Study of numerical treatment of functional first-kind Volterra integral equations

Hassanein Falah; Parviz Darania; Saeed Pishbin; Hassanein Falah; Parviz Darania; Saeed Pishbin

doi:10.3934/math.2024846

AIMS Mathematics

2024, Volume 9, Issue 7: 17414-17429. doi: 10.3934/math.2024846

Previous Article Next Article

Research article

Study of numerical treatment of functional first-kind Volterra integral equations

Department of Mathematics, Faculty of Sciences, Urmia University, P. O. Box 165, Urmia, Iran

Received: 26 March 2024 Revised: 27 April 2024 Accepted: 30 April 2024 Published: 21 May 2024
MSC : 45L05, 65R20

First-kind Volterra integral equations have ill-posed nature in comparison to the second-kind of these equations such that a measure of ill-posedness can be described by $\nu$ -smoothing of the integral operator. A comprehensive study of the convergence and super-convergence properties of the piecewise polynomial collocation method for the second-kind Volterra integral equations (VIEs) with constant delay has been given in ^[1]. However, convergence analysis of the collocation method for first-kind delay VIEs appears to be a research problem. Here, we investigated the convergence of the collocation solution as a research problem for a first-kind VIE with constant delay. Three test problems have been fairly well-studied for the sake of verifying theoretical achievements in practice.

Keywords:

Citation: Hassanein Falah, Parviz Darania, Saeed Pishbin. Study of numerical treatment of functional first-kind Volterra integral equations[J]. AIMS Mathematics, 2024, 9(7): 17414-17429. doi: 10.3934/math.2024846

Related Papers:

[1]	Huiyang Xu . Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials. Communications in Analysis and Mechanics, 2023, 15(2): 132-161. doi: 10.3934/cam.2023008
[2]	Tingfu Feng, Yan Dong, Kelei Zhang, Yan Zhu . Global existence and blow-up to coupled fourth-order parabolic systems arising from modeling epitaxial thin film growth. Communications in Analysis and Mechanics, 2025, 17(1): 263-289. doi: 10.3934/cam.2025011
[3]	Yuxuan Chen . Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity. Communications in Analysis and Mechanics, 2023, 15(4): 658-694. doi: 10.3934/cam.2023033
[4]	Yue Pang, Xiaotong Qiu, Runzhang Xu, Yanbing Yang . The Cauchy problem for general nonlinear wave equations with doubly dispersive. Communications in Analysis and Mechanics, 2024, 16(2): 416-430. doi: 10.3934/cam.2024019
[5]	Reinhard Racke . Blow-up for hyperbolized compressible Navier-Stokes equations. Communications in Analysis and Mechanics, 2025, 17(2): 550-581. doi: 10.3934/cam.2025022
[6]	Isaac Neal, Steve Shkoller, Vlad Vicol . A characteristics approach to shock formation in 2D Euler with azimuthal symmetry and entropy. Communications in Analysis and Mechanics, 2025, 17(1): 188-236. doi: 10.3934/cam.2025009
[7]	Mustafa Avci . On an anisotropic $\overset{\rightarrow }{p}(\cdot)$ -Laplace equation with variable singular and sublinear nonlinearities. Communications in Analysis and Mechanics, 2024, 16(3): 554-577. doi: 10.3934/cam.2024026
[8]	Fangyuan Dong . Multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb potential. Communications in Analysis and Mechanics, 2024, 16(3): 487-508. doi: 10.3934/cam.2024023
[9]	Ying Chu, Bo Wen, Libo Cheng . Existence and blow up for viscoelastic hyperbolic equations with variable exponents. Communications in Analysis and Mechanics, 2024, 16(4): 717-737. doi: 10.3934/cam.2024032
[10]	Ho-Sik Lee, Youchan Kim . Boundary Riesz potential estimates for parabolic equations with measurable nonlinearities. Communications in Analysis and Mechanics, 2025, 17(1): 61-99. doi: 10.3934/cam.2025004

Abstract

1. Introduction

Fractional calculus has been concerned with integration and differentiation of fractional (non-integer) order of the function. Riemann and Liouville defined the concept of fractional order intgro-differential equations ^[1]. Fractional calculus has developed an extensive attraction in current years in applied mathematics such as physics, medical, biology and engineering ^{[2,3,4,5,6,7,8]}. Whenever dealing with the fractional integro-differential equation many authors consider the terms Caputo fractional derivative, Riemann-Liouville and Grunwald-Letnikvo ^{[9,10,11,12,13]}. The subject fractional calculus has many applications in widespread and diverse field of science and engineering such as fractional dynamics in the trajectory control of redundant manipulators, viscoelasticity, electrochemistry, fluid mechanics, optics and signals processing etc.

Fractional integro-differential equations having some uncertainties in the form of boundary conditions, initial conditions and so on ^[14,15,16]. To resolve these type of uncertainties mathematicians introduced some concepts fuzzy set theory is one of them.

Zadeh introduced the concept of fuzzy set theory ^{[17,18,19,20]}. Later on Prade and Dubois ^[21,22], Nahmias ^[23], Tanaka and Mizumoto ^[24]. All of them experienced that the fuzzy number as a location of r-cut $0 \leqslant r \leqslant 1$ .

Many authors investigated some numerical techniques related to these problem which include the existence of the solution for discontinuous ^[25], reproducing kernel algorithm ^[26], integro-differential under generalized Caputo differentiability ^[27], A domain decomposition method ^[28], fractional differential transform method ^[29], Jacobi polynomial operational matrix ^[30], global solutions for nonlinear fuzzy equations ^[31], radioactivity decay model ^[32], Caputo-Katugampola fractional derivative approach ^[33], two-dimensional legendre wavelet method ^[34], fuzzy Laplace transform ^[35], fuzzy sumudu transform ^[36]. Further we can see ^{[37,38,39,40]}

Optimal Homotopy Asymptotic Method (OHAM) is one of the powerful techniques introduced by Marinca at al. ^[41,42,43] for approximate solution of differential equations. OHAM attracted an enormous importance in solving various problems in different field of science. Iqbal et al. applied this technique to Klein-Gordon equations and singular Lane-Emden type equation ^[44]. Sheikholeslami et al. used the proposed method for investigation of the laminar viscous flow and magneto hydrodynamic flow in a permeable channel ^[45]. Hashmi et al. obtained the solution of nonlinear Fredholm integral equations using OHAM ^[46]. Nawaz at al. applied the proposed method for solution of fractional order integro-differential equations ^[47], fractional order partial differential equations ^[48] and three-dimensional integral equations ^[49].

Aim of our study is to extend OHAM for solution of system of fuzzy Volterra integro differential equation of fractional order of the following form

$D_x^\alpha u(x) = h(x) + \mathop \smallint \limits_a^x k(x, t)u(t)dt, \, \, \, \, \, 0 \leqslant \alpha \leqslant 1, \, \, x \in [0, 1],$

(1.1)

with the given initial condition

${u^k}(0), \, \, \, \, {u_0}^k(x), \, \, \, k = 1, 2, 3, ...., \eta - 1, \, \, \, \eta - 1 < \alpha < \eta , \, \, \, \, \eta \in N,$

Where $D_x^\alpha$ represents the fuzzy fractional derivative in Caputo sense for fractional order of $\alpha$ with respect to $x$ , $h:[a, b] \to {\mathbb{R}_{\cal{F}}}$ is fuzzy valued function, $k(x, t)$ is arbitrary kernel ${u_0}(x) \in {\mathbb{R}_{\cal{F}}}$ is an unknown solution. ${\mathbb{R}_{\cal{F}}}$ represent set of all fuzzy valued function on real line.

The remaining paper is structured as follows: A brief overview on some elementary concept, notations and definitions of fuzzy calculus and fuzzy fractional calculus are discussed in section 2. Analysis of the technique is presented in section 3. Proposed method is applied to solve fuzzy fractional order Volterra integro-differential equations in section 4. Result and discussion of the paper is given in section 5 and section 6 is the conclusion of the paper.

