Mathematical Methods in Engineering

We have proposed regularization of infinite dimensional integral via fractional calculus. It is done on a Hilbert space H equipped with a Schatten class operator G. The

ζ

-function

ζ

(

G, s

) of G is assumed to be holomorphic at

s

=0. Regularization is done by using

ζ

(

G, s

). After reviewing this regularization, it is shown regularized Cauchy kernel of a Hilbert space with the determinant bundle exists if and only if

v

=

ζ

(

G

, 0) is an integer. Regularized residue on an infinite dimensional space is obtained as an application of regularized Cauchy kernel.

The structure of the fractional spaces

E

α,q

,(

L

q

[0, 1],

A

x

) generated by the positive differential operator

A

x

defined by the formula

A

x

u

= −

a

(

x

)

d

2

u

/

dx

2

+

δu

, with domain

D

(

A

x

) = {

u

∈

C

(2)

[0, 1] :

u

(0) =

u

(1),

u

′(0) =

u

′(1)} is investigated. It is established that for any 0 <

α

< 1/2 the norms in the spaces

E

α,q

(

L

q

[0, 1],

A

x

) and

W

q

2

α

[0, 1] are equivalent. The positivity of the differential operator

A

x

in

W

q

2α

[0, 1](0 ≤

α

< 1/2) is established. The discrete analogy of these results for the positive difference operator

A

h

x

a second order of approximation of the differential operator

A

x

, defined by the formula

$$ A_h^x u^h = \left\{ { - a\left( {x_k } \right)\frac{{u_{k + 1} - 2u_k + u_{k - 1} }} {{h^2 }} + \delta u_k } \right\}_1^{M - 1} ,u_h = \left\{ {u_k } \right\}_0^M ,Mh = 1 $$

with

u

0

=

u

M

and −

u

2

+ 4

u

1

− 3

u

0

=

u

M

−2

− 4

u

M

−1

+ 3

u

M

is established. In applications, the coercive inequalities for the solutions of the nonlocal boundary-value problem for two-dimensional elliptic equation and of the second order of accuracy difference schemes for the numerical solution of this problem are obtained.

The time-fractional diffusion equation is obtained by generalizing the standard diffusion equation by using a proper time-fractional derivative of order 1 —

β

in the Riemann-Liouville (R-L) sense or of order

β

in the Caputo (C) sense, with

β

∈ (0, 1). The two forms are equivalent and the fundamental solution of the associated Cauchy problem is interpreted as a probability density of a self-similar non-Markovian stochastic process, related to a phenomenon of sub- diffusion (the variance grows in time sub-linearly). A further generalization is obtained by considering a continuous or discrete distribution of fractional time-derivatives of order less than one. Then the two forms are no longer equivalent. However, the fundamental solution still is a probability density of a non-Markovian process but one exhibiting a distribution of time-scales instead of being self-similar: it is expressed in terms of an integral of Laplace type suitable for numerical computation. We consider with some detail two cases of diffusion of distributed order: the double order and the uniformly distributed order discussing the differences between the R-L and C approaches. For these cases we analyze in detail the behaviour of the fundamental solutions (numerically computed) and of the corresponding variance (analytically computed) through the exhibition of several plots. While for the R-L and for the C cases the fundamental solutions seem not to differ too much for moderate times, the behaviour of the corresponding variance for small and large times differs in a remarkable way.

Let

F

be a distribution in

$$ \mathcal{D}' $$

and let

f

be a locally summable function. The neutrix composition

F

(

f

(

x

)) is said to exist and be equal to the distribution

h

if the neutrix limit of the sequence {

F

n

(

f

(

x

))} is

h

, where

F

n

(

x

) =

F

(

x

)*

δ

n

(

x

) for

n

= 1, 2, . . . and {

δ

n

(

x

)} is a certain regular sequence converging to the Dirac delta funcion. In particular, the composition

F

(

f

(

x

)) is said to exist and be equal to the distribution

h

if the sequence {

F

n

(

f

(

x

))} converges to

h

in the normal sense. Some results are proved.

The problem of defining products of distributions has been open and an active research area since Schwartz introduced the theory of distribution around 1950. The inherent difficulties of obtaining products have never prevented their appearance in literature, as they are needed in quantum field and differential equations with distribution involved. The objective of this paper is to recollect various approaches, which include sequential and complex analysis methods, to tackling products of distributions in one or multiple variables, as well as particular generalized functions defined on certain manifolds.

