.30)

where,

h(x, t) = f(x, t)− ζ1(x, t)u0(t) + ζ1(x, t)u1(t) + (1− x)ζ0(x, t)u0(t)

+ xζ0(x, t)u1(t)− (1− x)u
′′′

0 (t)− xu
′′′

1 (t).

Now, we need to redefine some spaces as follows:

Assign H = B2(0, 1), H1 = W 2
2 (0, 1) ∩

◦
W 1

2 (0, 1). Notice that the W 2
2 (0, 1) is denoted to

the vector norm Sobolev type space, the subspace
◦

W 1
2 (0, 1) = {V(x) ∈ W 2

2 (0, 1),V(0) =
V(1) = 0} and W 3

2 (H,H1) = W 2,3
2 (Q). As ∂

∂x
χ(x, t) ∈ C(Q), provided that | ∂

∂x
χ(x, t) |≤

χ1, where χ1 is a constant greater than zero.
On H1 we identify the following operators

Φ(t) =
∂

∂x

(
χ(x, t)

∂

∂x

)
−ΘI, S(t) = ζ1(x, t)

∂

∂x
+ (ζ0(x, t)−Θ) I,

where the constant Θ <
χ2
1

χ0
and domain S(t) ⊃ domainΦ(t).

For problem (4.28) - (4.30), we take Ψ = d2

dx2 . Distinctly, It can be seen that the assumptions
(i)− (v) are fully satisfied. Consequently, simply we can verify that S(t) is α− subordinate
to Φ(0) with α = 1

2
.

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



APPROXIMATE SOLUTIONS AND ESTIMATE OF GALERKIN METHOD 101

In the following step,, we’ll show that the operators Φ(t) and Ψ fulfill the acute-angle
inequality (2.4). For each element y(x) ∈ H1, we have

(Φ(t)y(x),Ψy(x))B2(0,1)

≥ χ0

∫ 1

0

(
d2y(x)

dx2

)2

dx−Θ

∫ 1

0

(
dy(x)

dx

)2

dx+ χ1

∫ 1

0

| dy(x)
dx

|| d
2y(x)

d2x
| dx

≥ χ0

∫ 1

0

(
d2y(x)

dx2

)2

dx+
χ1ε

2

∫ 1

0

(
d2y(x)

dx2

)2

dx

+
χ1

2ε

∫ 1

0

(
dy(x)

dx

)2

dx−Θ

∫ 1

0

(
dy(x)

dx

)2

dx.

We choose ε = χ0

2χ1
. Then we get

(Φ(t)y(x),Ψy(x))B2(0,1) ≥
(
χ2
1

χ0

−Θ

)∫ 1

0

(
dy(x)

dx

)2

dx ≥ c∥y(x)∥2W 2
2 (0,1)

, (4.31)

Comparing (4.31) with (2.4), we get the fact that Φ(t) and Ψ verified the acute-angle
condition.
As well, Ψ satisfies the eigenvalue equation Ψer(x) = λrer(x), r = 1, 2, . . . , where er(x) =
sin rπx√

2
are eigenfunction solutions, which are normalizable and λr = (rπ)2, r = 1, 2, . . .

are the corresponding eigenvalues.
All hypotheses of theorems 3.1 and 3.2 are exactly fulfilled. Thus, for the Galerkin solution
of problem (4.28) - (4.30) and hence problem (4.24) - (4.27), holds the following estimate

max
0≤t≤T

∥un(x, t)− u(x, t)∥B2(0,1) ≤ c
1√
n
.

5. CONCLUSION

For a third-order multi-variable operator differential equations with initial-boundary
conditions, the existence and uniqueness theorem of a strong solution was proved using
Galerkin method. The rate of convergence of the Galerkin solution (approximate solutions)
to the exact one was estimates. The results of Galerkin method enabled us to present an
application and prove the solvability of a boundary value problem for non-classical higher
order differential equations of composite-mixed type with smooth coefficients.

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102 A.B.I. AHMED

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Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)


	Introduction
	Problem statement and auxiliary assertions 
	 Galerkin method
	Application
	Conclusion