An Asymptotic Model for Solving Mixed Integral Equation in Position and Time

Jan, A. R.

doi:https://doi.org/10.1155/2022/8063971

Journal of Mathematics

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Research Article | Open Access

Volume 2022 | Article ID 8063971 | https://doi.org/10.1155/2022/8063971

An Asymptotic Model for Solving Mixed Integral Equation in Position and Time

A. R. Jan¹

Academic Editor: Xian-Ming Gu

Received26 May 2022

Revised19 Jun 2022

Accepted19 Jul 2022

Published30 Aug 2022

Abstract

In this paper, we considered a mixed integral equation (MIE) of the second kind in the space The kernel of position has a singularity and takes some different famous forms, while the kernels of time are positive and continuous. Using an asymptotic method of separating the variables, we have a Fredholm integral equation (FIE) in position with variable parameters in time. Then, using the Toeplitz matrix method (TMM), we obtain a linear algebraic system (LAS) that can be solved numerically. Some applications with the aid of the maple 18 program are discussed when the kernel takes Coleman function, Cauchy kernel, Hilbert kernel, and a generalized logarithmic function. Also the error estimate, in each case, is computed.

1. Introduction

The theory of IEs has many wide applications in different sciences, for example, in the contact displacement problems, in the theory of elasticity (see Abdou [1]), in the spectrum of mathematical physics problems (see Abdou and Alharbi [2]), in the thermomechanics (see Chirita et al., [3]) and in models (see Chirita [4]). For this aim, many authors have been interested to establish different kinds of analytical and numerical methods for solving the IEs of different types and different kinds with the continuous or discontinuous kernel. In [5], Ladopulas used the collocation evaluation method to discuss the solution of a singular integro-differential equation in Banach space. In [6], Mirzaee and Hoseini used the Fibonacci collocation method for solving Volterra–Fredholm integral equations with continuous kernels. In [7], Khairullina and Makletsov used the wavelet-collocation method to solve an integral equation with a singular kernel. Mirzaee and Hadadiyan[8] discussed the solution of the nonlinear class of mixed Volterra–Fredholm integral equations using an operational matrix. In addition, the same authors Mirzaee and Hadadiyan in [9], used three-dimensional triangular functions and their applications for solving nonlinear mixed Volterra–Fredholm integral equations. Mosa, et al. [10] applied the Adomian decomposition method to discuss the solution of a mixed integral equation in position and time with a phase-lag term in the position. In [11], Zaheer, et al. applied the meshless method to discuss the solution of the Volterra integral equation in one dimensional with an oscillatory kernel. Abd El-Rahman [12] considered a mixed integral equation with a weak singular kernel.

In this paper, we review a mixed integral equation in position and time in a general form, so that all previously solved equations of this type are considered special cases of this work. In addition, we consider the kernel of position as a singular term. Numerical applications have been made when the kernel of the position takes many forms of singular form, which are famous in different sciences. Due to the difficulty of solving the singular integral equations, the Toeplitz matrix method will be used. This method is one of the best methods to solve the singular integral equations, where the singular of integral term disappeared, and we have directed the numerical results.

Consider, in the space of position and time , the following MIE:where are known functions, while is an unknown function. The constant defines the kind of the IE, while the constant , may be complex and have several physical meanings.

We can adapt equation (1) in the integral operator form

In the remaining part of the paper, using the separation of variables method, we have quadratic SFIEs. Then, after using the TMM, as a famous numerical method for solving singular integral equations, see Abdou, et al. [13], we have LAS which can be solved numerically. Some numerical results are calculated, and the error in each case is computed.

2. Existence of a Unique Solution

To prove the existence of a unique solution of equation (1), we assume the following:(i)The kernel of position satisfies the discontinuity condition(ii)The two kernels of Volterra belong to the class , , respectively, and satisfy for two constants N and M, and the conditions are as follows:(iii)The given function is continuous in with its partial derivatives with respect to position and time , and its norm is defined as When , the existence of a unique solution of equation (1) can be proved using two different methods: Banach fixed point theorem or Picard method while Picard’s method fails when In this case, we use Banach fixed point theorem. In Banach’s fixed point theorem, we must prove that the integral operator of equation (2) is bounded and continuous. And then, if the constant of continuity is less than one, the integral operator is contracting. Hence, we have a unique solution.

