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Abstract

In this article, we present an efficient method for solving nonlinear Fredholm integral equations of the second kind. The proposed method is based on the Galerkin method and transformations of shifted Chebyshev polynomials. This method is simple and computationally very attractive. Finally, illustrative examples and also the application of the proposed method to solve a functional differential equation are presented to show the validity and applicability of the technique.

Keywords

Hammerstein equations nonlinear Fredholm integral equation shifted Chebyshev polynomials Taylor expansion Galerkin method

Introduction

Integral equations play a very vital role in science, such as numerous problems in mathematics and engineering (see, for example,^1–5 and the references therein). For instance, one of the most important domains of applications of the ideas and methods of nonlinear functional analysis and also the theory of nonlinear operators of monotone type is integral equations of the Fredholm–Hammerstein.⁶ Furthermore, this kind of integral equations appears in nonlinear physical phenomena such as electro-magnetic fluid dynamics, reformulation of boundary value problems (BVPs) with a nonlinear boundary condition.⁷ This equation is as follows

u (t) - \int_{a}^{b} K (t, y) {[u (y)]}^{m} dy = f (t)

(1)

where $f (t) \in L^{2} [a, b] and K (t, y) \in L^{2} [a, b] \times [a, b]$ are known functions, $u (t)$ is the unknown function to be determined, $t \in I = [a, b]$ , and $m > 1$ is a positive integer.⁸ Since we use the shifted Chebyshev polynomials and these polynomials are an orthogonal set on the interval $[0, 1]$ , throughout this article, without loss of generality, we assume that $a = 0 and b = 1$ .

Numerous numerical methods have been proposed for approximating the solution of above Fredholm–Hammerstein integral equations. For example, Tricomi⁹ (section 4.6) introduced the classical method of successive approximations. Kumar and Sloan¹⁰ have considered the solution of the Fredholm–Hammerstein integral equations and presented a collocation-type method to solve it. Brunner¹¹ used this method for the numerical solution of nonlinear Volterra integral and integro-differential equations. Guoqiang¹² obtained the asymptotic error expansion of this method and showed that the Richardson’s extrapolation can be performed on the solution and this will greatly increase the accuracy of numerical solution. Elnagar and Kazemi¹³ investigated the Chebyshev spectral method to an equivalent equation of nonlinear Volterra–Hammerstein integral equations and discussed some convergence results. Hernandez et al.¹⁴ applied a one-parametric family of secant-type iterations for equation (1) and established a semilocal convergence result for these iterations by means of a technique based on a new system of recurrence relations. Kaneko et al.¹⁵ developed the Petrov–Galerkin method and the iterated Petrov–Galerkin method for equation (1) and established a framework for fast algorithms to obtaining approximate solutions based on Alpert’s Wavelets. Contea and Prete¹⁶ proposed discrete collocation methods for Volterra integral equations of Hammerstein type, where the Laplace transform of the kernel rather than the convolution kernel itself is known a priori. For details of the application of the spectral methods for solving some differential equations,^17–21 In recent years, numerous numerical methods have been proposed to find Hammerstein integral equations. These methods include Nyström type methods,^22,23 projection methods,^24,25 methods of special functions,^25–29 wavelets,^30–41 homotopy techniques, the Adomian decomposition method,^42,43 Toeplitz matrix method,⁴⁴ polynomial interpolation procedures,^45–47 and multigrid methods.⁴⁸ The main problem for solution of equation (1) is $[u (y)]^{m}$ . Some methods used quadrature formula methods and spline approximations (e.g. see the studies by Maleknejad and Lotfi⁴⁵). Some methods considered $[u (y)]^{m}$ as an independent variable (e.g. see equation (7) in Babolian and Shahsavaran³⁹). Some methods applied equivalent equation as follows

{[u (y)]}^{m} = ψ (y, u (y)) = ψ (y, \sum_{i = 1}^{n} c_{i} \int_{a}^{b} K (t, y) b_{i} (y) dy + f (y))

These require a huge number of arithmetic operators, high computational costs, and a large storage capacity.^13,15,34

Some methods also used operational vector. For example, in Mahmoudi,³³ equation (1) is investigated using Legendre wavelets basis. The method proposed there only works under the condition that the operational vector $Y^{*}$ is a vector function of Y; moreover, this function must be explicitly known (see equation (4.2) in Mahmoudi³³). Generally, it is difficult to meet this requirement in practice.