The incomplete Gamma function γ(α,

x

) is defined for α > 0 and

x

≥ 0 by

$$ \gamma \left( {\alpha ,x} \right) = \int_0^x {u^{\alpha - 1} e^{ - u} du} $$

and by using the recurrence formula

$$ \gamma \left( {\alpha + 1,x} \right) = \alpha \gamma \left( {\alpha ,x} \right) - x^\alpha e^{ - x} $$

the definition of γ(α,

x

) can be extended to negative, non integer value of α. Recently Fisher et al. [

FJK03

] defined γ(−

m

,

x

) for

m

= 0, 1, 2, . . . . In this paper we consider the derivatives of the incomplete Gamma function γ(α,

x

) and the derivatives of locally summable function γ(α,

x

+

) =

H

(

x

)γ(α,

x

) for negative integers, where

H

(

x

) denotes the Heaviside function.

In this study, the numerical solution of one-dimensional wave equation in multilayered cylindrical media is investigated. The multilayered medium consists of

N

different layers of Functionally Graded Material, i.e., it is assumed that the stiffness and the density of each layer are varying continuously in the radial direction but isotropic and homogeneous in the circumferential and axial directions. The inner surface of the layered medium is assumed to be subjected to a uniform dynamic in-plane time-dependent normal stress; whereas, the outer surface of the layered medium is assumed free of surface traction or fixed. The method of characteristics is employed to obtain the numerical solutions of this initial-boundary value problem. The obtained numerical results reveal clearly the scattering effects caused by the reflections and refractions of waves at the boundaries and at the interfaces of the layers and the effects of non-homogeneity in the wave profiles. Furthermore, based on the results obtained from this paper, one may conclude that when the inner surface is stiffer than the outer surface, the stress-wave levels throughout the functionally graded cylindrical layers become less than the load applied at the inner surface.

A piecewise constant control that solves the boundary value problem for linear impulsive differential systems is considered. We establish a necessary and sufficient conditions for the existence of such control. Moreover, a result that explicitly characterizes the solving control is presented.

A numerical method is proposed for solving the hyperbolic-parabolic partial differential equations with nonlocal boundary condition. The first and second order of accuracy difference schemes are presented. The method is illustrated by numerical examples.

In this paper we consider a linear signal system together with the two linear observation systems. The observation systems differ from each other by the noise processes. The noise of one of them is a constant shift in time of the signal noise. In the other one the shift is neglected. Respectively, we consider two best estimates of the signal corresponding to two different observation systems. The following problem is investigated: whether the effect of the shift on the best estimate becomes negligible as time increases. This leads to a comparison of the asymptotical behaviors of the solutions of respective Riccati equations. It is proved that under a certain relation between the parameters, the effect of the shift is not negligible.

Ordinary elliptic curves over fields of characteristic 3 can be represented by

y

2

=

x

3

+

ax

2

+

b

where

a

,

b

≠ 0 ∈

$$ F_{q = 3^n } $$

. In this paper we count the number of different isomorphism classes of ordinary elliptic curves over finite fields of characteristic three. We show there are (2

q

−2) different isomorphism classes.

Classification of the invariants of two - dimensional superintegrable systems is presented. The hidden symmetries associated to the existence of Killing - Yano tensors are investigated.

High accurate difference-analytical method of solving the mixed boundary value problem for Laplace’s equation on graduated polygons (which can have broken sections and be multiply connected) is described and justified. The uniform estimate for the error of the approximate solution is of order

O

(

h

4

), where

h

is the mesh step, for the errors of derivatives of order

p, p

= 1, 2, ..., in a finite neighbourhood of re-entrant vertices, of order

O

(

h

4

/

r

j

p

−

λj

), where

r

j

is the distance from the current point to the vertex in question,

λ

j

= 1/

α

j

or

λ

j

= 1/2

α

j

depending on the types of boundary conditions,

α

j

π

is the value of the angle. The last part of the paper is devoted to illustrate numerical experiments.

