#### Abstract

The goal of this paper is study the mixed integral equation with singular kernel in two-dimensional adding to the time in the Volterra integral term numerically. We established the problem from the plane strain problem for the bounded layer medium composed of different materials that contains a crack on one of the interface. Also, the existence of a unique solution of the equation proved. Therefore, a numerical method is used to translate our problem to a system of two-dimensional Fredholm integral equations (STDFIEs). Then, Toeplitz matrix (TMM) and the Nystrom product methods (NPM) are used to solve the STDFIEs with Cauchy kernel. Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the methods. The codes were written in Maple.

#### 1. Introduction

Many problems of engineering, mathematical physical, and contact problems in the theory of elasticity lead to singular integral equations. Integral equations provide an important tool for solving the ordinary and partial deferential equations. Therefore, many different methods are used to obtain the solution of the linear and nonlinear integral equations. Brunner and Kauthen [1] introduced collocation and iterated collocation methods for solving the two-dimensional Volterra integral equation (T-DVIE). In [2], authors proposed a class of explicit Runge-Kutta-type methods of order 3 for solving nonlinear T-DVIE. In [3], authors studied the approximate solution of T-DVIEs by the two-dimensional differential transform method. Abdou, in [4, 5], used different methods to obtain the solution of F-VIE of the first and second kinds in which the Fredholm integral term is considered in position while the Volterra integral term is considered in time. EL-Borai et al., in [6], studied the numerical solution for the T-DFIE with weak singular kernel, but they have studied the problem on a rectangular path of the parties only. AL-Bugami in [7] studied and discussed the solution of the T-DFIE with applications in contact problems. AL-Bugami in [8] studied and discussed the solution of the two-dimensional singular Fredholm integral equation (T-DFIE) with time. The solution of a large of mixed boundary value problem of a great variety of contact and crack problems in solid mechanics, physical, and engineering can be related to a system of the singular IEs have a simple Cauchy-type singularity (Ref. [9]). In [10], the authors studied the linear two-dimensional Volterra integral equation with continuous kernel numerically. In [11], the authors discussed continuous Fredholm-Volterra integral equation. Al-Bugami, in [12], studied the singular Hammerstein-Volterra integral equation and solved numerically. In [13], the author studied the surface cracks of finite layers of fractional materials.

In this work, we consider a mixed integral equation with singular kernel in two-dimensional (MIE)

Formula (1) is called the MIE with singular kernel in two-dimensional of the second kind with Cauchy kernel in , where the FI term is considered in position with singular kernel, and the VI term is considered in time with a positive and continuous kernel. is known function, while is unknown function to be determined. The numerical coefficient *λ* is called the parameter of the IE.

#### 2. The Basic Formulas of the Problem

Consider the plane strain problem for the bounded layer medium (Figure 1), composed of three different materials. Let the medium material contain a crack on one of the interface. Without any loss in generality, the half length of the crack is assumed to be unity. We will consider with the effect of the ratio of the layer thickness to the crack length on the stress, intensity factors, and the strain energy release rate. For interesting the disturbed stress state, while is variable also with time, caused by the crack.

We assume that the overall stress distribution , in the imperfection free medium, is known. The stress state , in the cracked medium, may be expressed as

where is the disturbed state, which may be obtained by using the tractions which are the only external loads applied to the medium. The problem can always be expressed as the sum of a symmetric component and an antisymmetric component. The tractions are , where

The solution of the antisymmetric problem requires only a slight modification. Let be the and components of the displacement vector in the *-*th materials and satisfy the filed equations in the form:

Then, assume the displacement functions in the following:

is known function of . Hence, using (7) and (8) in (5) and (6), we get

Formula (11) has a solution

For solving the two formulas (9) and (10), we use the Fourier integral transform:

Then, we have

After solving the system of Eqs. (15) and (16), and then using the two formulas (13) and (14), we get

where has physical meaning and for plane strain and for generalized plane stress, is Poisson’s coefficient for each materials, and are functions of *α* which can be determined from the boundary conditions. After obtaining, the values of , the stresses may be evaluated by Hooks law.