2. Preliminaries

In literature there exist various definitions of fuzzy calculus and fuzzy fractional calculus ^[50]. Some elementary concept, notations and definitions of fuzzy calculus and fuzzy fractional calculus related to this study are provided in this section.

Definition 2.1. The Riemann-Liouville fractional integral operator $I_x^\alpha$ of order $\alpha$ is ^[50]:

$I_x^\alpha u(x) = \, \left\{ \begin{array}{l} \frac{1}{{\Gamma (\alpha )}}\mathop \smallint \limits_0^x {(x - t)^{\alpha - 1}}u(t)dt = 0, \, \, \, \, \alpha > 0, \, \, \, \hfill \\ u(x), \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \alpha = 0.\, \, \hfill \\ \end{array} \right. \, \, \, \, \, \, \, \, \,$

(2.1)

Definition 2.2. Caputo partial fractional Derivative operator $D_x^\alpha$ of order $\alpha$ with respect to $x$ is defined as follow ^[50]:

$D_x^\alpha u(x) = \, \left\{ \begin{array}{l} \frac{1}{{\Gamma (\eta - \alpha )}}\mathop \smallint \limits_0^x {(x - t)^{\eta - \alpha - 1}}{u^{(n)}}(t)dt = 0, \, \, \, \, \, \, \, \, \, \, \, \eta - 1 < \alpha \leqslant \eta , \, \, \, \hfill \\ \frac{{{d^\eta }\, u(x)}}{{d{x^\eta }}}, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \alpha = \eta \in N.\, \, \hfill \\ \end{array} \right.$

(2.2)

which clearly shows that

$D_x^\alpha I_x^\alpha u(x) = u(x)$

(2.3)

Definition 2.3. A fuzzy number $\sigma$ is a mapping $\sigma :\mathbb{R} \to [0, 1]$ , satisfy the following property:

a. $\sigma$ is normal that is, $\exists \, \, {x_0} \in \mathbb{R}$ with $u({x_0}) = 1$ ^[51,52].

b. $\sigma$ is a convex fuzzy set that is, $u(\lambda x + (1 - \lambda )y) \geqslant \min \left\{ {u(x), u(y)} \right\}$ for all $x, y \in \mathbb{R}, {\text{ }}\lambda \in [0, 1].$

c. $\sigma$ is upper semi-continuous in $\mathbb{R}.$

d. $\overline {\left\{ {x \in \mathbb{R}:u(x) > 0} \right\}}$ is compact.

Definition 2.4. Parametric form of fuzzy number $\sigma$ represented by an order pair $(\underline \sigma , \bar \sigma )$ of the function $\left( {\underline \sigma (r), \bar \sigma (r)} \right)$ , satisfies the following conditions ^[52,53]:

a. $\underline \sigma (r)$ is bounded monotonic increasing left continuous $\forall \, \, r \in [0, 1].$

b. $\bar \sigma (r)$ is bounded monotonic decreasing left continuous $\forall \, \, r \in [0, 1].$

c. $\underline \sigma (r) \leqslant \bar \sigma (r)\, \, \forall \, \, r \in [0, 1].$

Definition 2.5. Addition and scalar multiplication of fuzzy number is given as:

a. $\left( {{\sigma _1} \oplus {\sigma _2}} \right) = \left( {{{\underline \sigma }_1}(r) + {{\underline \sigma }_2}(r), {{\bar \sigma }_1}(r) + {{\bar \sigma }_2}(r)} \right)$

b. $\left( {k \otimes \sigma } \right) = \left\{ \begin{array}{l} \left( {\underline \sigma (r), \bar \sigma (r)} \right), \, \, \, \, \, k \geqslant 0, \hfill \\ \left( {\underline \sigma (r), \bar \sigma (r)} \right), \, \, \, \, \, k < 0. \hfill \\ \end{array} \right.$

Definition 2.6. A fuzzy real valued function ${\sigma _1}, {\sigma _2}:[a, b] \to \mathbb{R},$ then in ^[54]:

${D_U}({\sigma _1}, {\sigma _2}) = \sup \left\{ {D({\sigma _1}(x), {\sigma _2}(x))|x \in [a, b]} \right\}.$

Definition 2.7. Assume $u:[a, b] \to {\mathbb{R}_{\cal{F}}}.$ For every partition ${\rm P} = \left\{ {{\sigma _0}, {\sigma _1}, {\sigma _2}, {\sigma _3}, ...., {\sigma _n}} \right\}$ and arbitrary ${\ell _i}:{\sigma _{i - 1}} \leqslant {\ell _i} \leqslant {\sigma _i}, {\text{ }}2 \leqslant i \leqslant n$ consider

${\mathbb{R}_p} = \mathop \Sigma \limits_{i = 2}^n u({\ell _j})({\sigma _i} - {\sigma _{i - 1}}).$ The definite integral of $u(x)$ over $[\alpha , \beta ]$ is

$\mathop \smallint \limits_\alpha ^\beta u(x)dx = \lim {\mathbb{R}_\rho },$

which show existence of limit in metric ^[55].

Definite integral exist if $u(x)$ is continuous in metric $D$ ^[51]:

$\begin{gathered} \left( {\underline {\mathop \smallint \limits_\alpha ^\beta u(x)dx} } \right) = \mathop \smallint \limits_\alpha ^\beta \underline u (x)dx, \hfill \\ \left( {\overline {\mathop \smallint \limits_\alpha ^\beta u(x)dx} } \right) = \mathop \smallint \limits_\alpha ^\beta \overline u (x)dxt. \hfill \\ \end{gathered}$

3. Application of OHAM

By considering definition 2.4. as discussed in section 2, Eq (1.1) becomes:

$\left\{ \begin{array}{l} D_x^\alpha \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (x, r) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} (x, r) - \mathop \smallint \limits_a^x k(x, t)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (t, r)dt = 0, \hfill \\ D_x^\alpha \bar u(x, r) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} (x, r) - \mathop \smallint \limits_a^x k(x, t)\bar u(t, r)dt = 0, \hfill \\ \end{array} \right. \, \, \, \, \, 0 \leqslant \alpha \leqslant 1, \, 0 \leqslant r \leqslant 1, \, \, \, x \in [0, 1],$

(3.1)

with the given initial condition

${\left[ {{u^k}(0)} \right]^r}, \, \, \, \, \left( {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}^k(x, r), {{\bar u}_0}^k(x, r)} \right), \, \, \, k = 1, 2, 3, ...., \eta - 1, \, \, \, \eta - 1 < \alpha < \eta , \, \, \, \, \eta \in N,$

(3.2)

The homotopy of OHAM ^[41,42,43], constructed as follow:

$\left\{ \begin{array}{l} (1 - \rho )\left( {\frac{{{\partial ^\alpha }\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } (x, r;\rho )}}{{\partial {t^\alpha }}} - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} (x, r)} \right) = H(\rho )\left( {\frac{{{\partial ^\alpha }\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } (x, r;\rho )}}{{\partial {t^\alpha }}} - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} (x, r) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\delta } (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } , r)} \right), \hfill \\ (1 - \rho )\left( {\frac{{{\partial ^\alpha }\bar \upsilon (x, r;\rho )}}{{\partial {t^\alpha }}} - \bar h(x, r)} \right) = H(\rho )\left( {\frac{{{\partial ^\alpha }\bar \upsilon (x, r;\rho )}}{{\partial {t^\alpha }}} - \bar h(x, r) - \bar \delta (\bar \upsilon , r)} \right). \hfill \\ \end{array} \right.$

(3.3)

where $\rho \in [0, 1]$ , $H(\rho ) = \sum\limits_{m\, \, \geqslant 1} {{c_m}{\rho ^m}}$ for all $\rho \ne 0$ is an auxiliary function, if $\rho = 0$ then $H(0) = 0$ where