The existing results presented above for solving the Hammerstein integral equations motivate the study of this type of functional integral equations. Therefore, the main contribution of this article is to correct these models, with regard to the computational costs. In this article, we present an efficient algorithmic method which is new and different from all the existing methods. Our method is based on Galerkin methods and transformations of orthogonal polynomials. This method is very simple to apply and offers several advantages in reducing computational costs.

This article is organized as follows. In sections “Properties of shifted Chebyshev polynomials” and “Galerkin method,” we, respectively, give an overview of shifted Chebyshev polynomials and Galerkin method with their relevant properties needed hereafter. In section “Main results,” the way of constructing the proposed method for solving Hammerstein integral equations is described. Numerical experiments and also the application of the proposed method to solve a BVP are presented in section “Illustrative examples.” Finally, conclusions are given in section “Conclusion.”

Solving the Hammerstein integral equations

Properties of shifted Chebyshev polynomials

Chebyshev polynomials are important in approximation theory and numerical analysis and in some quadrature rules based on these polynomials such as Gauss–Chebyshev rule that appears in the theory of numerical integration.^49–51

Consider the well-known shifted Chebyshev polynomials of order $n, T_{n} (x),$ which is derived from the following recursive formula (for $x \in [0, 1]$ )

\begin{matrix} T_{0} (x) = 1 \\ T_{1} (x) = 2 x - 1 \\ T_{n} (x) = 2 (2 x - 1) T_{n - 1} - T_{n - 2} \end{matrix}

(2)

Moreover, these polynomials are orthogonal with respect to the weight function $ω (x) = \frac{1}{\sqrt{x (1 - x)}}$ on the interval [0, 1] and satisfy

\int_{0}^{1} \frac{T_{i} T_{j}}{\sqrt{x (1 - x)}} = ϒ_{i} δ_{ij}

where $δ_{ij}$ is the Kronecker delta and

ϒ_{i} = {\begin{matrix} π / 2, i \neq 0 \\ π, i = 0 \end{matrix}

Furthermore, for $x^{n}$ in terms of shifted Chebyshev polynomials, we have the following formula

x^{n} = 2^{1 - 2 n} {\frac{1}{2} (\begin{matrix} 2 n \\ n \end{matrix}) T_{0} (x) + \sum_{k = 0}^{n - 1} (\begin{matrix} 2 n \\ k \end{matrix}) T_{n - k} (x)}, n \geq 1

(3)

The first few powers in terms of shifted Chebyshev polynomials are (see the work by Mason and Handscomb (section 2.3)⁵⁰ and Synder⁵¹)

\begin{matrix} 1 = T_{0} (x) \\ x = \frac{1}{2} [T_{0} (x) + T_{1} (x)] \\ x^{2} = \frac{1}{8} [3 T_{0} (x) + 4 T_{1} (x) + T_{2} (x)] \\ x^{3} = \frac{1}{32} [10 T_{0} (x) + 15 T_{1} (x) + 6 T_{2} (x) + T_{3} (x)] \\ x^{4} = \frac{1}{128} [35 T_{0} (x) + 56 T_{1} (x) + 28 T_{2} (x) + 8 T_{3} (x) + T_{4} (x)] \\ ⋮ \end{matrix}

For some recent application of these polynomials, see Doha et al.⁵² and Bhrawy and Alofi.⁵³

Galerkin method

The basic idea of the weighted residual method is to assume that the unknown function $u (t)$ in equation (1) can be approximated by the sum of $N + 1$ trail functions $b_{i} (t)$ , that is

u (t) \approx u_{app} = \sum_{i = 0}^{N} c_{i} b_{i} (t)

(4)

And by assuming that the function $u_{app}$ is a linear combination of $b_{i} (t)$ , its expansion coefficients $c_{i}$ are to be determined uniquely.