Theorem 1. The MIE (1) has a unique solution in the space under the conditionThe idea is to construct the solution as a sequence of functions as , and we assume the solution in the formwhereandHence, the proof of the theorem comes as a result of the following two lemmas.

Lemma 1. Besides the conditions (i)–(iii), the infinite series is uniformly convergent to a continuous solution function

Proof. From (8), we pick up two sequences to construct the following:Adapting (12), with the aid of (9), in the formFor , after applying Cauchy–Schwarz inequality and using the conditions (i)-(iii), formula (13) yieldsBy induction, we haveThis bound, according to condition (7), makes the sequence converges uniformly. In addition, the result of inequality (15) leads us to decide that the sequence has a convergent solution. So, we haveSince each is continuous, therefore, is also continuous, and the infinite series of equation (16) is uniformly continuous. Hence, the lemma is proved.

Lemma 2. The continuous solution of formula (16) represents a unique solution of equation (1).

Proof. Assume and are two different solutions of (1). Hence, we haveafter applying Cauchy–Schwarz inequality, and using the conditions (i)–(iii), formula (17) yieldsSince , we deduce that the solution of (1) must be unique.

3. Separation of Variables Method

One of the most important ways to solve mathematical physics problems is the method of separating variables. The importance of this method comes when the unknown function is related to the position and time variables. This helps the researcher to determine the appropriate form to choose the time function and the appropriate time to be used, and accordingly, we study the behavior of the position function.

For this, consider the following approximations:

It is preferable to choose the time function in the form of a polynomial. From the constants of polynomials, the function of time can be classified in the form of an exponential, trigonometric, hyperbolic, and so on. This choice depends on the basis that the time period of the experiment (solving the problem) is always less than one and that the time, in the beginning, is not equal to zero. Based on that, we assume that

From equation (20), we can deduce the following:(1), These results lead us to say that the function of time is bounded.(2)If the function of time becomes . While .

As for different values of , we have different time functions.

Using (19) in (1), we can arrive at the following equation:

It is clear by separating the variables that the mixed integral equation is transformed directly into equation (21) and has become a Fredholm equation in position with constants related to time .

The existence of a unique solution of equation (21) is satisfied under the condition:

3.1. Convergence Analysis of Integral equation (21)

To discuss the convergence analysis of equation (21), assume the sequence of solutions in the following form:

Let and are two arbitrary distinct partial sums in the sequence of solution . Using the definition of metric space,

Hence, we have

As and is fixed so, we conclude that the sequence is a Cauchy sequence in a complete metric space. Then, the series is convergent.

4. Toeplitz Matrix Method

Toeplitz matrix method is considered one of the best methods for solving the singular integral equations, where the singular term directly disappears, and we have a linear system of algebraic equations. For this, rewrite the integral term of equation (21) in the following form:

Here in (26), and are arbitrary functions, and is the estimated error.

In the light of TMM, we can determine the values of and , respectively, in the following forms:where

Assuming and then using the following notations:

Hence, the integral equation (21) yields the following LAS of (2N + 1) equations:

The matrices can be written in the following Toeplitz matrix form:

It is clear from (31) that we have two kinds of matrices: the first is called the Toeplitz matrix of order while represents a matrix of the order whose elements are zeros except the first and the last rows (columns). Moreover, it is worth mentioning that such a Toeplitz-like linear system (31) can be solved by using the preconditioned Krylov subspace methods of operations and memory cost, where M = 2N + 1; see [14,15] for details.

The error term can be determined from the following formula:

5. Convergence Analysis of Linear Algebraic System (30)

To discuss and prove the convergence of LAS (30) in the Banach space , we write it in the operator form

Then, consider the following.

Lemma 3. For if the kernel of equation (1) satisfies the conditionsthen, we have(i), ( is constant)(ii)

Proof. Applying Cauchy–Schwarz inequality for the first formula of (27) and summing from to and then using the first condition of (34), we have . Since each term of is bounded above, hence, for , we getSimilarly, we consider a small constant , such thatFrom (35) and (36), we deduce thatHence, case (i) holds.
For the second case of (26), consider . After applying Hölder inequality, and summing from to , we haveFinally, case (ii) of Lemma 3 is held.

