In the paper, the approximate solution for the two-dimensional linear and nonlinear Volterra-Fredholm integral equation (V-FIE) with singular kernel by utilizing the combined Laplace-Adomian decomposition method (LADM) was studied. This technique is a convergent series from easily computable components. Four examples are exhibited, when the kernel takes Carleman and logarithmic forms. Numerical results uncover that the method is efficient and high accurate.
Singular Integral Equation Linear and Nonlinear V-FIE Adomian Decomposition Method (ADM) Carleman Kernel Logarithmic Kernel1. Introduction
The V-FIE arises from parabolic boundary value problems. In practical applications one frequently encounters the V-FIE with singular kernel of the form
The functions, and are given and called the kernel of Fredholm integral term, Volterra integral term and the free term respectively and is a real parameter (may be complex and has physical meaning). Also, Ω is the domain of integration with respect to position, and the time t,. While is the unknown function to be determined in the space. In [1] Abdou et al. studied the existence and uniqueness of solution of V-FIE.
There are several techniques that have been utilized to handle the integral Equation (1), in [2] - [5] a few techniques, for example, the projection method, time collocation method, the trapezoidal Nystrom method, and furthermore analytical or numerical techniques were utilized to treated this equation, but this techniques experienced troubles as far as computational work utilized. In [6] treated Maleknejad and Hadizadeh Equation (1) by using the ADM presented in [7] - [9] .
Many authors have studied solutions of two-dimensional linear and nonlinear integral equations by utilizing different techniques, such as Abdou et al. in [10] discussed the solution of linear and nonlinear Hammerstien integral equations with continuous kernel and used two different methods (Adomian decomposition method and homotopy analysis method). Abdou et al. in [11] - [13] considered the integral equation with singular kernel and used Toeplitz matrix and product Nystrom methods to obtain the solution. In [14] El-Kalla and Al-Bugami used ADM and degenerate kernel method for solving nonlinear V-FIE with continuous kernel.
In this paper, we will discuss the combined (LADM) to approximate solutions with high degree of accuracy for V-FIE with a generalized singular kernel.
2. The Adomian Decomposition Method for Solving V-FIE
Consider the integral equation
The (ADM) introduces the following expression
for the solution of Equation (2), where the components will be determined recurrently. Moreover, the method defines the nonlinear function by an infinite series of polynomials
where are the so-called Adomian polynomials that represent the nonlinear term and can be calculated for various classes of nonlinear operators according to specific algorithms set by Adomian [8] [9] . A new algorithm for calculating these polynomials was established by Wazwaz [15] [16] .
Substituting Equation (3) and Equation (4) into Equation (2) yields
The components are computed using the following recursive relations
Relations (6,7) will enable us to determine the components recurrently, and as a result, the series solution of is readily obtained.
3. Laplace Adomian Decomposition Method Applied to V-FlE with Singular Kernel3.1. Carleman Kernel
We assume that the kernel of Equation (1) takes the form, [17] then integral Equation (1) can be expressed as:
Applying the Laplace transform to both sides of Equation (8) gives:
The ADM can be used to handle Equation (9). We represent the linear term from Equation (3) and the nonlinear term will be represented by the Adomian polynomials from Equation (4).
Substituting Equation (3) and Equation (4) into Equation (9) leads to
The ADM introduces the recursive relation
Applying the inverse Laplace transform to the first part of Equation (11) gives. Utilizing will empower us to evaluate, and so on. This will prompt the complete determination of the components of upon utilizing the second part of Equation (11). The series solution follows promptly after utilizing Equation (3). The obtained series solution may converge to an exact solution if such a solution exists.
3.2. Logarithmic Kernel
We assume that the kernel of Equation (1) takes the form, [17] then integral Equation (1) can be expressed as:
Applying the Laplace transform to both sides of Equation (12) gives:
Using the same method we shall find at the end the required solution by the inverse of Laplace transform.
