Integral equations arise in various fields of science and engineering. Nowadays integral equations and their applications is an important subject in applied mathematics. In some cases, it is difficult to solve them, especially analytically. There are many analytical approaches were introduced, such as A domian decomposition method, successive substitutions, Laplace transformation method, Picard’s method, etc. [1]]. The analytical approaches have limited applicability either because the analytical solutions of some of the problems those arise in application do not exists or because they are more time consuming. Therefore, many integral equations arising in mathematical modelling of physical world problem demands efficient numerical techniques. A number of numerical techniques such as Galerkin method [2]], collocation method [2]], Nystrom interpolation [3]], etc. have been proposed by various authors. In numerical analysis, solving integral equations is reduced to solving a system of algebraic equations. Iterative techniques are available to solve a system of algebraic equations, such as Newton, method, Jacobi iterative method, Gauss-Seidel method, etc. In the past several years, considerable attention has been made for obtaining solutions of nonlinear integral equations. Nonlinearity is one of the interesting topics among the physicists, mathematicians, engineers, etc. Since most physical systems are inherently nonlinear in nature.
The full-approximation scheme (FAS) is largely applicable in increasing the efficiency of the iterative methods used to solve nonlinear system of algebraic equations. In the historical three decades the development of effective iterative solvers for nonlinear systems of algebraic equations has been a significant research topic in numerical analysis. Nowadays it is recognized that FAS iterative solvers are highly efficient for nonlinear differential equations introduced by Brandt [4]]. For a detailed treatment of FAS is given in Briggs et al. [5]]. An introduction of FAS is found in Hackbusch and Trottenberg [6]], Wesseling [7]] and Trottenberg et al. [8]]. Many authors applied the FAS to some class of differential equations; Lubrecht [9]], Venner and Lubrecht [10]], Zargari [11]] and others have significant contributions in EHL problems. Lee [12]] has introduced a multigrid method for solving the nonlinear Urysohn integral equations. In this paper, we introduce the full-approximation scheme (FAS) for the numerical solution of nonlinear Volterra integral and integro-differential equations.
Wavelet based numerical methods are used for solving the system of equations with faster convergence and lower computational cost. Since 1991, the various types of wavelet based methods have been applied for the numerical solution of different kinds of integral equations, a detailed survey on these papers can be found in [13]–15]. In recent years, wavelet analysis is gaining considerable attention in the numerical solution of differential equations. Recently, many authors (Leon [16]], Bujurke et al. [17]–19], Avudainayagam and Vani [32]]) have worked on wavelet multigrid method for the numerical solution of differential equations. Wang et al. [21]] have applied a fast wavelet multigrid algorithm for the solution of electromagnetic integral equations. In this paper, we introduce the modified wavelet full-approximation scheme (MWFAS) for the numerical solution of nonlinear Volterra integral and integro-differential equations. Thus, the proposed scheme can be applied to a wide range of science and engineering problems.
The organization of this paper is as follows. In Section 2, Daubechies wavelets is given. In Section 3 intergrid operators are discussed. In Section 4, method of solution is discussed. In Section 5, presents the numerical experiments and results are given. Finally, conclusion of the proposed work is given in Section 6.
The refinement relation of scaling function
where
Based on the scaling function
where {
In this paper, we use Daubechies filter coefficients for
Brandt [4]] was one of the first to introduce nonlinear multigrid, which seeks to use concepts from the linear multigrid iteration and apply them directly in the nonlinear setting. Since the early application to elliptic partial differential equations, multigrid methods have been applied successfully to a large and growing class of problems. Classical multigrid begins with a two-grid process. First, iterative relaxation is applied, whose effect is to smooth the error. In this paper, we describe how to apply multigrid to nonlinear problems. Applying multigrid method directly to the nonlinear problems by employing the method so-called Full Approximation Scheme (FAS). Full approximation scheme suitable for nonlinear problems is the FAS [5],8] which treats directly the nonlinear equations on finer and coarser grids. In FAS, a nonlinear iteration, such as the nonlinear Gauss-Seidel method is applied to smooth the error. In FAS, the residual is passed from the fine grids to the coarser grids. Vectors from fine grids are transferred to coarser grids with Restriction operator (
Using this matrix author’s [24]] used the restriction operator and prolongation operators respectively given in Section 4.2.
Using this matrix we defined the new restriction operator and new prolongation operators respectively given in Section 4.3.
Consider the Nonlinear Volterra integral equation of the second kind,
where
It has
Now, we are deliberating about the Full-Approximation Scheme (FAS) of solutions given by Briggs et. al [5]] is as follows the procedure,
Reduce the matrices in the finer level to coarsest level using Restriction operator, i.e.,
and then construct the matrices back to finer level from the coarsest level using Prolongation operator, i.e.,
Similarly,
and
Solving (6) with initial guess 0, we get
Similarly,
and
Solving (7) with initial guess 0, we get
Similarly,
and
Solving (8), we get
and so on we have,
This is the required solution of the given integral equation.