Substituting the approximate solution given by equation (4) into equation (1), the result is the residual function defined by

E (t, c_{i}) = L [u_{app}] - f (t)

(5)

where

L [u_{app}] = u_{app} - \int_{a}^{b} K (t, y) {[u_{app}]}^{m} dy

(6)

Since the residual function is identically equal to zero for the exact solution, the challenge is to choose the coefficients $c_{i}$ so that the residual function is minimized. In integral form, this can be achieved with the following condition

\int_{a}^{b} ω_{i} (t) E (t, c_{i}) dt = 0

(7)

where $ω_{i} (t)$ is the weight function. The Galerkin approach makes the residual $E (t, c_{i})$ orthogonal to $N + 1$ , given the independent function on the domain $[a, b]$ . In this approach, the weighting function is chosen to be identical to the trail functions.

Main results

Consider the approximate solution given by equation (4) which is $b_{i} (t) = t^{i}$ , that is

u_{app} = \sum_{i = 0}^{N} c_{i} t^{i}

(8)

Then, by equation (1), we have

ψ (y, u (y)) = [u_{app}]^{m} = {[\sum_{i = 0}^{N} c_{i} y^{i}]}^{m}

(9)

Now, by Taylor expansion of $ψ (t, y)$ about zero, we can write

\begin{matrix} ψ (y, u (y)) = ψ (y = 0, u (y)) + (\frac{\partial}{\partial y} ψ (y = 0, u (y))) y + (\frac{1}{2!} \frac{\partial^{2}}{\partial y^{2}} ψ (y = 0, u (y))) y^{2} \\ \begin{matrix} + \dots + (\frac{1}{N!} \frac{\partial^{N}}{\partial y^{N}} ψ (y = 0, u (y))) y^{N} + \dots + (\frac{1}{(D)!} \frac{\partial^{D}}{\partial y^{D}} ψ (y = 0, u (y))) y^{D} \end{matrix} \end{matrix}

(10)

where $D = m \cdot N$ .

Then, we have

\begin{matrix} ψ (y, u (y)) = {({[\sum_{i = 0}^{N} c_{i} y^{i}]}^{m})}_{y = 0} + {(\frac{\partial}{\partial y} {[\sum_{i = 0}^{N} c_{i} y^{i}]}^{m})}_{y = 0} y + {(\frac{1}{2!} \frac{\partial^{2}}{\partial y^{2}} {[\sum_{i = 0}^{N} c_{i} y^{i}]}^{m})}_{y = 0} y^{2} \\ \begin{matrix} + \dots + {(\frac{1}{N!} \frac{\partial^{N}}{\partial y^{N}} {[\sum_{i = 0}^{N} c_{i} y^{i}]}^{m})}_{y = 0} y^{N} + \dots + {(\frac{1}{D!} \frac{\partial^{D}}{\partial y^{D}} {[\sum_{i = 0}^{N} c_{i} y^{i}]}^{m})}_{y = 0} y^{D} \end{matrix} \end{matrix}

(11)

Moreover, since

{(\frac{\partial^{s}}{\partial y^{s}} {[\sum_{i = s + 1}^{N} c_{i} y^{i}]}^{m})}_{y = 0} = 0

(12)

We obtain the following expansion for the above relation

\begin{matrix} ψ (y, u (y)) = ([c_{0}]^{m}) + {(\frac{\partial}{\partial y} {[c_{0} + c_{1} y]}^{m})}_{y = 0} y + {(\frac{1}{2!} \frac{\partial^{2}}{\partial y^{2}} {[\sum_{i = 0}^{2} c_{i} y^{i}]}^{m})}_{y = 0} y^{2} + {(\frac{1}{3!} \frac{\partial^{3}}{\partial y^{3}} {[\sum_{i = 0}^{3} c_{i} y^{i}]}^{m})}_{y = 0} y^{3} \\ + \dots + {(\frac{1}{N!} \frac{\partial^{N}}{\partial y^{N}} {[\sum_{i = 0}^{N} c_{i} y^{i}]}^{m})}_{y = 0} y^{N} + {(\frac{1}{(N + 1)!} \frac{\partial^{N + 1}}{\partial y^{N + 1}} {[\sum_{i = 0}^{N} c_{i} y^{i}]}^{m})}_{y = 0} y^{N + 1} + \dots + {(\frac{1}{D!} \frac{\partial^{D}}{\partial y^{D}} {[\sum_{i = 0}^{N} c_{i} y^{i}]}^{m})}_{y = 0} y^{D} \end{matrix}

