Modified Wavelet Full-Approximation Scheme for the Numerical Solution of Nonlinear Volterra integral and integro-differential Equations

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 ϕ(t) is given by,

ϕ(t)=2∑k=0L−1hkϕ(2t−k)$$\begin{array}{} \displaystyle \phi (t) = \sqrt 2 \mathop \Sigma \limits_{k = 0}^{L - 1} {h_k}\phi (2t - k) \end{array}$$(1)

where ϕ(t) is normalized: ∫−∞∞ϕ(t)dt=1$\begin{array}{} \displaystyle \int_{-\infty}^{\infty} \phi(t) dt =1 \end{array}$.

Based on the scaling function ϕ(t) , the wavelet function can be written as, The refinement relation of wavelet function ψ(t) is given by,

ψ(t)=2∑k=0L−1gkϕ(2t−k)$$\begin{array}{} \displaystyle \psi(t)=\sqrt{2}\sum\limits_{k=0}^{L-1} g_{k} \phi(2t-k) \end{array}$$(2)

where {h_k}_kεZ and {g_k}_kεZ are known respectively as the low and high pass filter coefficients, and they are related by g_k = (−1)^kh_L−k for k = 0, 1,...,L − 1. where, L is an even integer, is the order of the wavelet. The compactly supported wavelet bases were introduced by Daubechies [22]]. They are an orthonormal bases for functions in L²(R). A family of orthogonal Daubechies wavelets with compactly supported property has been constructed by Daubechies in [23]]. Due to excellent properties of orthogonality and minimum compact support, Daubechies wavelets can be useful and convenient in a wide variety of situations.

In this paper, we use Daubechies filter coefficients for L = 4 which are, h₀ = 0.4830, h(1) = 0.8365, h(2) = 0.2241, h(3) = −0.1294 are low pass filter coefficients and g₀ = 0.1294, g(1) = −0.2241, g(2) = 0.8365, g(3) = −0.4830 are the high pass filter coefficients.

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,

Emax=max|ue−ua|,$$\begin{array}{} \displaystyle E_{max}=max \vert u_{e} - u_{a} \vert, \end{array}$$

where u_e and u_a are the exact and approximate solution respectively.

Solving (10) through the iterative method, we get the approximate solution ν of u. i.e., u = e + ν ⇒ ν = u − e, where e is (8 × 1 matrix) error to be determined. The implementation of MWFAS is discussed in Section 4 is as follows,

From (10), we find the residual as

r8×1=[b]8×1−[A]8×8[ν]8×1$$\begin{array}{} \displaystyle r_{8 \times 1}=[b]_{8 \times 1}-[A]_{8 \times 8}[\nu]_{8 \times 1} \end{array}$$(11)

we get, = r_8×1[0] 0 2.30e-06 2.01e-05 6.32e-05 8.89e-05 3.98e-05 -2.03e-07] we reduce the matrices in the finer level to coarsest level using Restriction operator MW and then construct the matrices back to finer level from the coarsest level using Prolongation operator MW^T .

From (11),

r4×1=[MW]4×8[r]8×1$$\begin{array}{} \displaystyle r_{4 \times 1}=[MW]_{4 \times 8}[r]_{8 \times 1} \end{array}$$(12)

Similarly,

ν4×1=[MW]4×8[ν]8×1$$\begin{array}{} \displaystyle \nu_{4 \times 1}=[MW]_{4 \times 8}[\nu]_{8 \times 1} \end{array}$$

and

A(ν4×1+e4×1)−A(ν4×1)=r4×1.$$\begin{array}{} \displaystyle A(\nu_{4 \times 1}+ e_{4 \times 1})-A(\nu_{4 \times 1})=r_{4 \times 1}. \end{array}$$(13)

Solving (13) with initial guess 0, we get e_4×1.

From (12),

r2×1=[MW]2×4[r]4×1$$\begin{array}{} \displaystyle r_{2 \times 1}=[MW]_{2 \times 4}[r]_{4 \times 1} \end{array}$$(14)

Similarly,

ν2×1=[MW]2×4[ν]4×1$$\begin{array}{} \displaystyle \nu_{2 \times 1}=[MW]_{2 \times 4}[\nu]_{4 \times 1} \end{array}$$

and

A(ν2×1+e2×1)−A(ν2×1)=r2×1.$$\begin{array}{} \displaystyle A(\nu_{2 \times 1}+ e_{2 \times 1})-A(\nu_{2 \times 1})=r_{2 \times 1}. \end{array}$$(15)

Solving (15) with initial guess 0, we get e_2×1.