The components of the stress vector at the interfaces and boundaries may be expressed as

On the boundaries, the medium may have formally any one of the following four groups of homogeneous boundary conditions

The continuity requires that on the interfaces, the stress and displacement vectors in the adjacent layers be equal, i.e.,

Now, to obtain the integral equation, we first assume that at , the bond between the two adjacent layers is perfect except for the dislocations at and defied by

where the superscripts + and – refer to the limiting values of the displacement as approaches zero from + and – sides, respectively. In addition to (21), on the interface , we have the following conditions

The components of the stress vector at and may be expresses as

where is the Heaviside functions, and is the Fourier transforms of defined as follows:

The constants depend on the elastic properties of the materials adjacent to the crack only and are given by

where is the shear modules, and *λ*’s is Lame’s constants.

Note that once the dislocations on the interface are specified, formulas (23)and (24) give the stresses for all values of . The crack problem under consideration is zero for and is unknown for . On the other hand, the stress vector on the interface is unknown for that is given by the following known functions for , i.e.,

Then,

Hence, we obtain

Evaluating the infinite integrals in (30), passing to the Cauchy theorems, we have where

The two formulas of (31) represent a system of MIE with Cauchy kernel. For one layer, we can have the following MIE, on noting the difference notations.

In general, we can write Eq. (33) in the form:

#### 3. The Existence and Uniqueness of the Solution

We write this formula in the integral operator form

We assume the following conditions:
(1)The singular kernel of FI term satisfies in the discontinuity condition(2)The kernel of VI term *ζ*(*t*, *τ*) is continuous in the Banach space and satisfies(3)The continuous kernel (4) in the space, , behaves as the known function

Theorem 1. *Eq. (34) has an exact unique solution in , under the condition
*

Lemma 1. *The integral operator maps into itself.*

*Proof. *From (35) and (40), the normality of the integral operator takes the forms
Applying Cauchy-Schwarz inequality, we have
Using the definition of the norm in the space , we get
Then, using condition (1), we obtain
Also, the term takes the form
Using condition (3), we get
Hence,
In the same manner, we can write
Using condition (2), we obtain
Thus, one has
Hence, with the aid of conditions (5), (44), (47), and (50), Eq. (41) takes the form
The inequality (51) involves the boundedness of the operators and .☐

Lemma 2. *The integral operator (35) under the condition (40) is continuous and contraction operator.*

*Proof. *For the functions in the space , formula (35) yields
Hence, we have
Using formula (53) with the conditions (1), (2), and (3), then applying Cauchy-Schwarz inequality, we obtain
Hence, is a continuous operator in the space , and under the condition , is a contraction operator.☐

#### 4. THE STDFIEs

Consider Eq. (34). In this section, we divide the interval where , to get

Using the quadrature formula, the Volterra term becomes

is the weight, where denotes the constant step size for integration. Using (56) in (55), we have

Formula (57) can be adapted in the form

Then, the general form of Eq. (58) can be represented as where

Formula (59) represents a linear system of TDFIEs of the second kind, which contains equation of unknown functions of corresponding to the time interval [0,*T*].

#### 5. Some Numerical Methods

##### 5.1. The TMM

We present the TMM to obtain numerical solution of TDFIE of the second kind with Cauchy form, which it expresses in the form which it may be adapted as where

Then, write the integral term in Eq. (62) as the form

Formula (64) reduces as

Then,

Then, we put in Eq. (66), and then we obtain where Eq. (65) becomes where

Thus, the IE (62) becomes

If we put then we get where

The matrix may be written as where is the TM of order , and the matrix

However, the solution of the system can be obtained in the form where is the identity matrix and .

##### 5.2. The PNM

Consider where

where and are badly behaved and well-behaved functions of their arguments, respectively. We approximate the integral term in (77) when by where is the weights. Also, we approximate the integral term in (77) in the form:

where with and even. Now, if we approximate the nonsingular part of the integrand over each interval , , by the second degree Lagrange interpolation polynomial that interpolates, we find where .

If we define