4. Numerical Examples4.1. Application for Carleman Kernel and Logarithmic Kernel
We consider two examples for the integral equation
, the exact solution
We consider the linear and nonlinear cases: respectively, for the Carleman kernel
and the computing results are obtained when where is called Poisson’s coefficient, , while the kernel in the second example takes the logarithmic kernel and the results are computing, using Maple 17 at and.
Example 1 [(see 1)]: Consider the V-FIE with Carleman kernel
, the exact solution
Using Maple 17, we obtain Table 1 and Table 2
Example 2 [(see 1)]: Consider the V-FIE with logarithmic kernel
, the exact solution
Using Maple 17, we obtain Table 3 and Table 4
Results obtained for example 1 and error (Linear case,<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/6-1100541x62.png" xlink:type="simple"/></inline-formula>)
Error
App.
Error
App.
Exact
0.000000000E+00
1.00000000E−06
0.000000000E+00
1.00000000E−06
1.000000000E−06
−1.00E+00
0.000000000E+00
2.500000000E−07
0.000000000E+00
2.500000000E−07
2.50000000E−07
−5.00E−01
0.000000000E+00
2.500000000E−07
0.000000000E+00
2.500000000E−07
2.500000000E−07
5.00E−01
3.000000000E−16
9.999999997E−07
3.000000000E−16
9.999999997E−07
1.000000000E−06
1.00E+00
8.402300000E−05
4.900840230E−01
8.148970000E−05
4.900814897E−01
4.900000000E−01
−1.00E+00
1.31121000E−05
1.225131121E−01
1.025640000E−05
1.225102564E−01
1.225000000E−01
−5.00E−01
2.447000000E−05
1.224755300E−01
1.026020000E−05
1.224897398E−01
1.225000000E−01
5.00E−01
1.568011000E−04
4.898431989E−01
8.150950000E−05
4.899184905E−01
4.900000000E−01
1.00E+00
Results obtained for example 1 and error (Nonlinear case,<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/6-1100541x71.png" xlink:type="simple"/></inline-formula>)
Error
App.
Error
App.
Exact
4.690000000E−13
9.999995310E−07
5.073000000E−13
9.999994927E−07
1.000000000E−06
−1.00E+00
1.464000000E−13
2.499998536E−07
1.277000000E−13
2.499998723E−07
2.50000000E−07
−5.00E−01
2.732000000E−13
2.499997268E−07
1.278000000E−13
2.499998722E−07
2.500000000E−07
5.00E−01
8.754000000E−13
9.999991246E−07
5.076000000E−13
9.999994924E−07
1.000000000E−06
1.00E+00
1.609239000E−04
4.898390761E−01
1.739859000E−04
4.898260141E−01
4.900000000E−01
−1.00E+00
5.022500000E−05
1.224497750E−01
4.379890000E−05
1.224562011E−01
1.225000000E−01
−5.00E−01
9.366060000E−05
1.224063394E−01
4.38206000E−05
1.224561794E−01
1.225000000E−01
5.00E−01
3.001406000E−04
4.896998594E−01
1.740717000E−04
4.898259283E−01
4.900000000E−01
1.00E+00
Results obtained for example 2 and error (Linear case,<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/6-1100541x80.png" xlink:type="simple"/></inline-formula>)
Error
App.
Exact
6.000000000E−16
9.999999994E−07
1.000000000E−06
−1.00E+00
1.000000000E−16
2.499999999E−07
2.50000000E−07
−5.00E−01
1.000000000E−16
2.500000001E−07
2.500000000E−07
5.00E−01
1.000000000E−15
1.000000001E−06
1.000000000E−06
1.00E+00
1.467841000E−04
4.898532159E−01
4.900000000E−01
−1.00E+00
2.527690000E−05
1.224747231E−01
1.225000000E−01
−5.00E−01
2.527990000E−05
1.225252799E−01
1.225000000E−01
5.00E−01
1.467682000E−04
4.901467682E−01
4.900000000E−01
1.00E+00
4.2. Application for a Generalized Carleman Kernel and Logarithmic Kernel
Example 3 [(see 11, 13)]: Consider the V-FIE with generalized Carleman kernel
, the exact solution
Using Maple 17, we obtain Table 5.