From (14),

r1×1=[MW]1×2[r]2×1$$\begin{array}{} \displaystyle r_{1 \times 1}=[MW]_{1 \times 2}[r]_{2 \times 1} \end{array}$$(16)

Similarly,

ν1×1=[MW]1×2[ν]2×1$$\begin{array}{} \displaystyle \nu_{1 \times 1}=[MW]_{1 \times 2}[\nu]_{2 \times 1} \end{array}$$

and

A(ν1×1+e1×1)−A(ν1×1)=r1×1.$$\begin{array}{} \displaystyle A(\nu_{1 \times 1}+ e_{1 \times 1})-A(\nu_{1 \times 1})=r_{1 \times 1}. \end{array}$$(17)

Solving (17), we get e_1×1.

From e_1×1 , Interpolate error up to the finer level, i.e.,

e2×1=[MWT]2×1[e]1×1,e4×1=[MWT]4×2[e]2×1,$$\begin{array}{} \displaystyle \begin{array}{*{20}{l}} {{e_{2 \times 1}}}& = &{{{[M{W^T}]}_{2 \times 1}}{{[e]}_{1 \times 1}},}\\ {{e_{4 \times 1}}}& = &{{{[M{W^T}]}_{4 \times 2}}{{[e]}_{2 \times 1}},} \end{array} \end{array}$$

and lastly we have,

e8×1=[MWT]8×4[e]4×1,$$\begin{array}{} \displaystyle e_{8 \times 1} = [MW^T]_{8 \times 4}[e]_{4 \times 1}, \end{array}$$(18)

we get e_8×1 = [1].31e − 06 0 2.28e − 06 2.06e − 05 3.04e − 05 0 − 1.75e − 05 0]

From (18), correct the solution with error u_8×1 = ν_8×1 + e_8×1.

Lastly, we get u_8×1 is the required solution of (9). The numerical solutions of the given equation is obtained through the method as explained in Section 4 and are presented in comparison with the exact solution are shown in Table 1 and in Figure 1 for N = 64. Maximum error analysis and CPU time is shown in Table 2.

Table 1

Numerical results of test problem 5.1, for N = 8.

Fig. 1

Table 2

Maximum error and CPU time (in seconds) of the methods of test problem 5.1.

Test problem 5.2

Next, consider [29]]

u(t)=sin(t)+18sin(2t)−14t+12∫0tu2(s)ds,0≤t≤1$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {u(t) = sin(t) + \frac{1}{8}sin(2t) - \frac{1}{4}t + \frac{1}{2}\int_0^t {{u^2}} (s)ds,}&{0 \le t \le 1} \end{array} \end{array}$$(19)

which has the exact solution u(t) = sin(t). The numerical solutions of (19) are presented in Table 3 for N = 16 and in Figure 2 for N = 64. Maximum error analysis and CPU time is shown in Table 4.

Table 3

Numerical results of test problem 5.2, for N = 16.

t	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

Fig. 2

Table 4

Maximum error and CPU time (in seconds) of the methods of test problem 5.2.

N	Methods	Emax	Setup time	Running time	Total time
16	FASWFASMWFAS	1:38e − 021:28e − 021:28e − 02	0.06240.00480.0019	0.06360.04430.0256	0.12600.04910.0275
32	FASWFASMWFAS	7:08e − 036:80e − 036:80e − 03	0.00990.00310.0019	0.14570.14690.1475	0.15560.14990.1494
64	FASWFASMWFAS	3:56e − 033:49e − 033:49e − 03	0.24630.00310.0019	1.17611.13851.1456	1.42251.14161.1475
128	FASWFASMWFAS	1:79e − 031:77e − 031:77e − 03	0.18190.01250.0019	10.25919.53019.3220	10.44109.54269.3239

Test problem 5.3

Next, consider the Nonlinear Volterra-Hammerstein integral equations [30]],

u(t)=1+sin2(t)−∫0t3sin(t−s)u2(s)ds,0≤t≤1$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {u(t) = 1 + si{n^2}(t) - \int_0^t 3 sin(t - s){u^2}(s)ds,}&{0 \le t \le 1} \end{array} \end{array}$$(20)

which has the exact solution u(t) = cos(t). The numerical solutions of (20) are presented in Table 5 for N = 16 and in Figure 3 for N = 64. Maximum error analysis and CPU time is shown in Table 6.