Processes governed by Partial Differential Equations (PDE) display very rich dynamical behavior, which is continuous spatially. Influencing the behavior of PDE systems through boundaries is an interesting research as it is involves the handling of infinite dimensionality, due to which the traditional tools of control theory do not apply directly. This study demonstrates how a nonlinear PDE is converted into a reasonably descriptive Ordinary Differential Equation (ODE) model. The approach is based on Proper Orthogonal Decomposition (POD), which separates the temporal and spatial components of the dynamics. The finite term expansion of the solution results in an autonomous ODE and this paper demonstrates how the external excitations are made explicit in the dynamical model. 2D Burgers equation is used to illustrate the effectiveness of the approach and a finite dimensional dynamical model is shown to be capable of capturing the essential response.

In this paper, we consider a boundary value problem (BVP) for second order nonlinear partial difference equations on the lattice rectangles. Some explicit conditions are established that ensure existence and uniqueness or solely existence of solution to the BVP under consideration.

The mapping and modified mapping methods, with a new mapping relation, have been developed to derive some new exact doubly periodic solutions of the (2+1)-D Kadomtsev-Petviashvili equation in terms of squares of Jacobian elliptic functions. The corresponding limit solutions such as triangular solutions, solitary wave solutions, and singular solutions in the case of the modulus of the elliptic function approaching 0 and 1 have also been derived.

We translate in semi-group theory Varadhan estimates, lower bound, got by ourself by using the Malliavin Calculus for hypoelliptic heat-kernels.

We solve Dirichlet problem of the two dimensional nonstationary heat conduction problem for orthotropic bounded cylinder with boundary conditions of the first and second kinds of circular discontinuity of temperature density of heat flow. The solution of such problem was derived with the help of Laplace and Hankel transformations.

A class of monotone conservative schemes is derived for the boundary value problem for second order differential equation with discontinuous coefficient. The necessary condition for conservativeness of the finite difference scheme is obtained. The examples are presented for different discontinuous coefficients and the theoretical statements for the conservativeness conditions are supported by the results of numerical experiments.

A hyperbolic type equation with certain initial and boundary conditions, appropriate for application of the Mikusiński calculus, is considered. Similar problems appeared as mathematical models of the shock between a solid body and a viscoelastic bar.

The exact solution of the corresponding problem in the field of Mikusiński operators is constructed, and the character and regularity of the operational function solution of the problem is analyzed. Then the solution of the starting problem is obtained as a finite sum of continuous functions. An algorithm for constructing an approximate solution is given, and an example is presented.

There are various scenarios proposed in literature for transition in plane channel (Poiseuille) flow. In this work, one of these scenarios, namely, streak break-down, is tested numerically using a Karhunen-Loeve (K-L) based model. The K-L basis was empirically generated earlier using a numerical database representing the flow. This basis is modified in this work to include the mean flow. A K-L basis provides an optimal parametrization of the underlying flow in energy norm. Since it is specific to the flow, each basis element carries an independent characteristic of the flow and has physical interpretation. A system of model amplitude equations is then obtained by Galerkin projection of the governing equations onto the space spanned by the K-L basis. The physical interpretation of the basis elements is used to truncate the resulting system to obtain a low dimensional model.

Wavelet transform method was successfully applied to the multicomponent analysis of the binary mixtures containing different colorants in commercial food product. In this application, wavelet transform method is suitable for the quantitative resolution of the mixtures of these colorants and this hybrid approach doesn’t require any separation and extraction steps. The method was tested by using various synthetic ternary mixtures and applied to the sample and successfully results were obtained.

In the spectral analysis, the continuous wavelet transform or very recently developed fractional wavelet transform are powerful tools for the data reduction, de-noising, compressing and baseline correction of the analytical signals and resolution of multicomponent overlapping signals. Recently, continuous wavelet transform in combination of zero-crossing approach and spectral ratio treatment has been used for the quantitative resolution and the prediction of multi-mixtures in the presence of the original overlapping signals. This combined approach provides a short time analysis, accurate, precision, rapid and low cost for the quality control and routine analysis of the commercial products containing active compounds. This hybrid approach indicates that this technique is perfectly suitable for the multicomponent analysis of the overlapping analytical signals in the various fields of the analytical chemistry. In addition, the wavelet transform method are an alternative and promising signal analysis approach for the elimination or reduction of the disadvantageous of the classical spectral derivative methods for the analytical purposes. This review presents briey the theoretical basis of the applications of continuous wavelet transform and fractional wavelet transform with the classical analytical approaches and reports some of their analytical applications.