Therefore, we can write

[u_{app}]^{m} = \sum_{i = 0}^{D} P_{i} y^{i}

(13)

where

P_{s} = {\begin{matrix} {(\frac{1}{s!} \frac{\partial^{s}}{\partial y^{s}} {[\sum_{i = 0}^{s} c_{i} y^{i}]}^{m})}_{y = 0}, & for s = 0, 1, \dots, N \\ {(\frac{1}{s!} \frac{\partial^{s}}{\partial y^{s}} {[\sum_{i = 0}^{N} c_{i} y^{i}]}^{m})}_{y = 0}, & for s = N + 1, \dots, D \end{matrix}

(14)

Now, by transformations of orthogonal polynomials based on formula (3), we will obtain an efficient method to solve equation (1). This method is as follows.

Algorithm 1

Step 1. Choose $u_{app} = \sum_{i = 0}^{N} c_{i} t^{i}$ and $[u_{app}]^{m} = \sum_{i = 0}^{D} P_{i} y^{i}$ .

Step 2. Use formula (3) and set $u_{app} = \sum_{i = 0}^{N} α_{i} T_{i} (t)$ and $[u_{app}]^{m} = \sum_{i = 0}^{D} β_{i} T_{i} (y)$

Step 3. Apply Galerkin method and solve the following nonlinear equations:

ϒ_{j} α_{j} - \sum_{i = 0}^{D} β_{i} K_{i, j} = f_{j}, \begin{matrix} j = 0, \dots, N, \begin{matrix} i = 0, \dots, D \end{matrix} \end{matrix}

(15)

where,

f_{j} = \int_{0}^{1} ω (t) f (t) P_{j} (t) dt

(16)

K_{i, j} = \int_{0}^{1} \int_{0}^{1} ω (t) K (t, y) P_{i} (y) P_{j} (t) dydt

(17)

Illustrative examples

In this section, we give some numerical experiments to illustrate the results obtained in previous sections.

Example 1

Consider the following nonlinear Fredholm–Hammerstein integral equation

u (t) = - \frac{1}{4} (t - 4 e^{t} + t e^{2}) + \int_{0}^{1} ty {[u (y)]}^{2} dy

with exact solution $u (t) = e^{t}$ .

To solve the above problem using our method, we do the following steps:

Let us consider $N = 2$ . Then, we construct trail space as follows

u_{app} = \sum_{i = 0}^{2} c_{i} t^{i} = c_{0} + c_{1} x + c_{2} x^{2}

and using formula (3), we have

u_{app} = \sum_{i = 0}^{2} α_{i} T_{i} (t)

where

α_{0} = c_{0} + \frac{1}{2} c_{1} + \frac{3}{8} c_{2}, α_{1} = \frac{1}{2} c_{1} + \frac{1}{2} c_{2}, α_{2} = \frac{1}{8} c_{2}

Similarly,

[u_{app}]^{2} = \sum_{i = 0}^{4} P_{i} y^{i}

where

P_{0} = c_{0}^{2}, P_{1} = 2 c_{0} c_{1}, P_{2} = c_{1}^{2} + 2 c_{0} c_{2}, P_{3} = 2 c_{1} c_{2}, P_{4} = c_{1}^{2}

and using formula (3), we have

[u_{app}]^{2} = \sum_{i = 0}^{4} β_{i} T_{i} (y)

where

\begin{matrix} β_{0} = P_{0} + \frac{1}{2} P_{1} + \frac{3}{8} P_{2} + \frac{10}{32} P_{3} + \frac{35}{128} P_{4} \\ β_{1} = \frac{1}{2} P_{1} + \frac{1}{2} P_{2} + \frac{15}{32} P_{3} + \frac{56}{128} P_{4} \\ β_{2} = \frac{1}{8} P_{2} + \frac{6}{32} P_{3} + \frac{28}{128} P_{4} \\ β_{3} = \frac{1}{32} P_{3} + \frac{8}{128} P_{4} \\ β_{4} = \frac{1}{128} P_{4} \end{matrix}