${\left\{ \begin{array}{l} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } (x, r, 0) = {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}(x, r)\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } (x, r;1) = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (x, r), \hfill \\ \bar \upsilon (x, r, 0) = {{\bar u}_0}(x, r)\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \bar \upsilon (x, r;1) = \bar u(x, r). \hfill \\ \end{array} \right. _{\, \, }}\,$

and ${c_m}$ represent auxiliary constants. Using Taylor's series to expand $\upsilon (x, r;\rho )$ about $\rho$ we get

$\left\{ \begin{array}{l} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } (x, r;\rho ) = {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}(x, r) + \sum\limits_{m\, \, \geqslant 1} {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_m}(x, r){\rho ^m}} , \hfill \\ \bar \upsilon (x, r;\rho ) = {{\bar u}_0}(x, r) + \sum\limits_{m\, \, \geqslant 1} {{{\bar u}_m}(x, r){\rho ^m}} . \hfill \\ \end{array} \right.$

(3.4)

Inserting Eq (3.4) into Eq (3.3) we get series of the problems by comparing the like power of $\rho$ given as follow:

${\rho ^0}:\, \, \, \left\{ \begin{array}{l} \, {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}(x, r) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} (x, r) = 0, \hfill \\ \, {{\bar u}_0}(x, r) - \bar h(x, r) = 0. \hfill \\ \end{array} \right.$

(3.5)

${\rho ^1}:\, \, \, \, \left\{ \begin{array}{l} {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_1}(x, r) + {c_1}\delta ({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}) + (1 + {c_1}) + {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}(x, r) = 0, \hfill \\ {{\bar u}_1}(x, r) + {c_1}\delta ({{\bar u}_0}) + (1 + {c_1}) + {{\bar u}_0}(x, r) = 0. \hfill \\ \end{array} \right.$

(3.6)

${\rho ^2}:\, \, \, \, \left\{ \begin{array}{l} {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_2}(x, r) + {c_1}\delta ({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_1}) + {c_2}\delta ({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}) + {c_2}(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} (x, r) - {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}(x, r)) \hfill \\ - (1 + {c_1}){{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_1}(x, r) = 0, \hfill \\ {{\bar u}_2}(x, r) + {c_1}\delta ({{\bar u}_1}) + {c_2}\delta ({{\bar u}_0}) + {c_2}(\bar h(x, r) - {{\bar u}_0}(x, r)) \hfill \\ - (1 + {c_1}){{\bar u}_1}(x, r) = 0. \hfill \\ \end{array} \right.$

(3.7)

${\rho ^n}:\, \, \, \, \left\{ \begin{array}{l} {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_n}(x, r) + {c_1}\delta ({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_n}) + {c_2}\delta ({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_{n - 1}}) + {c_3}(h + \delta ({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}))... - {c_2}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_{n - 1}}(x, r) - (1 + {c_1}){{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_n}(x, r) = 0, \hfill \\ {{\bar u}_n}(x, r) + {c_1}\delta ({{\bar u}_n}) + {c_2}\delta ({{\bar u}_{n - 1}}) + {c_3}(h + \delta ({{\bar u}_0}))... - {c_2}{{\bar u}_{n - 1}}(x, r) - (1 + {c_1}){{\bar u}_n}(x, r) = 0. \hfill \\ \end{array} \right.$

(3.8)

For calculating the constants ${c_1}, {c_2}, {c_3}...$ , m^th order optimum solution becomes

$\left\{ \begin{array}{l} {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }^m}(x, r, {c_l}) = {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}(x, r) + \sum\limits_{k = 1}^m {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_k}(x, r, {c_l}), } \, \, \, \, l = 1, 2, 3, ...m, \hfill \\ {{\bar u}^m}(x, r, {c_l}) = {{\bar u}_0}(x, r) + \sum\limits_{k = 1}^m {{{\bar u}_k}(x, r, {c_l}), } \, \, \, \, l = 1, 2, 3, ...m. \hfill \\ \end{array} \right.$

(3.9)

Putting Eq (3.9) into Eq (3.1), we can found our residual given as follow:

$\left\{ \begin{array}{l} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{R} \left( {x, r;{c_l}} \right) = {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }^m}(x, r;{c_l}) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} (x, r) - \delta (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} ), \, \, l = 1, 2, ... \hfill \\ \bar R\left( {x, r;{c_l}} \right) = {{\bar u}^m}(x, r;{c_l}) - \bar h(x, r) - \delta (\bar u), \, \, l = 1, 2, ... \hfill \\ \end{array} \right.$

(3.10)

If $R\left( {x, r;{c_l}} \right) = 0,$ then ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} ^m}(x, r;{c_l})\, \, \& \, \, {\bar u^m}(x, r;{c_l})$ will be the exact solutions.

Optimum solution contains some auxiliary constants; the optimal values of these constants are obtained through various techniques. In the present work, we have used the least square method ^[56,57]. The method of least squares is a powerful technique for obtaining the values of auxiliary constants. By putting the optimal values of these constants in Eq (8), we obtain the OHAM solution.

4. Application and accuracy

Problem 4.1. Consider system of fuzzy fractional order Volterra integro-differential equation as ^[58]:

$\left\{ \begin{array}{l} D_x^\alpha \underline u (x, r) = \left( {r - 1} \right) + \int\limits_0^x {\underline u (t, r)dt} \hfill \\ D_x^\alpha \bar u(x, r) = \left( {1 - r} \right) + \int\limits_0^x {\bar u(t, r)dt} \hfill \\ \end{array} \right. , \, \, \, \, 0 < \alpha \leqslant 1, \, \, x \in [0, 1],$

(4.1)

subject to the fuzzy initial condition ${\left[ {u(0)} \right]^r} = \left[ {r - 1, 1 - r} \right],$ and for $\alpha = 1$ fuzzy fractional order Volterra integro-differential equations the exact solution is ${\left[ {u(x)} \right]^r} = \left[ {r - 1, 1 - r} \right]Sinh(x)$ and $0 \leqslant r \leqslant 1.$

By follow the technique as discussed in section 3, we get series of problems and their solutions as:

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _0}(x, r) + (1 - r) = 0, \hfill \\ {D_x}^\alpha {\overline u _0}(x, r) + (r - 1) = 0. \hfill \\ \end{array} \right.$

(4.2)

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _1}(x, r) - 1 + r - {c_1} + r{c_1} + \left( {\mathop \smallint \limits_0^x {{\underline u }_0}(t, r)dt} \right){c_1} - {D_x}^\alpha {\underline u _0}(x, r) - {c_1}{D_x}^\alpha {\underline u _0}(x, r) = 0, \hfill \\ {D_x}^\alpha {{\bar u}_1}(x, r) + 1 - r + {c_1} - r{c_1} + \left( {\mathop \smallint \limits_0^x {{\bar u}_0}(t, r)dt} \right){c_1} - {D_x}^\alpha {{\bar u}_0}(x, r) - {c_1}{D_x}^\alpha \bar u(x, r) = 0. \hfill \\ \end{array} \right.$

(4.3)

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _2}(x, r) + \left( {\mathop \smallint \limits_0^x {{\underline u }_1}(t, r)dt} \right){c_1} - {c_2} + r{c_2} + \left( {\mathop \smallint \limits_0^x {{\underline u }_0}(t, r)dt} \right){c_2} - {c_2}{D_x}^\alpha {\underline u _0}(x, r) - \hfill \\ {D_x}^\alpha {\underline u _1}(x, r) - {c_1}{D_x}^\alpha {\underline u _0}(x, r) = 0, \hfill \\ {D_x}^\alpha {{\bar u}_2}(x, r) + \left( {\mathop \smallint \limits_0^x {{\bar u}_1}(t, r)dt} \right){c_1} + {c_2} - r{c_2} + \left( {\mathop \smallint \limits_0^x {{\bar u}_0}(t, r)dt} \right){c_2} - {c_2}{D_x}^\alpha {{\bar u}_0}(x, r) - \hfill \\ {D_x}^\alpha {{\bar u}_1}(x, r) - {c_1}{D_x}^\alpha {{\bar u}_0}(x, r) = 0. \hfill \\ \end{array} \right.$