Wavelet analysis is successfully applied to the quantitative determination of the components in the binary mixture. This mathematical application is based on the use of the division of the absorption signals by the standard absorption signal and the transformation of the ratio signals. Calibration functions are obtained by measuring the continuous wavelet amplitudes corresponding to the minimum points of the wavelengths. The method is validated and applied to one example of binary mixture analysis.

This paper presents an Improved Incremental Self-Organizing Map (I2SOM) network that utilizes automatic threshold (AT) value for the segmentation of ultrasound (US) images. I2SOM network has been compared with the well-known unsupervised Kohonen’s SOM network (KSOM) and a supervised Grow and Learn (GAL) network in terms of classification accuracy, learning time and number of nodes. For the feature extraction process, two-dimensional discrete cosine transform (2D-DCT) and 2D continuous wavelet transform (2D-CWT) were individually considered and were comparatively investigated to form the feature vectors of US breast and phantom images.

It is observed that the proposed automatic threshold scheme has significantly enhanced the robustness of I2SOM algorithm. Obtained results show that I2SOM can segment US images as good as Kohonen’s network.

Continuous wavelet transform (CWT) is a new powerful tool for removing noise, irrelevant information and signal baseline correction of voltammograms. In this application, morlet continuous wavelet transforms (ML-CWT) for signal treatments were found to be suitable among the wavelet families. MLCWT approach was applied to the peak current data vectors consisting of 139 data points in the potential range of (−1004) – (−1556) mV versus Ag/AgCl reference electrode. Peak current data for the calibration and prediction steps in the concentration range of 83.0–375.0

µg

=

m

L zafirlukast were obtained by using Osteryoung Square Wave Adsorption Stripping Voltammetry (OSWAdSV).

Three different calibration models namely mean centering calibration (MCC), principal component regression (PCR) and partial least squares (PLS) were constructed by using the relationship between concentration set and CWT-coefficients of the peak current data. The proposed methods were validated by analyzing the synthetic samples and standard addition samples. These methods were successfully applied to the quantitative analysis of zafirlukast in tablets and satisfactory results were reported.

A couple of numerical examples, which justify one way of determining the threshold in dependence of the allowed relative error, are given. Specificity of the mentioned way is the fact that geometric interpretation of the pyramidal algorithm and basic laws of the theory of probability were used in determining the threshold.

Computerized Ionospheric Tomography (CIT) is a method to reconstruct ionospheric electron density image by computing Total Electron Content (TEC) values from the recorded GPS signals. Due to the multi-scale variability of the ionosphere and inherent biases and errors in the computation of TEC, CIT constitutes an underdetermined ill-posed inverse problem. In this study, CIT is performed by using a Bayesian approach with Gaussian random field priors. The 3-D mean and the covariance of the assumed Gaussian random field priors can either be obtained from ionospheric models such as IRI or they can be estimated by an iterative algorithm from the GPS measurements. Given sparse and non-uniform TEC measurements, the electron field is obtained from mean square estimation where the Gaussian random field structure provides regularization. Geographical and temporal variations of ionosphere can be observed by obtaining tomographic reconstructions of electron density distribution from Earth-based GPS stations for both quiet and disturbed days of ionosphere. 2-D slices that will be obtained from 3-D reconstructions can be compared with the model based reconstructions or with the available Global Ionospheric Maps from IGS centers.

In this study, we investigated modeling performances of two popular nonlinear system identification methods, namely fuzzy modeling and Volterra series. In literature a general approach to nonlinear structure modeling does not exist, therefore both fuzzy models and Volterra series are interesting and widely used as they can approximate a large class of nonlinear functions. In fuzzy modeling, a dynamic system is modeled using a set of fuzzy membership functions and rules. The fuzzy model parameters are trained using optimization techniques. In Volterra series approach, the dynamic system is modeled using a set of kernel functions that represent the first and higher order convolutions. The kernel functions are typically estimated using an orthogonal expansion technique using a set of suitable basis functions such as Laguerre. We compared the modeling performance of these approaches on a hypothetical test system whose kernels or structure is known priori and observed that the Volterra modeling based on Laguerre basis expansion of kernels offers better performance.

Finding the location of an object is one of the important features of robotics applications, electronic warfare positioning, and the 3G/4G wireless communication systems. Many valuable location based services can be enabled by this new feature. Position estimation from Time Difference of Arrival (TDOA) measurements is one of the commonly used methods. This approach is based on intersections of hyperbolic curves defined by the time differences of arrival of signals received from different sources. The location is determined using standard complex computation methods that are usually implemented in software. In this paper, we propose new hardware-oriented algorithms that use only simple add and shift operations in the computation and therefore can be easily implemented in hardware.