6. The Existence and Uniqueness Solution of the LAS (21) of Toeplitz Matrix

To discuss the unique solution of the LAS (30), we consider the following assumptions: ( is a constant) For the constants , the points satisfy

Theorem 2. For all values of time , the LAS of equation (30), in the Banach space , has a unique solution under the following condition:To prove this theorem, we must consider the following lemmas.

Lemma 4. If the conditions (vi)–(iv) are verified, then the operator defined by equation (33) maps the space into itself.

Proof. Let U be the set of all functions in such that , . Define the norm of the operator in the Banach space byFrom formula (43), and with the aid of the conditions (vi)–(iv), we haveThe above inequality can be adapted in the following form:The inequality (43) shows that the operator maps the set U into itself, whereSince , therefore, we have . Also, the inequality (43) involves that the operator T is bounded, where . Hence, the operator is bounded.

Lemma 5. Under the two conditions (vi) and (iv), is a contraction operator in the space .

Proof. For the two functions and in , formula (33) leads toIn the light of conditions (vi) and (iv), and in view of (41), we obtainThe inequality (46) shows that the operator is continuous in the space , and then, is a contraction operator under the condition .
In the light of Lemmas 4 and 5, the operator defined by (33) is contractive in the Banach space . Hence, has a unique fixed point which is the unique solution of the linear algebraic system in .

Definition 1. The TMM is said to be convergent of order in the interval , if and only if for sufficiently large , there exists a constant independent of such that

7. Applications

We solve equation (1) numerically using TMM with the aid of the maple 18 program, when the kernel of Fredholm term takes some weakly singular and singular forms, and at different times, the approximate solution and the estimated error, in each case, are calculated. In general, consider the following general form of a mixed integral equation with its exact solution

Example 1. When the kernel of (26) takes the Carleman kernel,The Carleman kernel is called in the sense of contact problem weakly kernel. The importance of the Carleman kernel came from the works of Russian scholes in the nonlinear theory of plasticity; see Popov [16] and Aleksandrovsk and Covalence [17]. In this case, the elements of the Toeplitz coefficient matrix of (31) take the following form:The estimated error becomesIn Table 1, we calculate numerically, the numerical solution of (35), using TMM for two different values of Poisson’s coefficient ʋ at time and the estimating error in each case is computing.
Figures 1(a) and 1(b) describe the numerical solution of (41) with the Carleman kernel at T = 0.3.

(a)

(b)

Example 2. With the Cauchy kernel,The integral equations with Cauchy kernels have wide applications in mathematical physics problems and in the different sciences, see Abdou [1], Constanda [18], and Frankel [19].
In this case, the elements of the Toeplitz matrix of (31), at time , take the following form:The estimated error, at time , takes the following form:See Table 2 and Figure 2.

Example 3. Consider the Hilbert kernelThe importance of the integral equation with the Hilbert kernel with its application in the crack problem in the theory of elasticity can be found in Kamalyan, et al., [20].
To obtain the elements of the Toeplitz matrix, we must use the famous integral formula, see Gradstein and Ryzhik [21].where B2s are Bernoulli numbers.
After using the famous integral of Bernoulli, we have the elements of the Toeplitz coefficient matrix in the following form:See Table 3 and Figure 3.

Example 4. When the kernel takes the following generalized logarithmic function form:Several problems have been considered when the kernel takes the logarithmic form. , for example, see Abdou, et al., [22,23].
The importance of logarithmic function came from its form and its derivatives form, where we can derive the following cases:(i) (u (x, y) is continuous function)(ii) Cauchy kernelIn addition, we can establish a strong singular kernel and so on. In our application, we use the result of the following integral, see Gradstein and Ryzhik [21]In the generalized logarithmic kernel, the elements of the Toeplitz matrix becomeThe result of estimate error is (4-1): We have the logarithmic kernel. We use TMM to solve numerically equation (21) at the time interval . See Table 4 and Figure 4. (4-2): We have a strong singular logarithmic kernel of power five at the interval of time with the exact solution See Table 5 and Figure 5.