(4.4)

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _3}(x, r) + \left( {\mathop \smallint \limits_0^x {{\underline u }_2}(t, r)dt} \right){c_1} + \left( {\mathop \smallint \limits_0^x {{\underline u }_1}(t, r)dt} \right){c_2} - {c_3} + r{c_3} + \left( {\mathop \smallint \limits_0^x {{\underline u }_0}(t, r)dt} \right){c_3} - {c_3}{D_x}^\alpha {\underline u _0}(x, r) \hfill \\ - {c_2}{D_x}^\alpha {\underline u _1}(x, r) - {D_x}^\alpha {\underline u _2}(x, r) - {c_1}{D_x}^\alpha {\underline u _2}(x, r) = 0, \hfill \\ {D_x}^\alpha {{\bar u}_3}(x, r) + \left( {\mathop \smallint \limits_0^x {{\bar u}_2}(t, r)dt} \right){c_1} + \left( {\mathop \smallint \limits_0^x {{\bar u}_1}(t, r)dt} \right){c_2} + {c_3} - r{c_3} + \left( {\mathop \smallint \limits_0^x {{\bar u}_0}(t, r)dt} \right){c_3} - {c_3}{D_x}^\alpha {{\bar u}_0}(x, r) \hfill \\ - {c_2}{D_x}^\alpha {{\bar u}_1}(x, r) - {D_x}^\alpha {{\bar u}_2}(x, r) - {c_1}{D_x}^\alpha {{\bar u}_2}(x, r) = 0. \hfill \\ \end{array} \right.$

(4.5)

Their solutions are

$\left\{ \begin{array}{l} {\underline u _0}(x, r) = \frac{{( - 1 + r){x^\alpha }}}{{\alpha \, \Gamma (\alpha )}} \hfill \\ {{\bar u}_0}(x, r) = - \frac{{( - 1 + r){x^\alpha }}}{{\alpha \, \Gamma (\alpha )}} \hfill \\ \end{array} \right. ,$

(4.6)

$\left\{ \begin{array}{l} {\underline u _1}(x, r) = - \frac{{( - 1 + r){x^{1 + 2\alpha }}{c_1}}}{{\Gamma (2 + 2\alpha )}}, \hfill \\ {{\bar u}_1}(x, r) = \frac{{( - 1 + r){x^{1 + 2\alpha }}{c_1}}}{{\Gamma (2 + 2\alpha )}}. \hfill \\ \end{array} \right.$

(4.7)

$\left\{ \begin{array}{l} {\underline u _2}(x, r) = ( - 1 + r){x^{1 + 2\alpha }}\left( {\frac{{{x^{1 + \alpha }}c_1^2}}{{\Gamma (3 + 3\alpha )}} - \frac{{{c_1} + c_1^2 + {c_2}}}{{\Gamma (2 + 2\alpha )}}} \right), \hfill \\ {{\bar u}_2}(x, r) = ( - 1 + r){x^{1 + \alpha }}\left( { - \frac{{{x^{1 + \alpha }}c_1^2}}{{\Gamma (3 + 3\alpha )}} + \frac{{{c_1} + c_1^2 + {c_2}}}{{\Gamma (2 + 2\alpha )}}} \right). \hfill \\ \end{array} \right.$

(4.8)

$\left\{ \begin{array}{l} {\underline u _3}(x, r) = ( - 1 + r){x^{1 + 2\alpha }}\left( { - \frac{{{x^{2 + 2\alpha }}c_1^3}}{{\Gamma (4 + 4\alpha )}} + \frac{{2{x^{1 + \alpha }}{c_1}({c_1} + c_1^2 + {c_2})}}{{\Gamma (3 + 3\alpha )}} - \frac{{{c_1} + 2c_1^2 + c_1^3 + {c_2} + 2{c_1}{c_2} + {c_3}}}{{\Gamma (2 + 2\alpha )}}} \right), \hfill \\ {{\bar u}_3}(x, r) = ( - 1 + r){x^{1 + 2\alpha }}\left( {\frac{{{x^{2 + 2\alpha }}c_1^3}}{{\Gamma (4 + 4\alpha )}} - \frac{{2{x^{1 + \alpha }}{c_1}({c_1} + c_1^2 + {c_2})}}{{\Gamma (3 + 3\alpha )}} + \frac{{{c_1} + 2c_1^2 + c_1^3 + {c_2} + 2{c_1}{c_2} + {c_3}}}{{\Gamma (2 + 2\alpha )}}} \right). \hfill \\ \end{array} \right.$

(4.9)

Adding (4.6), (4.7), (4.8) and (4.9), one can construct $\underline u (x, r)$ & $\bar u(x, r)$ :

$\left\{ \begin{array}{l} \underline u (x, r) = ( - 1 + r){x^\alpha }\left( \begin{gathered} \frac{1}{{\Gamma (1 + \alpha )}} - \frac{{{x^{3 + 3\alpha }}c_1^3}}{{\Gamma (4 + 4\alpha )}} + \frac{{{x^{2 + 2\alpha }}{c_1}({c_1}(3 + 2{c_1}) + 2{c_2})}}{{\Gamma (3 + 3\alpha )}} - \hfill \\ \frac{{{x^{1 + \alpha }}(2{c_2} + {c_1}(3 + {c_1}(3 + {c_1}) + 2{c_2}) + {c_3})}}{{\Gamma (2 + 2\alpha )}} \hfill \\ \end{gathered} \right), \hfill \\ \bar u(x, r) = ( - 1 + r){x^\alpha }\left( \begin{gathered} - \frac{1}{{\Gamma (1 + \alpha )}} + \frac{{{x^{3 + 3\alpha }}c_1^3}}{{\Gamma (4 + 4\alpha )}} - \frac{{{x^{2 + 2\alpha }}{c_1}({c_1}(3 + 2{c_1}) + 2{c_2})}}{{\Gamma (3 + 3\alpha )}} + \hfill \\ \frac{{{x^{1 + \alpha }}(2{c_2} + {c_1}(3 + {c_1}(3 + {c_1}) + 2{c_2}) + {c_3})}}{{\Gamma (2 + 2\alpha )}} \hfill \\ \end{gathered} \right). \hfill \\ \end{array} \right.$

(4.10)

Values of ${c_1}, \, \, {c_2}$ and ${c_3}$ contain is in Eq (4.10)

Substituting the values from into Eq (4.10), the approximate solutions for $\underline u (x, r)$ & $\bar u(x, r)$ at different values of $\alpha$ taking $r = 0.75$ respectively is as follow

$\alpha = 0.7$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.2751369{x}^{0.7}-0.08602097{x}^{2.4}-0.25{x}^{2.4}(-0.00904995+0.0376716{x}^{1.7})\\ -0.25{x}^{2.4}(0.000425859-0.00198165{x}^{1.7}+0.0021734872{x}^{3.4}), \\ \overline{u}(x, r)\approx 0.2751369{x}^{0.7}+0.08602097{x}^{2.4}-0.25{x}^{2.4}(0.00904995-0.0376716{x}^{1.7})-\\ 0.25{x}^{2.4}(-0.000425859+0.00198165{x}^{1.7}-0.0021734872{x}^{3.4}).\end{array} \right.$

(4.11)

$\alpha = 0.8$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.2684178{x}^{0.8}-0.0685589{x}^{2.6}-0.25{x}^{2.6}(-0.005391327+0.023297809{x}^{1.8})\\ -0.25{x}^{2.6}(0.000197513-0.0009160448{x}^{1.8}+0.001008410504{x}^{3.6}), \\ \overline{u}(x, r)\approx 0.2684178{x}^{0.8}+0.0685589{x}^{2.6}-0.25{x}^{2.6}(0.0053913-0.023297809{x}^{1.8})\\ -0.25{x}^{2.6}(-0.000197513+0.0009160448{x}^{1.8}-0.001008410504{x}^{3.6}).\end{array} \right.$