We consider a duopoly model with unknown costs. The firms’ aims are to maximize their profits by choosing the levels of their outputs. The chooses are made simultaneously by both firms.

In this paper, we suppose that each firm has two different technologies, and uses one of them following a probability distribution. The utilization of one or the other technology affects the unitary production cost. We show that this game has exactly one Bayesian Nash equilibrium. We analyze the advantages, for firms and for consumers, of using the technology with highest production cost versus the one with cheapest production cost. We also analyze the expected total quantity produced in each situation, which is of particular importance in the case that scanty natural resources are used in the production.

In this paper, we consider a linear price setting duopoly competition with differentiated goods and with unknown costs. The firms’ aims are to choose the prices of their products according to the well-known concept of perfect Bayesian Nash equilibrium. There is a firm (

F

1

) that chooses first the price

p

1

of its good; the other firm (

F

2

) observes

p

1

and then chooses the price

p

2

of its good.

We suppose that each firm has two different technologies, and uses one of them following a probability distribution. The utilization of one or the other technology affects the unitary production cost. We show that there is exactly one perfect Bayesian Nash equilibrium for this game. We analyze the advantages, for firms and for consumers, of using the technology with highest production cost

versus

the one with cheapest production cost.

The aim of this work is to predict a future value of the daily mean discharge of the river Paiva. Several approaches are considered. Methods from Dynamical Systems and Stochastic Processes are applied. The Takens embedding shows an intermittent dynamical behaviour of the river Paiva where the laminar phase occurs in the absence of rainfall. The forcing of the system is non-deterministic and is due to the precipitation occurrence. Good predictability is found in the laminar regime.

A XY Heisenberg spin chain model with two perpendicular spins par site is mapped onto a Kirchhoff thin elastic rod. It is shown that in the case of constant curvature the Euler—Lagrange equation leads to the static sine-Gordon equation. The case of a double-helical DNA-like configuration corresponds to two interacting Heisenberg spin chains and the corresponding Euler—Lagrange equation gives a system of coupled static sine-Gordon-type equations. The kink-antikink type and periodical static solutions for these models are derived. The soliton dynamics and the the nonlinear excitations of the systems are investigated. The interplay between curvature and nonlinear excitations is analyzed as well.

We are given (

Ω

,

$$ \mathcal{F} $$

,

P

) as a complete probability space with right continuous complete

σ

-algebra filtration

$$ \left( {\mathcal{F}_t } \right)_{t \in \left[ {0,T} \right]} $$

, generated by the infinite sequence of independent Brownian motions (

W

i

)

i

≥1

. Let, for every

t

∈ [0,

T

],

L

2

(

Ω

,

$$ \mathcal{F}_t $$

,

R

) be the Hilbert space of all

$$ \mathcal{F}_t $$

-measurable, and square-integrable variables in

R

, and

L

2

(

∈

,

C

([0,

T

],

R

)) be the space of all square integrable and a.e. continuous functions on

R

equipped with the norm |

X

| = (

E

sup

t

∈[0,

T

]

|

X

(

t

)|

2

)

1/2

.

$$ L_2^\mathcal{F} $$

([0,

T

],

R

) denotes the Hilbert space of all square-integrable and

$$ \mathcal{F}_t $$

-adapted processes with values in

R

. Define the sequence

σ

(

x

) = (

σ

i

(

x

))

i

≥1

, where for each

i

≥ 1,

σ

i

(

x

) ∈

C

([0,

T

],

R

) and that

σ

(

x

) ∈ ℓ

2

, i.e. |

σ

(

x

)|

2

=

$$ \Sigma _{i = 1}^\infty $$

|

σ

i

(

x

)|

2

< ∞. In this paper we study the approximate controllability of the one-dimensional semi-linear stochastic differential equation

$$ \left\{ \begin{gathered} dX\left( t \right) = \left[ {AX\left( t \right) + Bu\left( t \right) + b\left( {X\left( t \right)} \right)} \right]dt + \sum\limits_{i = 1}^\infty {\sigma _i \left( {X\left( t \right)} \right)dW^i \left( t \right)} \hfill \\ X\left( 0 \right) = X_0 ,t \in \left[ {0,T} \right], \hfill \\ \end{gathered} \right. $$

where

A, B

∈

R

, and

u

∈

$$ L_2^\mathcal{F} $$

([0,

T

],

R

) is a control. We obtain sufficient conditions for approximate controllability of the above system when coefficients

b

, and

σ

satisfy non-Lipschitz conditions.