Example 4 [(see 11,13)]: Consider the V-FIE with generalized logarithmic kernel
, the exact solution,
Using Maple 17, we obtain Table 6.
Results obtained for example 2 and error (Nonlinear case,<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/6-1100541x91.png" xlink:type="simple"/></inline-formula>)
Error
App.
Exact
7.500000000E−13
1.000000750E−06
1.000000000E−06
−1.00E+00
2.741000000E−13
2.500002741E−07
2.50000000E−07
−5.00E−01
2.741000000E−13
2.500002741E−07
2.500000000E−07
5.00E−01
7.500000000E−13
1.000000750E−06
1.000000000E−06
1.00E+00
2.2570318000E−04
4.902570318E−01
4.900000000E−01
−1.00E+00
9.399840000E−05
1.225939984E−01
1.225000000E−01
−5.00E−01
9.407250000E−05
1.225940725E−01
1.225000000E−01
5.00E−01
2.573282000E−04
4.902573282E−01
4.900000000E−01
1.00E+00
Results obtained for example 3 and error
Nonlinear
Linear
Exact
Error
App.
Error
App.
0.000000000E+00
−4.665600000E−14
2.285000000E−20
−4.665602285E−14
−4.665600000E−14
−1.00E+00
0.000000000E+00
−1.45800000E−15
4.980000000E−22
−1.458000498E−15
−1.45800000E−15
−5.00E−01
0.000000000E+00
1.45800000E−15
7.929000000E−21
1.457992071E−15
1.45800000E−15
5.00E−01
0.000000000E+00
4.665600000E−14
3.638500000E−19
4.665563615E−14
4.665600000E−14
1.00E+00
2.646100000E−08
−4.095973539E−03
4.223060000E−06
−4.100223060E−03
−4.096000000E−03
−1.00E+00
1.800000000E−11
−1.279999820E−04
1.230657000E−07
−1.281230657E−04
−1.280000000E−04
−5.00E−01
2.862000000E−10
1.279997138E−04
3.023951300E−06
1.249760487E−04
1.280000000E−04
5.00E−01
4.202460000E−07
4.095579754E−03
1.374088740E−04
3.958591126E−03
4.096000000E−03
1.00E+00
Results obtained for example 4 and error
Nonlinear
Linear
Exact
Error
App.
Error
App.
0.000000000E+00
−4.665600000E−14
1.828950000E−18
−4.665417105E−14
−4.665600000E−14
−1.00E+00
0.000000000E+00
−1.45800000E−15
3.666200000E−20
−1.457963338E−15
−1.45800000E−15
−5.00E−01
0.000000000E+00
1.45800000E−15
3.666300000E−20
1.458036663E−15
1.45800000E−15
5.00E−01
0.000000000E+00
4.665600000E−14
1.828990000E−18
4.665782899E−14
4.665600000E−14
1.00E+00
1.965280000E−06
−4.097965280E−03
7.262218110E−04
−3.369778189E−03
−4.096000000E−03
−1.00E+00
1.179900000E−09
−1.280011799E−04
1.400159720E−05
−1.139984028E−04
−1.280000000E−04
−5.00E−01
1.179900000E−09
1.280011799E−04
1.594017510E−05
1.439401751E−04
1.280000000E−04
5.00E−01
1.966603000E−06
4.097966603E−03
8.51706301E−04
4.947706301E−03
4.096000000E−03
1.00E+00
5. Conclusion
In this study, we considered linear and nonlinear integral equations of type Volterra-Fredholm with singular kernel. We have proven that the (LADM) is effective and useful technique for solving these kinds of integral equations with singular kernel and many nonlinear problems, efficiency and accuracy of the introduced method are illustrated by four numerical examples which showed simplicity of this method.
Acknowledgements
The authors would like to thank the King Abdulaziz city for science and technology.