(4.12)

$\alpha = 0.9$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.2599385{x}^{0.9}-0.0540269{x}^{2.8}-0.25{x}^{2.8}(-0.0031639+0.01418901{x}^{1.9})\\ -0.25{x}^{2.8}(0.00008989-0.0004154745{x}^{1.9}+0.000458881{x}^{3.8}), \\ \overline{u}(x, r)\approx 0.2599385{x}^{0.9}+0.0540269{x}^{2.8}-0.25{x}^{2.8}(0.0031639-0.014189097{x}^{1.9})\\ -0.25{x}^{2.8}(-0.00008989+0.0004154745{x}^{1.9}-0.000458881{x}^{3.8}).\end{array} \right.$

(4.13)

$\alpha = 1$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.25x-0.042114377{x}^{3}-0.25{x}^{3}(-0.001829736+0.0085133796{x}^{2})\\ -0.25{x}^{3}(0.0000401535-0.0001849397{x}^{2}+0.00020487753{x}^{4}), \\ \overline{u}(x, r)\approx 0.25x+0.042114377{x}^{3}-0.25{x}^{3}(0.001829736-0.0085133796{x}^{2})\\ -0.25{x}^{3}(-0.0000401535+0.0001849397{x}^{2}-0.0002048775{x}^{4}).\end{array} \right.$

(4.14)

Table 1. at r = 0.75.

$\alpha$	${\underline c _1}$ & ${\overline c _1}$	${\underline c _2}$ & ${\overline c _2}$	${\underline c _3}$ & ${\overline c _3}$
0.7	−1.0257850714449026	5.298291106236844×10⁻⁴	−3.040859671410477×10⁻⁵
0.8	−1.0193406988378892	3.249294721058776×10⁻⁴	−1.5364294422415488×10⁻⁵
0.9	−1.014446487354385	1.9694362983845834×10⁻⁴	−7.63055570435551×10⁻⁶
1	−1.0107450504316333	1.1791102776455743×10⁻⁴	−3.779171763451589×10⁻⁶

| Show Table

DownLoad: CSV

Substituting the values from into Eq (4.10), the approximate solutions for $\underline u (x, r)$ & $\bar u(x, r)$ at different values of $\alpha$ taking $r = 0.5$ respectively is as follow

$\alpha = 0.7$

$\left\{ \begin{array}{l} \underline u (x, r) \approx - 0.5502737{x^{0.7}} - 0.172041940{x^{2.4}} - 0.5{x^{2.4}}( - 0.00904995 + 0.0376716{x^{1.7}}) \hfill \\ - 0.5{x^{2.4}}(0.0004258594 - 0.0019816476{x^{1.7}} + 0.0021734872{x^{3.4}}), \hfill \\ \bar u(x, r) \approx 0.5502737{x^{0.7}} + 0.172041940{x^{2.4}} - 0.5{x^{2.4}}(0.00904995 - 0.03767164{x^{1.7}}) \hfill \\ - 0.5{x^{2.4}}( - 0.0004258594 + 0.0019816476{x^{1.7}} - 0.0021734872{x^{3.4}}). \hfill \\ \end{array} \right.$

(4.15)

$\alpha = 0.8$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.53683564{x}^{0.8}-0.137117858{x}^{2.6}-0.5{x}^{2.6}(-0.00539133+0.02329781{x}^{1.8})\\ -0.5{x}^{2.6}(0.00019751-0.0009160448{x}^{1.8}+0.0010084105{x}^{3.6}), \\ \overline{u}(x, r)\approx 0.53683564{x}^{0.8}+0.137117858{x}^{2.6}-0.5{x}^{2.6}(0.00539133-0.02329781{x}^{1.8})\\ -0.5{x}^{2.6}(-0.00019751+0.0009160448{x}^{1.8}-0.0010084105{x}^{3.6}).\end{array} \right.$

(4.16)

$\alpha = 0.9$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.5198771{x}^{0.9}-0.10805378{x}^{2.8}-0.5{x}^{2.8}(-0.0031640+0.0141891{x}^{1.9})\\ -0.5{x}^{2.8}(0.0000898945-0.0004154745{x}^{1.9}+0.000458881{x}^{3.8}), \\ \overline{u}(x, r)\approx 0.5198771{x}^{0.9}+0.10805378{x}^{2.8}-0.5{x}^{2.8}(0.0031640-0.0141891{x}^{1.9})\\ -0.5{x}^{2.8}(-0.0000898945+0.0004154745{x}^{1.9}-0.000458881{x}^{3.8}).\end{array} \right.$

(4.17)

$\alpha = 1$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.5x-0.0842288{x}^{3}-0.5{x}^{3}(-0.0018298+0.008513{x}^{2})-0.5{x}^{3}\\ (0.0000401612-0.00018494622{x}^{2}+0.0002048777{x}^{4}), \\ \overline{u}(x, r)\approx 0.5x+0.0842288{x}^{3}-0.5{x}^{3}(0.0018298-0.008513{x}^{2})-0.5{x}^{3}\\ (-0.000040162+0.00018494622{x}^{2}-0.0002048777{x}^{4}).\end{array} \right.$

(4.18)

Table 2. at r = 0.5.

$\alpha$	${\underline c _1}$ & ${\overline c _1}$	${\underline c _2}$ & ${\overline c _2}$	${\underline c _3}$ & ${\overline c _3}$
0.7	−1.0257850714449026	5.298291106236844×10⁻⁴	−3.040859671410477×10⁻⁵
0.8	−1.0193406988378892	3.249294721058776×10⁻⁴	−1.5364294422415488×10⁻⁵
0.9 1	−1.014446487354385 −1.0107453381292266	1.9694362983845834×10⁻⁴ 1.1800167363027721×10⁻⁴	−7.630555570435551×10⁻⁶ −3.726389827252244×10⁻⁶

| Show Table

DownLoad: CSV

Problem 4.2. Consider system of fuzzy fractional order Volterra integro-differential equation as ^[59]:

$D_x^\alpha u(x, r) + \int\limits_0^t {u(t, r)dt = 0} , \, \, \, \, 0 < \alpha \leqslant 1, \, \, x \in [0, 1],$

(4.19)

subject to the fuzzy initial condition ${\left[ {u(0)} \right]^r} = \left[ {r - 1, 1 - r} \right],$ and the exact solution is $\underline u (x, r) = \left( {r - 1} \right){E_{\alpha + 1}}\left( { - {t^{\alpha + 1}}} \right), \bar u(x, r) = \left( {1 - r} \right){E_{\alpha + 1}}\left( { - {t^{\alpha + 1}}} \right),$

where ${E_{\alpha + 1}}$ is a Mittag-Leffler function and $0 \leqslant r \leqslant 1.$

By follow the technique as discussed in section 3, we get series of problems and their solutions as:

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _0}(x, r) = 0, \hfill \\ {D_x}^\alpha {{\bar u}_0}(x, r) = 0. \hfill \\ \end{array} \right.$

(4.20)

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _1}(x, r) + \left( { - \mathop \smallint \limits_0^x {{\underline u }_0}(t, r)dt} \right){c_1} - {D_x}^\alpha {\underline u _0}(x, r) - {c_1}{D_x}^\alpha {\underline u _0}(x, r) = 0, \hfill \\ {D_x}^\alpha {{\bar u}_1}(x, r) + \left( { - \mathop \smallint \limits_0^x {{\bar u}_0}(t, r)dt} \right){c_1} - {D_x}^\alpha {{\bar u}_0}(x, r) - {c_1}{D_x}^\alpha \bar u(x, r) = 0. \hfill \\ \end{array} \right.$

(4.21)