So, we have

\sum_{i = 0}^{2} α_{i} T_{i} (t) - \int_{0}^{1} ty [\sum_{i = 0}^{4} β_{i} T_{i} (y)] dy = - \frac{1}{4} (t - 4 e^{t} + t e^{2})

Now, we multiply both sides of the above relation with ${ω (t) T_{j} (t)}_{j = 0}^{2}$ ; from the orthogonality of shifted Chebyshev polynomials, we obtain the following nonlinear equations

\begin{array}{l} [\begin{matrix} π α_{0} \\ \frac{π}{2} α_{1} \\ \frac{π}{2} α_{2} \end{matrix}] - [\begin{matrix} π / 4 & π / 12 & - π / 12 & - π / 20 & - π / 60 \\ π / 8 & π / 24 & - π / 24 & - π / 40 & - π / 120 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}] \\ [\begin{matrix} β_{0} \\ β_{1} \\ β_{2} \\ β_{3} \\ β_{4} \end{matrix}] = [\begin{matrix} 2741 / 1238 \\ - 828 / 2659 \\ 98 / 593 \\ 93 / 6788 \\ 17 / 19915 \end{matrix}] \end{array}

Therefore, we get

c_{0} = \frac{983}{975}, c_{1} = \frac{533}{628}, c_{2} = \frac{2440}{2899}

Table 1 and Figure 1 show the absolute values of error for N = 2, 3, and 4.

Table 1.

Absolute errors for Example 1.

$t_{i}$	N = 2	N = 3	N = 4
0.0	8.204694227216480e−003	9.399459517538400e−004	5.264158677142916e−005
0.1	3.676928165183391e−003	2.104507185856086e−004	2.103582406087767e−005
0.2	9.786081585507978e−003	3.966020978720675e−004	8.800910186401723e−006
0.3	1.128605335619359e−002	1.269715662108872e−004	1.416823584632354e−005
0.4	9.462474795363329e−003	2.123214919784733e−004	1.710423227496705e−005
0.5	5.736188247173812e−003	3.703687183107540e−004	7.833777755372751e−007
0.6	1.677467349557205e−003	2.456930481757880e−004	1.669461628006630e−005
0.7	9.782671394225062e−004	9.803546290454435e−005	1.562245218034164e−005
0.8	3.130786673284369e−004	4.146309650350233e−004	7.946079886966828e−006
0.9	5.792680466306877e−003	2.561964602918465e−004	2.300074817274123e−005
1.0	1.968158325660552e−002	1.048090343907404e−003	5.763926725288471e−005

Figure 1.

Comparison plot of exact and approximation solution of Example 1, for N = 2, 3, and 4.

Example 2

As the second example, consider the following integral equation

u (t) = \sin (\frac{π}{2} t) - \frac{11 t}{6} + \int_{0}^{1} 4 π ty \sin (π y) {[u (y)]}^{4} dy

with exact solution $u (t) = \sin (\frac{π}{2} t)$ . Table 2 and Figure 2 show the numerical results for Example 2.

Table 2.

Absolute errors for Example 2.

$t_{i}$	N = 2	N = 3	N = 4
0.0	2.432494696309375e−002	2.258436808356286e−003	2.330110502653885e−004
0.1	5.775167127636260e−004	4.625305463742124e−004	9.265274986605676e−005
0.2	1.033037376100277e−002	9.358829988277417e−004	2.997961756262546e−005
0.3	1.215581272280830e−002	2.707275950365462e−004	6.516615217438959e−005
0.4	8.468529014208581e−003	6.111880420142102e−004	6.812408155287830e−005
0.5	2.562993482382980e−003	1.063374283716279e−003	2.461061340164328e−006
0.6	2.622701825846230e−003	7.957203018177372e−004	6.576006851799487e−005
0.7	4.579189241174175e−003	5.316035441582656e−005	5.110486925519187e−005
0.8	1.287614431945583e−003	8.541407980687810e−004	4.022250977953057e−005
0.9	8.730652702006791e−003	4.378699107373540e−004	8.527905068522745e−005
1.0	2.637760924900290e−002	2.941636437828521e−003	1.969480360283216e−004

Figure 2.