8. Conclusions

From the above work, we can deduce the following:(1)From Equation (1), we can establish the following famous special cases:(i)Let in (1), to obtain The above formula (62) represents the Fredholm–Volterra integral equation of the second kind with discontinuous kernels. Many authors established the solution of (62), using different numerical methods, see Khairullina, et al., [7] and Bazm [24](ii)Fredholm–Volterra integral equations of the first kind, with different singular kernels, are obtained directly from (62), when , for the solution of these cases with some different methods, see Abdou et al., [2,25].(iii)Let in (1) , to have Formula (63) represents the Volterra–Fredholm integral equation of the second kind. The solution of (63), using some different numerical methods, is discussed by many authors, for example, see Abdou, et al., [22,23], Mirzaee and Hoseini [6], and Mirzaee and Samadyar [26](iv)If in (3) , we have V-FIE of the first kind. Many spectral relationships, when the kernel of position has a singular term and the kernel of time is continuous, have been obtained and discussed in Mirzaee and Hoseini [6].(2)We notice the following from the results of the programs in all previous cases.(i)In Table 1, the error increases with increasing of , where is called the Poisson ratio, and in the theory of elasticity and when , the atomic bond between the particles of the material is normal, while ; the atomic bond becomes strong, and for this, the error may be large. Also, it was found that the maximum error value is obtained when Moreover, the error decreases gradually and has less value when . The maximum error is at , when The minimum error is at , when . In all the studied situations, the error value increases when it gets closer to the end points . It also decreases at the middle when it gets closer to zero.(ii)In Table 2, the maximum error is at , and when the values of T are increasing, the error values increase slowly.(3)The TMM is considered as one of the best methods for solving the singular integral equation, where the singularity disappears and the solution can be obtained directly.

Data Availability

All data are taken from the results which are calculated using mathematical programs.

Additional Points

In the future work, we will try to consider the mixed integral equation in (2 + 1)—dimensional with potential kernel.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The researcher extends all thanks and gratitude to Prof. Dr. M. A. Abdou for his important scientific advice and guidance while working on this research.