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _2}(x, r) + \left( {\mathop \smallint \limits_0^x {{\underline u }_1}(t, r)dt} \right){c_1} - \left( {\mathop \smallint \limits_0^x {{\underline u }_0}(t, r)dt} \right){c_2} - {c_2}{D_x}^\alpha {\underline u _0}(x, r) - {D_x}^\alpha {\underline u _1}(x, r) - {c_1}{D_x}^\alpha {\underline u _0}(x, r) = 0, \hfill \\ {D_x}^\alpha {{\bar u}_2}(x, r) + \left( { - \mathop \smallint \limits_0^x {{\bar u}_1}(t, r)dt} \right){c_1} - \left( {\mathop \smallint \limits_0^x {{\bar u}_0}(t, r)dt} \right){c_2} - {c_2}{D_x}^\alpha {{\bar u}_0}(x, r) - {D_x}^\alpha {{\bar u}_1}(x, r) - {c_1}{D_x}^\alpha {{\bar u}_0}(x, r) = 0. \hfill \\ \end{array} \right.$

(4.22)

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _3}(x, r) - \left( {\mathop \smallint \limits_0^x {{\underline u }_2}(t, r)dt} \right){c_1} - \left( {\mathop \smallint \limits_0^x {{\underline u }_1}(t, r)dt} \right){c_2} - \left( {\mathop \smallint \limits_0^x {{\underline u }_0}(t, r)dt} \right){c_3} - {c_3}{D_x}^\alpha {\underline u _0}(x, r) \hfill \\ - {c_2}{D_x}^\alpha {\underline u _1}(x, r) - {D_x}^\alpha {\underline u _2}(x, r) - {c_1}{D_x}^\alpha {\underline u _2}(x, r) = 0, \hfill \\ {D_x}^\alpha {{\bar u}_3}(x, r) - \left( {\mathop \smallint \limits_0^x {{\bar u}_2}(t, r)dt} \right){c_1} - \left( {\mathop \smallint \limits_0^x {{\bar u}_1}(t, r)dt} \right){c_2} - \left( {\mathop \smallint \limits_0^x {{\bar u}_0}(t, r)dt} \right){c_3} - {c_3}{D_x}^\alpha {{\bar u}_0}(x, r) \hfill \\ - {c_2}{D_x}^\alpha {{\bar u}_1}(x, r) - {D_x}^\alpha {{\bar u}_2}(x, r) - {c_1}{D_x}^\alpha {{\bar u}_2}(x, r) = 0. \hfill \\ \end{array} \right.$

(4.23)

And their solutions are

$\left\{ \begin{array}{l} {\underline u _0}(x, r) = r - 1, \hfill \\ {{\bar u}_0}(x, r) = 1 - r. \hfill \\ \end{array} \right.$

(4.24)

$\left\{ \begin{array}{l} {\underline u _1}(x, \underline r ) = \frac{{( - 1 + r){x^{1 + \alpha }}{c_1}}}{{(\alpha + {\alpha ^2})\Gamma (\alpha )}}, \hfill \\ {\overline u _1}(x, \overline r ) = - \frac{{( - 1 + r){x^{1 + \alpha }}{c_1}}}{{\alpha (1 + \alpha )\Gamma (\alpha )}}, \hfill \\ \end{array} \right.$

(4.25)

$\left\{ \begin{array}{l} {\underline u _2}(x, r) = ( - 1 + r){x^{1 + \alpha }}\left( {\frac{{{x^{1 + \alpha }}c_1^2}}{{\Gamma (3 + 2\alpha )}} + \frac{{{c_1} + c_1^2 + {c_2}}}{{\Gamma (2 + \alpha )}}} \right), \hfill \\ {{\bar u}_2}(x, r) = ( - 1 + r){x^{1 + \alpha }}\left( { - \frac{{{x^{1 + \alpha }}c_1^2}}{{\Gamma (3 + 2\alpha )}} - \frac{{{c_1} + c_1^2 + {c_2}}}{{\Gamma (2 + \alpha )}}} \right). \hfill \\ \end{array} \right.$

(4.26)

$\left\{ \begin{array}{l} {\underline u _3}(x, r) = \frac{{( - 1 + r){x^{1 + \alpha }}\left( {{c_2} + {c_1}\left( {1 + \frac{{{x^{2 + 2\alpha }}c_1^2}}{{\Gamma (4 + 2\alpha )}} + {c_1}(2 + {c_1}) + 2{c_2} + \frac{{2{x^{1 + \alpha }}({c_1} + c_1^2 + {c_2})}}{{\Gamma (3 + 3\alpha )}}} \right) + {c_3}} \right)}}{{\Gamma (1 + \alpha )}}, \hfill \\ {{\bar u}_3}(x, r) = ( - 1 + r){x^{1 + \alpha }}\left( { - \frac{{{x^{2 + 2\alpha }}c_1^3}}{{\Gamma (4 + 3\alpha )}} - \frac{{2{x^{1 + \alpha }}{c_1}({c_1} + c_1^2 + {c_2})}}{{\Gamma (3 + 2\alpha )}} - \frac{{{c_2} + {c_1}({{(1 + {c_1})}^2} + 2{c_2}) + {c_3}}}{{\Gamma (2 + \alpha )}}} \right), \hfill \\ \end{array} \right.$

(4.27)

Adding (4.24), (4.25), (4.26) and (4.27), one can construct $\underline u (x, r)$ & $\bar u(x, r)$ :

$\left\{ \begin{array}{l} \underline u (x, r) = - 1 + r + ( - 1 + r){x^{1 + \alpha }}\left( \begin{gathered} \frac{{{x^{1 + \alpha }}c_1^2}}{{\Gamma (3 + 2\alpha )}} + \frac{{{x^{2 + 2\alpha }}c_1^3}}{{\Gamma (1 + \alpha )\Gamma (4 + 2\alpha )}} + \frac{{2{x^{1 + \alpha }}{c_1}({c_1} + c_1^2 + {c_2})}}{{\Gamma (1 + \alpha )\Gamma (3 + \alpha )}} \hfill \\ + \frac{{{c_1}(2 + {c_1}) + {c_2}}}{{\Gamma (2 + \alpha )}} + \frac{{{c_2} + {c_1}({{(1 + {c_1})}^2} + 2{c_2}) + {c_3}}}{{\Gamma (1 + \alpha )}} \hfill \\ \end{gathered} \right), \hfill \\ \bar u(x, r) = 1 - r + ( - 1 + r){x^{1 + \alpha }}\left( \begin{gathered} - \frac{{{x^{2 + 2\alpha }}c_1^3}}{{\Gamma (4 + 3\alpha )}} - \frac{{{x^{1 + \alpha }}{c_1}({c_1}(3 + 2{c_1}) + 2{c_2})}}{{\Gamma (3 + 2\alpha )}} - \hfill \\ \frac{{2{c_2} + {c_1}(3 + {c_1}(3 + {c_1}) + 2{c_2}) + {c_3}}}{{\Gamma (2 + \alpha )}} \hfill \\ \end{gathered} \right). \hfill \\ \end{array} \right.$

(4.28)

Values of ${c_1}, \, \, {c_2}$ and ${c_3}$ contain in Eq (4.28)

Substituting the values from and into Eq (4.28), the approximate solutions for $\underline u (x, r)$ & $\bar u(x, r)$ at different values of $\alpha$ taking $r = 0.5$ is as follow

$\alpha = 0.2$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.5-0.5{x}^{1.2}(-0.905948+0.325139{x}^{1.2}-0.055807{x}^{2.4}), \\ \overline{u}(x, r)\approx 0.5-0.5{x}^{1.2}(0.905948-0.325139{x}^{1.2}+0.055807{x}^{2.4}).\end{array} \right.$

(4.29)