In this paper, we applied WFAS for the numerical solution of nonlinear Volterra integral equations. The same procedure is applied as explained in the FAS in the above Section 4.1. But replacing
and
In this paper, we introduced a modified wavelet full-approximation scheme (MWFAS) for the numerical solution of nonlinear Volterra integral equations. The same procedure is applied as explained in the FAS in Section 4.1, using
and
In this section, we implemented FAS, WFAS and MWFAS for the numerical solution of nonlinear Volterra integral and integro-differential equations and subsequently presented the efficiency of the MWFAS in the form of tables and figures, here error analysis is considered as,
where
Let us consider the nonlinear Volterra integral equation [28]],
which has the exact solution
Solving (10) through the iterative method, we get the approximate solution
From (10), we find the residual as
we get, =
From (11),
Similarly,
and
Solving (13) with initial guess 0, we get
From (12),
Similarly,
and
Solving (15) with initial guess 0, we get
From (14),
Similarly,
and
Solving (17), we get
From
and lastly we have,
we get
From (18), correct the solution with error
Lastly, we get
Numerical results of test problem 5.1, for Exact FAS WFAS MWFAS 0 0 0 0 0 0.1428 0.1224 0.1224 0.1224 0.1224 0.2857 0.2040 0.2043 0.2043 0.2043 0.4285 0.2448 0.2457 0.2457 0.2457 0.5714 0.2448 0.2463 0.2463 0.2463 0.7142 0.2040 0.2055 0.2055 0.2055 0.8571 0.1224 0.1231 0.1231 0.1231 1 0 0.0001 0.0001 0.0001
Maximum error and CPU time (in seconds) of the methods of test problem 5.1.Methods Setup time Running time Total time 16 FAS 7:52 0.0054 0.0321 0.0375 32 FAS 3:66 0.0099 0.1459 0.1558 64 FAS 1:80 0.1261 0.1581 1.2842 128 FAS 8:99 0.1307 0.2185 0.3492
Next, consider [29]]
which has the exact solution
Numerical results of test problem 5.2, for Exact FAS WFAS MWFAS 0 0 0 0 0 0.0666 0.0666 0.0667 0.0667 0.0667 0.1333 0.1329 0.1332 0.1332 0.1332 0.2000 0.1986 0.1994 0.1994 0.1994 0.2666 0.2635 0.2648 0.2648 0.2648 0.3333 0.3271 0.3291 0.3291 0.3291 0.4000 0.3894 0.3922 0.3922 0.3922 0.4666 0.4499 0.4537 0.4537 0.4537 0.5333 0.5084 0.5133 0.5130 0.5130 0.6000 0.5646 0.5708 0.5703 0.5703 0.6666 0.6183 0.6259 0.6253 0.6253 0.7333 0.6693 0.6784 0.6776 0.6776 0.8000 0.7173 0.7280 0.7270 0.7270 0.8666 0.7621 0.7746 0.7734 0.7734 0.9333 0.8036 0.8174 0.8164 0.8164 1 0.8414 0.8449 0.8441 0.8441
Maximum error and CPU time (in seconds) of the methods of test problem 5.2.Methods Setup time Running time Total time 16 FAS 1:38 0.0624 0.0636 0.1260 32 FAS 7:08 0.0099 0.1457 0.1556 64 FAS 3:56 0.2463 1.1761 1.4225 128 FAS 1:79 0.1819 10.2591 10.4410
Next, consider the Nonlinear Volterra-Hammerstein integral equations [30]],
which has the exact solution
Numerical results of test problem 5.3, for Exact FAS WFAS MWFAS 0 1 1.0357 1.0357 1.0357 0.0666 0.9977 0.9972 0.9972 0.9972 0.1333 0.9911 0.9901 0.9901 0.9901 0.2000 0.9800 0.9786 0.9786 0.9786 0.2666 0.9646 0.9628 0.9628 0.9628 0.3333 0.9449 0.9427 0.9427 0.9427 0.4000 0.9210 0.9185 0.9185 0.9185 0.4666 0.8930 0.8903 0.8903 0.8903 0.5333 0.8611 0.8582 0.8582 0.8582 0.6000 0.8253 0.8223 0.8223 0.8223 0.6666 0.7858 0.7829 0.7829 0.7829 0.7333 0.7429 0.7400 0.7400 0.7400 0.8000 0.6967 0.6939 0.6939 0.6939 0.8666 0.6473 0.6448 0.6448 0.6448 0.9333 0.5951 0.5929 0.5929 0.5929 1 0.5403 0.5383 0.5383 0.5383