Table 5

Numerical results of test problem 5.3, for N = 16.

t	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

Fig. 3

Table 6

Maximum error and CPU time (in seconds) of the methods of test problem 5.3.

N	Methods	Emax	Setup time	Running time	Total time
16	FASWFASMWFAS	3:57e − 023:57e − 023:57e − 02	0.00540.00310.0019	0.01210.01210.0108	0.01750.01520.0127
32	FASWFASMWFAS	1:66e − 021:66e − 021:66e − 02	0.00990.00310.0019	0.01720.01640.0156	0.02710.01950.0175
64	FASWFASMWFAS	8:06e − 038:06e − 038:06e − 03	0.02810.00310.0019	0.03700.03600.0357	0.06510.03900.0375
128	FASWFASMWFAS	3:96e − 033:96e − 033:96e − 03	0.10090.00440.0020	0.11680.13290.1300	0.21770.13730.1320

Test problem 5.4

Next, consider [31]]

u(t)=−1556t8+1314t7−1110t6+920t5+t2−t+∫0t(t+s)u3(s)ds,0≤t≤1$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {u(t) = - \frac{{15}}{{56}}{t^8} + \frac{{13}}{{14}}{t^7} - \frac{{11}}{{10}}{t^6} + \frac{9}{{20}}{t^5} + {t^2} - t + \int_0^t {(t + s)} {u^3}(s)ds,}&{0 \le t \le 1} \end{array} \end{array}$$(21)

which has the exact solution u(t) = t² − t. The numerical solutions of (21) are presented in Table 7 for N = 16 and in Figure 4 for N = 64. Maximum error analysis and CPU time is shown in Table 8.

Table 7

Numerical results of test problem 5.4, for N = 16.

t	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

Fig. 4

Table 8

Maximum error and CPU time (in seconds) of the methods of test problem 5.4.

N	Methods	Emax	Setup time	Running time	Total time
16	FASWFASMWFAS	5:72e − 045:60e − 045:60e − 04	0.00540.00300.0018	0.04040.03340.0315	0.04580.03650.0333
32	FASWFASMWFAS	2:81e − 042:77e − 042:76e − 04	0.01050.00300.0018	0.20840.20560.2057	0.21890.20860.2075
64	FASWFASMWFAS	1:38e − 041:37e − 041:37e − 04	0.03060.00300.0018	0.04840.04710.0456	0.07900.05020.0474
128	FASWFASMWFAS	6:87e − 056:85e − 056:85e − 05	0.11140.00440.0018	0.18100.17800.1804	0.29240.18230.1822

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.,

∫0t∫0t...∫0tu(t)dtn︸n−times=1(n−1)!∫0t(t−s)n−1u(s)ds$$\begin{array}{} \displaystyle \underbrace{\int_{0}^{t} \int_{0}^{t} . . . \int_{0}^{t} u(t) dt^n}_{n-times} = \dfrac{1}{(n-1)!} \int_{0}^{t} (t-s)^{n-1} u(s) ds \end{array}$$(23)

Integrating (22) on both sides from 0 to t and using the initial condition and also converting the double integral to the single integral, we obtain

u(t)=f(t)+∫0tk(t,s)u2(s)ds,$$\begin{array}{} \displaystyle u(t)=f(t)+\int_{0}^{t} k(t,s) u^2(s) ds, \end{array}$$(24)

where k(t, s) = (t − s) and f(t) = −t. The numerical solutions of (22) are presented in Table 9 for N = 32 and in Figure 5 for N = 64. Maximum error analysis and CPU time is shown in Table 10.

Table 9

Numerical results of test problem 5.5, for N = 32.

Fig. 5

Table 10

Maximum error and CPU time (in seconds) of the methods of test problem 5.5.

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 N used. The standard FAS and WFAS converge slowly with larger computational cost, whereas MWFAS does ensure such slower convergence with lesser computational cost. Test problems are justified through the error analysis, as the level of resolution N increases, larger the accuracy increases. Hence, the new scheme is very convenient and efficient than the existing standard methods.