$\alpha = 0.4$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.5-0.5{x}^{1.4}(-0.804545+0.210093{x}^{1.4}-0.025570{x}^{2.8}), \\ \overline{u}(x, r)\approx 0.5-0.5{x}^{1.4}(0.80454545-0.210093{x}^{1.4}+0.025570{x}^{2.8}).\end{array} \right.$

(4.30)

$\alpha = 0.6$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.5-0.5{x}^{1.6}(-0.699349+0.128177{x}^{1.6}-0.010450{x}^{3.2}), \\ \overline{u}(x, r)\approx 0.5-0.5{x}^{1.6}(0.699349-0.128177{x}^{1.6}+0.010450{x}^{3.2}).\end{array} \right.$

(4.31)

$\alpha = 0.8$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.5-0.5{x}^{1.8}(-0.596450+0.074561{x}^{1.8}-0.0038863{x}^{3.6}), \\ \overline{u}(x, r)\approx 0.5-0.5{x}^{1.8}(0.5964503-0.074561{x}^{1.8}+0.0038863{x}^{3.6}).\end{array} \right.$

(4.32)

$\alpha = 1$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.5-0.5{x}^{2}(-0.499992+0.04163089{x}^{2}-0.001336253{x}^{4}), \\ \overline{u}(x, r)\approx 0.5-0.5{x}^{2}(0.4999914-0.04163076{x}^{2}+0.001336247{x}^{4}).\end{array} \right.$

(4.33)

Table 3. at r = 0.5.

$\alpha$	${\underline c _1}$	${\underline c _2}$	${\underline c _3}$
0.2	−0.8038238618267683	7.6178003377104005×10⁻³	−1.334733828882206×10⁻³
0.4	−0.739676946061329	0.02530379950927192	−3.78307542584476×10⁻³
0.6	−0.6725325561865596	0.04771922184372877	−8.140310215705777×10⁻³
0.8	−0.6062340661192892	0.06889015391501904	−0.015746150088743013
1.0	−0.5432795308983783	0.08615024033359142	−0.026582381449582644

| Show Table

DownLoad: CSV

Table 4. at r = 0.5.

$\alpha$	${\overline c _1}$	${\overline c _2}$	${\overline c _3}$
0.2	−0.9072542694138958	3.5727393445527333×10⁻³	3.637119200533134×10⁻⁴
0.4	−0.9409187563211361	1.9803045412863643×10⁻³	1.7807647588483674×10⁻⁴
0.6	−0.9636043521097131	9.808463578935658×10⁻⁴	7.369503633679291×10⁻⁵
0.8	−0.9781782948007163	4.4492367269725435×10⁻⁴	2.6729929232143615×10⁻⁵
1.0	−0.9872029432879605	1.896941835601795×10⁻⁴	1.021455544474314×10⁻⁵

| Show Table

DownLoad: CSV

Substituting the values from and into Eq (4.28), the approximate solutions for $\underline u (x, r)$ & $\bar u(x, r)$ at different values of $r$ taking $\alpha = 0.5$ is as follow

$r = 0$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -1-{x}^{1.5}(-0.75199032254+0.165168588631{x}^{1.5}-0.016559344247{x}^{3.}), \\ \overline{u}(x, r)\approx 1-{x}^{1.5}(0.75199032256-0.165168588653{x}^{1.5}+0.016559344262{x}^{3.}).\end{array} \right.$

(4.34)

$r = 0.2$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.8-0.8{x}^{1.5}(-0.751990322550+0.165168588{x}^{1.5}-0.01655934425{x}^{3.}), \\ \overline{u}(x, r)\approx 0.8-0.8{x}^{1.5}(0.751990322554-0.165168588{x}^{1.5}+0.01655934426{x}^{3.}). \end{array} \right.$

(4.35)

$r = 0.4$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.6-0.6{x}^{1.5}(-0.7519903225+0.1651685886{x}^{1.5}-0.016559344250{x}^{3.}), \\ \overline{u}(x, r)\approx 0.6-0.6{x}^{1.5}(0.7519903225-0.1651685886{x}^{1.5}+0.0165593442496{x}^{3.}).\end{array} \right.$

(4.36)

$r = 0.6$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.4-0.4{x}^{1.5}(-0.751990322550+0.16516858863{x}^{1.5}-0.01655934425{x}^{3.}), \\ \overline{u}(x, r)\approx 0.4-0.4{x}^{1.5}(0.751990322554-0.16516858865{x}^{1.5}+0.01655934426{x}^{3.}).\end{array} \right.$

(4.37)

$r = 0.8$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.1999910-0.1999910{x}^{1.5}(-0.7519903+0.16516859{x}^{1.5}-0.0165593{x}^{3.}), \\ \overline{u}(x, r)\approx 0.19999910-0.19999910{x}^{1.5}(0.7519903-0.16516859{x}^{1.5}+0.0165593{x}^{3.}).\end{array} \right.$

(4.38)

Table 5. at

$\alpha$ = 0.5.

$r$	${\underline c _1}$	${\underline c _2}$	${\underline c _3}$
$0$	$- 0.7062087686601037$	$0.03638991875243272$	$- 5.6065011933001474 \times {10^{ - 3}}$
$0.2$	$- 0.7062087687083373$	$0.03638991875755982$	$- 5.606501189519195 \times {10^{ - 3}}$
$0.4$	$- 0.7062087686911187$	$0.03638991875566347$	$- 5.606501190876118 \times {10^{ - 3}}$
$0.6$	$- 0.7062087687083373$	$0.03638991875755982$	$- 5.606501189519195 \times {10^{ - 3}}$
$0.8$	$- 0.7062087686312865$	$0.036389918749298394$	$- 5.606501195566148 \times {10^{ - 3}}$

| Show Table

DownLoad: CSV

Table 6. at

$\alpha$ = 0.5.

$r$	${\overline c _1}$	${\overline c _2}$	${\overline c _3}$
$0$	$- 0.9534544876637709$	$1.4110239362094313 \times {10^{ - 3}}$	$1.1669890178958486 \times {10^{ - 4}}$
$0.2$	$- 0.9534544876175205$	$1.4110239407705756 \times {10^{ - 3}}$	$1.1669890245071123 \times {10^{ - 4}}$
$0.4$	$- 0.9534544874104965$	$1.4110239614371703 \times {10^{ - 3}}$	$1.1669890547775563 \times {10^{ - 4}}$
$0.6$	$- 0.9534544876175205$	$1.4110239407705756 \times {10^{ - 3}}$	$1.1669890245071123 \times {10^{ - 4}}$
$0.8$	$- 0.9534544876289686$	$1.4110239398903034 \times {10^{ - 3}}$	$1.1669890235330204 \times {10^{ - 4}}$

| Show Table

DownLoad: CSV

5. Result and discussion

– show the values of auxiliary constant at different values of $r$ & $\alpha$ for both lower and upper solution of OHAM for the solved problems. and show the comparison of absolute error of 3^rd order OHAM with Fractional Residual Power Series (FRPS) Method for 5-approximated solution and $k = 5$ for both lower and upper solutions of OHAM at different value of $\alpha$ for problem 1. Comparison of absolute error of 3^rd orders OHAM for both lower and upper solution of OHAM are shown in and . Numerical result show that OHAM provide more accuracy as compared to the other method and as $\alpha \to 1$ the approximate solution become very close to the exact solution. Graphical representation confirmed the convergence of fractional order solution towards the integer order solution. In graphical representation of OHAM at $\alpha = 0.7, \, \, 0.8, \, \, 0.9\, , \, \, 1, \, \, r = 0.75$ and $\alpha = 0.7, \, \, 0.8, \, \, 0.9\, , \, \, 1, \, \, r = 0.50$ are discussed for both $\underline u (x, r)$ & $\bar u(x, r)$ for problem 1. Figures 2 and 3 show the comparison of OHAM with the exact solution at different values of and taking r = 0.75 & r = 0.5 respectively for problem 1. represent the comparison of OHAM at $\alpha = 0.2, \, 0.4, \, \, 0.6, \, \, 0.8, \, \, 1, \, \, r = 0.5$ and $r = 0, \, \, 0.2, \, 0.4, \, \, 0.6, \, \, 0.8, \, \, \alpha \, = 0.5$ for both $\underline u (x, r)$ and $\bar u(x, r)$ for problem 2. Figure 5 shows the comparison of OHAM with the exact solution at different values of and r = 0.5 while Figure 6 shows the comparison of OHAM with the exact solution at different values of r and = 0.5 for problem 2.