Maximum error and CPU time (in seconds) of the methods of test problem 5.3.Methods Setup time Running time Total time 16 FAS 3:57 0.0054 0.0121 0.0175 32 FAS 1:66 0.0099 0.0172 0.0271 64 FAS 8:06 0.0281 0.0370 0.0651 128 FAS 3:96 0.1009 0.1168 0.2177
Next, consider [31]]
which has the exact solution
Numerical results of test problem 5.4, for Exact FAS WFAS MWFAS 0 0 0 0 0 0.0666 -0.0622 -0.0622 -0.0622 -0.0622 0.1333 -0.1155 -0.1155 -0.1155 -0.1155 0.2000 -0.1600 -0.1600 -0.1600 -0.1600 0.2666 -0.1955 -0.1957 -0.1957 -0.1957 0.3333 -0.2222 -0.2224 -0.2224 -0.2224 0.4000 -0.2400 -0.2403 -0.2403 -0.2403 0.4666 -0.2488 -0.2493 -0.2493 -0.2493 0.5333 -0.2488 -0.2494 -0.2494 -0.2494 0.6000 -0.2400 -0.2405 -0.2405 -0.2405 0.6666 -0.2222 -0.2227 -0.2227 -0.2227 0.7333 -0.1955 -0.1959 -0.1959 -0.1959 0.8000 -0.1600 -0.1602 -0.1602 -0.1602 0.8666 -0.1155 -0.1156 -0.1156 -0.1156 0.9333 -0.0622 -0.0622 -0.0622 -0.0622 1 0 0 0 0
Maximum error and CPU time (in seconds) of the methods of test problem 5.4.Methods Setup time Running time Total time 16 FAS 5:72 0.0054 0.0404 0.0458 32 FAS 2:81 0.0105 0.2084 0.2189 64 FAS 1:38 0.0306 0.0484 0.0790 128 FAS 6:87 0.1114 0.1810 0.2924
Lastly, consider the Nonlinear Volterra integro-differential equation [32]],
which has the exact solution
We convert the Volterra integro-differential equation to equivalent Volterra integral equation by using the well-known formula, which converts multiple integrals into a single integral. i.e.,
Integrating (22) on both sides from 0 to
where
Numerical results of test problem 5.5, for Exact FAS WFAS MWFAS 0 0 0 0 0 0.0322 -0.0322 -0.0322 -0.0322 -0.0322 0.0645 -0.0645 -0.0645 -0.0645 -0.0645 0.0967 -0.0967 -0.0967 -0.0967 -0.0967 0.1290 -0.1290 -0.1290 -0.1290 -0.1290 0.1612 -0.1612 -0.1612 -0.1612 -0.1612 0.1935 -0.1934 -0.1934 -0.1934 -0.1934 0.2258 -0.2255 -0.2255 -0.2255 -0.2255 0.2580 -0.2577 -0.2577 -0.2577 -0.2577 0.2903 -0.2897 -0.2897 -0.2897 -0.2897 0.3225 -0.3216 -0.3216 -0.3216 -0.3216 0.3548 -0.3535 -0.3535 -0.3535 -0.3535 0.3870 -0.3852 -0.3852 -0.3852 -0.3852 0.4193 -0.4167 -0.4168 -0.4168 -0.4168 0.4516 -0.4481 -0.4481 -0.4481 -0.4481 0.4838 -0.4793 -0.4793 -0.4793 -0.4793 0.5161 -0.5102 -0.5102 -0.5102 -0.5102 0.5483 -0.5409 -0.5409 -0.5409 -0.5409 0.5806 -0.5712 -0.5712 -0.5712 -0.5712 0.6129 -0.6012 -0.6013 -0.6013 -0.6013 0.6451 -0.6309 -0.6309 -0.6309 -0.6309 0.6774 -0.6601 -0.6601 -0.6601 -0.6601 0.7096 -0.6888 -0.6889 -0.6889 -0.6889 0.7419 -0.7171 -0.7172 -0.7172 -0.7172 0.7741 -0.7449 -0.7449 -0.7449 -0.7449 0.8064 -0.7720 -0.7721 -0.7721 -0.7721 0.8387 -0.7986 -0.7986 -0.7986 -0.7986 0.8709 -0.8244 -0.8245 -0.8245 -0.8245 0.9032 -0.8496 -0.8497 -0.8497 -0.8497 0.9354 -0.8740 -0.8741 -0.8741 -0.8741 0.9677 -0.8976 -0.8977 -0.8977 -0.8977 1 -0.9204 -0.9205 -0.9205 -0.9205
Maximum error and CPU time (in seconds) of the methods of test problem 5.5.Methods Setup time Running time Total time 16 FAS 2:81 0.0054 0.0125 0.0180 32 FAS 6:56 0.0100 0.0169 0.0268 64 FAS 1:57 0.0286 0.0368 0.0654 128 FAS 3:69 0.1042 0.1240 0.2281
In this paper, we proposed a modified wavelet full-approximation scheme for the numerical solution of nonlinear Volterra integral and integro-differential equations. The modified wavelet intergrid operators of prolongation and restrictions are used in this paper, a modified wavelet based FAS, has been shown to be effective and versatile. The numerical solutions obtained agree well with the exact ones. Convergence is also observed in the numerical solutions when the calculation is refined by increasing the number