Comparison plot of exact and approximation solution of Example 2, for N = 2, 3, and 4.

Example 3 (application to the BVP functional differential equations)

One of the important problems in science and engineering is a BVP for functional differential equations which is investigated by numerous researchers.^54,55 This problem can be converted, using Green’s function technique, into a Hammerstein integral equation.^56–59 Here, we consider an example of this problem as an application of our methods.

Consider the two-point BVP

{\begin{matrix} x ″ (t) = [x (t)]^{3}, 0 \leq t \leq 1 \\ x (0) = 0, \begin{matrix} x (1) = 1 \end{matrix} \end{matrix}

This BVP is equivalent with the following Hammerstein integral equation

u (t) - \int_{0}^{1} K (t, y) {[u (y)]}^{3} dy = t, \begin{matrix} t \in [0, 1] \end{matrix}

where

K (t, y) = {\begin{matrix} - y (1 - t), y \leq t \\ - t (1 - y), y \geq t \end{matrix}

is the Green function.

To solve this problem, we applied three methods.

Method I (using the shifted Chebyshev polynomials and Galerkin methods with approach of operational vector)

In this approach, first, we use the shifted Chebyshev polynomials as $Ψ (y) = [ψ_{1} (y), ψ_{2} (y), \dots, ψ_{N} (y)]^{T}$ .³³ Then, we approximate $[u (y)]^{m}$ as follows:

[u (y)]^{m} = [U^{T} Ψ (y)]^{m} = U_{m}^{* T} Ψ (y)

where $U_{m}^{* T}$ is a column vector, whose elements are nonlinear combinations of the elements of the vector U. $U_{m}^{* T}$ is called the operational vector of the mth power of the function $u (y)$ (see equation (4.2) in Mahmoudi³³).

Method II (using the shifted Chebyshev polynomials and Galerkin methods with the approach of the equivalent equation)

In this approach, first, by using the shifted Chebyshev polynomials, we obtain^13,15,34

u (y) \approx \sum_{i = 0}^{N} c_{i} b_{i} (t)

Then, we approximate $[u (y)]^{m}$ as follows

[u (y)]^{m} = ψ (y, u (y)) = ψ (y, \sum_{i = 1}^{n} c_{i} \int_{a}^{b} K (t, y) b_{i} (y) dy + f (y))

Method II (present method—Algorithm 1)

After testing these three methods on the mentioned example, we see that where the absolute errors in all three methods for Example 3 with different N, t is almost the same; however, the computational costs in these methods are different. In Table 3, we report the total CPU time for the corresponding methods.

Table 3.

CPU time for Example 3.

Method	N = 4	N = 8	N = 16	N = 32
Method I	4.6137	9.7566	21.2418	42.334
Method II	3.2564	7.9783	15.3990	27.6430
Present method	1.9705	5.0370	9.0375	18.2325

From the results, we can see that the present method works better than the other two existing approaches.

Conclusion

In this article, we have proposed a new algorithmic method for the solution of Hammerstein integral equation. This method is very simple to apply and to make an algorithm. Numerical examples are given to further compare the approximation solution of this method with the exact solution, which shows the feasibility and effectiveness of the method for solving Hammerstein integral equation.

Footnotes

Academic Editor: Yi Wang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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An efficient algorithmic method to solve Hammerstein integral equations and application to a functional differential equation

Abstract

Keywords

Introduction

Solving the Hammerstein integral equations

Properties of shifted Chebyshev polynomials

Galerkin method

Main results

Algorithm 1

Illustrative examples

Example 1

Example 2

Example 3 (application to the BVP functional differential equations)

Method I (using the shifted Chebyshev polynomials and Galerkin methods with approach of operational vector)

Method II (using the shifted Chebyshev polynomials and Galerkin methods with the approach of the equivalent equation)

Method II (present method—Algorithm 1)

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References