References

M. A. Abdou, “On asymptotic method for Fredholm – Volterra integral equation of the second kind in contact problems,” Journal of Computational and Applied Mathematics, vol. 154, pp. 431–446, 2003.
View at: Google Scholar
M. A. Abdou and F. M. Alharbi, “Generalized main theorem of spectral relationships for logarithmic kernel and its applications,” Journal of Computational and Theoretical Nanoscience, vol. 16, no. 7, pp. 2704–2711, 2019.
View at: Google Scholar
S. Chirita, M. Ciarletta, and V. Tibullo, “On the thermomechanical consistency of the time differential dual-phase-lag models of heat conduction,” International Journal of Heat and Mass Transfer, vol. 114, pp. 277–285, 2017.
View at: Publisher Site | Google Scholar
S. Chirita, “On the time differential dual-phase-lag thermoelastic model,” Meccanica, vol. 52, no. 1-2, pp. 349–361, 2017.
View at: Publisher Site | Google Scholar
E. G. Ladopoulos, “Non-linear singular integro-differential equations in Banach spaces by collocation evaluation Methods Univers,” Journal of Integral Equations and Applications, vol. 1, pp. 28–38, 2013.
View at: Google Scholar
F. Mirzaee and S. F. Hoseini, “Application of Fibonacci collocation method for solving Volterra-Fredholm integral equations,” Applied Mathematics and Computation, vol. 273, pp. 637–644, 2016.
View at: Publisher Site | Google Scholar
L. E. Khairullina and S. V. Makletsov, “Wavelet-collocation method of solving singular integral equation,” Indian Journal of Science and Technology, vol. 10, no. 1, pp. 1–5, 2017.
View at: Google Scholar
F. Mirzaee and E. Hadadiyan, “Using operational matrix for solving nonlinear class of mixed Volterra–Fredholm integral equations,” Mathematical Methods in the Applied Sciences, vol. 40, no. 10, pp. 3433–3444, 2017.
View at: Publisher Site | Google Scholar
F. Mirzaee and E. Hadadiyan, “Three-dimensional triangular functions and their applications for solving nonlinear mixed Volterra–Fredholm integral equations,” Alexandria Engineering Journal, vol. 55, no. 3, pp. 2943–2952, 2016.
View at: Publisher Site | Google Scholar
G. A. Mosa, M. A. Abdou, and A. S. Rahby, “Numerical solutions for nonlinear Volterra-Fredholm integral equations of the second kind with a phase lag,” AIMS Mathematics, vol. 6, no. 8, pp. 8525–8543, 2021.
View at: Publisher Site | Google Scholar
Z. Up-Din, S. U-I-Islam, and S. Zaman, “Meshless procedure for highly oscillatory kernel based one-dimensional Volterra integral equations,” Journal of Computational and Applied Mathematics, vol. 413, Article ID 114360, 2022.
View at: Publisher Site | Google Scholar
R. O. A. El-Rahman, “General formula of linear mixed integral equation with weak singular kernel,” IOSR Journal of Mathematics, vol. 12, no. 04, pp. 31–38, 2016.
View at: Publisher Site | Google Scholar
M. A. Abdou, M. M. El-Borai, and M. M. El-Kojok, “Toeplitz matrix method and nonlinear integral equation of Hammerstein type,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 765–776, 2009.
View at: Publisher Site | Google Scholar
X.-M. Gu, T.-Z. Huang, X.-L. Zhao, W.-R. Xu, H. B. Li, and L. Li, “Circulant preconditioned iterative methods for peridynamic model simulation,” Applied Mathematics and Computation, vol. 248, pp. 470–479, 2014.
View at: Publisher Site | Google Scholar
X.-M. Gu, T. –Z. Huang, X. –L. Zhao, H. B. Li, and L. Li, “Fast iterative solvers for numerical simulations of scattering and radiation on thin wires,” Journal of Electromagnetic Waves and Applications, vol. 29, no. 10, pp. 1281–1296, 2015.
View at: Publisher Site | Google Scholar
G. Y. Popov, Contact Problems for a Linearly Deformable Base, 1982.
V. M. Aleksandrovsk and E. V. Covalence, Problems in Mechanics Media with Mixed Boundary Conditions, Isdatel’stvo Nauka, Moscow, Russia, 1986.
C. Constanda, “Integral equations of the first kind in plane elasticity,” Quarterly of Applied Mathematics, vol. 53, no. 4, pp. 783–793, 1995.
View at: Publisher Site | Google Scholar
J. I. Frankel, “A Galerkin solution to a regularized Cauchy singular integro-differential equation,” Quarterly of Applied Mathematics, vol. 53, no. 2, pp. 245–258, 1995.
View at: Publisher Site | Google Scholar
A. G. Kamalyan, A. G. Stepanyan, and G. M. Topikyan, “Integral equations with Hilbert kernel,” Journal of Contemporary Mathematical Analysis, vol. 47, no. 2, pp. 51–61, 2012.
View at: Publisher Site | Google Scholar
I. S. Gradstein and I. M. Ryzhik, Tables of Integrals, Series and Product, Academic Press, New York, NY, USA, 5th edition, 1994.
M. A. Abdou, A. A. Soliman, and M. A. Abdel-Aty, “On a discussion of Volterra-Fredholm integral equation with discontinuous kernel,” Journal of the Egyptian Mathematical Society, vol. 28, no. 1, p. 11, 2020.
View at: Publisher Site | Google Scholar
M. A. Abdou and S. A. Raad, “Mixed integral equation in position and time with weakly kernels,” Int. J. Appl. Math. Comput., vol. 2, no. 2, pp. 57–67, 2010.
View at: Google Scholar
S. Bazm, “Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations,” Journal of Computational and Applied Mathematics, vol. 275, pp. 44–60, 2015.
View at: Publisher Site | Google Scholar
M. A. Abdou, M. N. Elhamaky, A. A. Soliman, and G. A. Mosa, “The behaviour of the maximum and minimum error for Fredholm-Volterra integral equations in two-dimensional space,” Journal of Interdisciplinary Mathematics, vol. 24, no. 8, pp. 2049–2070, 2021.
View at: Publisher Site | Google Scholar
F. Mirzaee and N. Samadyar, “Convergence of 2D- orthonormal Bernstein collocation method for solving 2D-mixed Volterra-Fredholm integral equations,” Transactions of A. Razmadze Mathematical Institute, vol. 172, no. 3, pp. 631–641, 2018.
View at: Publisher Site | Google Scholar

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Copyright © 2022 A. R. Jan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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