Cite this paper
Fatheah Ahmed Hendi,Manal Mohamed Al-Qarni, (2016) Numerical Treatment of Nonlinear Volterra-Fredholm Integral Equation with a Generalized Singular Kernel. American Journal of Computational Mathematics,06,245-250. doi: 10.4236/ajcm.2016.63025
ReferencesHendi F.A. ,et al. (2011)Laplace Adomian Decomposition Method for Solving the Nonlinear Volterra Integral Equation with Weakly Kernels Studies in Nonlinear Sciences 2, 129-134.Wazwaz, A.M. (2000) A New Algorithm for Calaculating Adomian Polynomials for Nonlinear Operators. Applied Mathematics and Computation, 111, 53-59. http://dx.doi.org/10.1016/S0096-3003(99)00063-6Wazwaz, A.M. (2002) A Reliable Treatment for Mixed Volterra-Fredholm Integral Equations. Applied Mathematics and Computation, 127, 405-414. http://dx.doi.org/10.1016/S0096-3003(01)00020-0El-Kalla I.L. and Al-Bugami A.M. ,et al. (2012)Numerical Solution for Nonlinear Volterra-Fredholm Integral Equation with Applications in Torsion Problems International Journal of Computational and Applied Mathematics 7, 403-418.Al-Bugami A.M. ,et al. (2013)Toeplitz Matrix Method and Volterra-Hammerstin Integral Equation with a Generalized Singular Kernel Progress in Applied Mathematics 6, 16-42.El-Kalla I.L. and Al-Bugami A.M. ,et al. (2011)Fredholm-Volterra Integral Equation with A Generalized Singular Kernel and Its Numerical Solutions IJRRAS 6, 341-352.Abdou, M.A., El-Kalla, I.L. and Al-Bugami A.M. (2011) Numerical Solution for Volterra-Fredholm Integral Equation with A Generalized Singular Kernel. Journal of Modern Methods in Numerical Mathematics, 2, 1-15. http://dx.doi.org/10.20454/jmmnm.2011.60Abdou M.A., El-Kalla I.L. and Al-Bugami A.M. ,et al. (2011)New Approach for Convergence of the Series Solution to a Class of Hammerstein Integral Equations International Journal of Applied Mathematics and Computation 3, 261-269.Adomian, G. (1994) Solving Frontier Problems of Physics: The Decomposition Method. Kluwer. http://dx.doi.org/10.1007/978-94-015-8289-6Adomian, G. (1991) A Review of the Decomposition Method and Some Recent Results for Nonlinear Equation. Computers & Mathematics with Applications, 21, 101-127. http://dx.doi.org/10.1016/0898-1221(91)90220-XCherruault, Y., Saccomandi, G. and Some, B. (1992) New Results for Convergence of Adomian’s Method Applied to Integral Equations. Mathematical and Computer Modelling, 16, 85-93. http://dx.doi.org/10.1016/0895-7177(92)90009-AMaleknejad, K. and Hadizadeh, M. (1999) A New Computational Method for Volterra-Fredholm Integral Equations. Computers & Mathematics with Applications, 37, 1-8. http://dx.doi.org/10.1016/S0898-1221(99)00107-8Brunner, H. (1990) On the Numerical Solution of Nonlinear Volterra-Fredholm Integral Equation by Collocation Methods. SIAM Journal on Numerical Analysis, 27, 987-1000. http://dx.doi.org/10.1137/0727057Han, G.Q. and Zhang, L.Q. (1994) Asymptotic Expansion for the Trapezoidal Nystrom Method of Linear Volterra-Fredholm Equations. Journal of Computational and Applied Mathematics, 51, 339-348. http://dx.doi.org/10.1016/0377-0427(92)00013-YKauthen, P.J. (1989) Continuous Time Collocation Methods for Volterra-Fredholm Integral Equations. Numerische Mathematik, 56, 409-424. http://dx.doi.org/10.1007/BF01396646Hacia L. ,et al. (1996)On Approximate Solution for Integral Equations of Mixed Type. ZAMM. Z. Angew. Math. Mech 76, 415-416.Abdou, M.A., Hendi, F.A. and Abu Alnaja, K.J.M. (2012) Toeplitz Matrix Method and Volterra-Hammerstin Integral Equation. Far East Journal of Applied Mathematics, 70, 103-121.