The FitzHugh-Nagumo (FHN) model was proposed as a simplification of the neuronal model and provided insight into the more complex neuronal models. Recently, an analytical approach has been proposed for determining the response of a neuron or of the activity in a network of connected neurons based on the FHN model with Gaussian white noise current. In this study, we investigate the synchronization between neuronal spiking activity and sub-threshold sinusoidal stimuli. For this purpose, we obtain the phase probability density of the spiking events for the sub-threshold stimuli. We show that the system exhibits the phase locking behaviour. We also show that the phase synchronization clusters the spiking activity on the positive phase of the sub- threshold sinusoidal driving for smaller frequencies while it shifts the spiking activity towards the negative phase for larger frequencies.

In the present report the dynamic behaviour of the one dimensional family of maps

$$ F_{a,b,c} \left( x \right) = c\left[ {\left( {1 - a} \right)x - b} \right]^{\frac{1} {{1 - a}}} $$

is examined, for different ranges of the control parametres

a, b

and

c

. These maps are of special interest, since they are solutions of

N

f

′

(

x

) =

a

, where

N

f

′

is the Newton’s method derivative. In literature only the case

N

f

′

(

x

) = 2 has been completely examined. Simultaneously, they may be viewed as solutions of normal forms of second order homogeneous equations,

F

″(

x

)+

p

(

x

)

F

(

x

) = 0, with immense applications in mechanics and electronics. The reccurent form of these maps,

$$ x_n = c\left[ {\left( {1 - a} \right)x_{n - 1} - b} \right]^{\frac{1} {{1 - a}}} $$

, after excessive iterations, shows an oscillatory behaviour with amplitudes undergoing the period doubling route to chaos. This behaviour was confirmed by calculating the corresponding Lyapunov exponents.

We consider dissipative soliton (dissipaton) of the second member of SL(2,R) AKNS hierarchy in 1+1 dimension and show that it describes nonlinear doubled damped oscillator in 0+1 dimensions, where the velocity field plays the role of an effective damping. Combined with the third member of the hierarchy it give also rise to the real 2+1 dimensional solitons of KP-II and for KN hierarchy, to solitons of the MKP-II. By the Hirota bilinear form for both flows, we find new bilinear system and two soliton solution, showing resonance behaviour with creation of four virtual solitons. Our approach allows one to interpret the resonance soliton as a composite object of two dissipative solitons in 1+1 dimensions.

Floating point operations, which find their applications in vast areas such as many mathematical optimization methods, digital signal and image processing algorithms, and Artificial Neural Networks (ANNs), require much area and time for ordinary implementation on Field Programmable Gate Arrays (FPGAs). However, meaningful floating point arithmetic implementation on FPGAs is quite difficult with low level design specifications due to mapping difficulties and the complexity of floating point arithmetic. Design and implementation of floating point arithmetic and mapping of this into an FPGA become easier with the emergence of new generation FPGAs and development of high level languages such as VHDL tools. This paper presents the implementation methodologies of various floating point arithmetic operations such as addition, subtraction, multiplication, and division using 32-bit IEEE 754 floating point format. The implementation is performed using Xilinxs Spartan 3 FPGAs. The algorithms and implementation steps used for different operations are discussed in detail. As an example, an ANN application is presented using these algorithms.

In the paper a new method of finding the time-dependent electric field in a layered inhomogeneous uniaxial anisotropic dielectric is suggested. This method is related to an initial value problem solving for finding the electric field. The permittivity

ɛ

is a diagonal matrix and the components of

ɛ

are smooth functions of the variable

x

3

only. The density of the electric current is the source of the electric waves. The Fourier transform of electric current density with respect to

x

1

and

x

2

variables is assumed to be a continuous function. The suggested method consists of finding the Fourier image with respect to

x

1

and

x

2

variables of the electric field. The problem of finding the Fourier image of the electric field is reduced to an operator integral equation. This operator integral equation is solved by successive approximations method. After that the time-dependent electric field is found by the inverse Fourier transform.