Table 7. Comparison of Absolute Error (Abs Err.) of 3^rd order OHAM for

$\underline u (x, r)$ and Fractional Residual Power Series (FRPS) ^[54] Method for 5-approximated solution and

$k = 5$ for problem 1.

$r$	$x$	FRPS ^[58] $\alpha = 0.7$	OHAM	FRPS ^[58] $\alpha = 0.8$	OHAM	FRPS ^[58] $\alpha = 0.9$	OHAM	FRPS ^[58] $\alpha = 1$	OHAM
0.75	0.2	0.042797	0.040621	0.025514	0.024764	0.011512	0.011321	6.35273×10⁻¹⁰	2.14676×10⁻⁹
	0.4	0.059664	0.051698	0.035840	0.032584	0.016392	0.015405	8.14507×10⁻⁸	1.00832×10⁻⁸
	0.6	0.075769	0.058997	0.045171	0.037635	0.020545	0.018031	1.39554×10⁻⁶	1.11336×10⁻⁸
	0.8	0.094364	0.066136	0.055863	0.04232	0.025182	0.020362	1.04955×10⁻⁵	6.21567×10⁻⁹
0.50	0.2	0.085595	0.081241	0.051027	0.049528	0.011321	0.022643	1.27055×10⁻⁹	4.25946×10⁻⁹
	0.4	0.119328	0.103396	0.071680	0.065167	0.015405	0.030811	1.62901×10⁻⁷	1.98877×10⁻⁸
	0.6	0.151537	0.117994	0.090342	0.075269	0.018031	0.036062	2.79107×10⁻⁶	2.12923×10⁻⁸
	0.8	0.188728	0.132271	0.111723	0.084640	0.020362	0.040725	2.09911×10⁻⁵	1.00120×10⁻⁸

| Show Table

DownLoad: CSV

Table 8. Comparison of Absolute Error (Abs Err.) of 3rd order OHAM for

$\bar u(x, r)$ and Fractional Residual Power Series (FRPS) ^[58] Method for 5-approximated solution and

$k = 5$ for problem 1.

$r$	$x$	FRPS ^[58] $\alpha = 0.7$	OHAM	FRPS ^[58] $\alpha = 0.8$	OHAM	FRPS ^[58] $\alpha = 0.9$	OHAM	FRPS ^[58] $\alpha = 1$	OHAM
0.75	0.2	0.085595	0.081242	0.025514	0.024764	0.011512	0.011321	6.35273×10⁻¹⁰	2.14676×10⁻⁹
	0.4	0.119328	0.103396	0.035840	0.032584	0.016392	0.015405	8.14507×10⁻⁸	1.00832×10⁻⁸
	0.6	0.151537	0.117994	0.045171	0.037635	0.020545	0.018031	1.39554×10⁻⁶	1.11336×10⁻⁸
	0.8	0.188728	0.132271	0.055862	0.04232	0.025182	0.020362	1.04955×10⁻⁵	6.21567×10⁻⁹
0.50	0.2	0.042797	0.040621	0.051027	0.049528	0.023025	0.022643	1.27055×10⁻⁹	4.25946×10⁻⁹
	0.4	0.059664	0.051698	0.071680	0.065167	0.032784	0.030811	1.62901×10⁻⁷	1.98877×10⁻⁸
	0.6	0.075769	0.058997	0.090342	0.075269	0.041090	0.036062	2.79107×10⁻⁶	2.12923×10⁻⁸
	0.8	0.094364	0.066136	0.111723	0.084640	0.050364	0.040725	2.09911×10⁻⁵	1.0012×10⁻⁸

| Show Table

DownLoad: CSV

Table 9. Comparison of Absolute Error (Abs Err.) of 3^rd order OHAM for

$\underline u (x, \underline r )$ &

$\bar u(x, r)$ at different values of

$\alpha$ taking

$r = 0.5$ for problem 2.

$x$	$\underline u (x, r)$ $\alpha = 0.4$	$\bar u(x, r)$	$\underline u (x, r)$ $\alpha = 0.6$	$\bar u(x, r)$	$\underline u (x, r)$ $\alpha = 0.8$	$\bar u(x, r)$	$\underline u (x, r)$ $\alpha = 1$	$\bar u(x, r)$
0.2	1.2736×10⁻⁵	1.2736×10⁻⁵	3.2527×10⁻⁶	3.2527×10⁻⁶	6.93075×10⁻⁷	6.93076×10⁻⁷	1.29476×10⁻⁷	1.44585×10⁻⁷
0.4	2.3337×10⁻⁶	2.3337×10⁻⁶	2.4426×10⁻⁶	2.4426×10⁻⁶	9.50573×10⁻⁷	9.50573×10⁻⁷	2.67538×10⁻⁷	3.26771×10⁻⁷
0.6	9.9993×10^-6	9.9993×10⁻⁶	1.8451×10⁻⁶	1.8451×10⁻⁶	3.8544×10⁻⁸	3.85438×10⁻⁸	1.10198×10⁻⁷	2.39031×10⁻⁷
0.8	4.1251×10⁻⁶	4.1251×10⁻⁶	1.3416×10⁻⁷	1.3416×10⁻⁷	6.55017×10⁻⁸	6.55017×10⁻⁸	9.64628×10⁻⁹	2.27883×10⁻⁷
1.0	1.312×10⁻⁵	1.312×10⁻⁵	1.8373×10⁻⁶	1.8373×10⁻⁶	5.49341×10⁻⁸	5.4934×10⁻⁸	7.7356×10⁻⁸	3.9735×10⁻⁷

| Show Table

DownLoad: CSV

Table 10. Comparison of Absolute Error (Abs Err.) of 3^rd order OHAM for

$\underline u (x, r)$ &

$\bar u(x, r)$ at different values of

$r$ taking

$\alpha = 0.5$ for problem 2.

$x$	$\underline u (x, r)$ $r = 0.4$	$\bar u(x, r)$	$\underline u (x, r)$ $r = 0.6$	$\bar u(x, r)$	$\underline u (x, r)$ $r = 0.8$	$\bar u(x, r)$	$\underline u (x, r)$ $r = 1$	$\bar u(x, r)$
0.	0.	0.	0.	0.	0.	0.	0.	0.
0.2	1.0578×10⁻⁵	1.0578×10⁻⁵	7.9338×10⁻⁶	7.9338×10⁻⁶	5.28921×10⁻⁶	5.28921×10⁻⁶	2.64461×10⁻⁶	2.64461×10⁻⁶
0.4	4.8940×10⁻⁶	4.8940×10⁻⁶	3.6705×10⁻⁶	3.6705×10⁻⁶	2.44698×10⁻⁶	2.44698×10⁻⁶	1.22349×10⁻⁶	1.22349×10⁻⁶
0.6	7.4825×10⁻⁶	7.4825×10⁻⁶	5.6119×10⁻⁶	5.6119×10⁻⁶	3.74125×10⁻⁶	3.74125×10⁻⁶	1.87062×10⁻⁶	1.87062×10⁻⁶
1.8	1.8038×10⁻⁶	1.8038×10⁻⁶	1.35291×10⁻⁶	1.35291×10⁻⁶	9.01939×10⁻⁷	9.01939×10⁻⁷	4.5097×10⁻⁷	4.5097×10⁻⁷

| Show Table

DownLoad: CSV

Figure 1. Solution plot of OHAM for

$\underline u (x, r)$ &

$\bar u(x, r)$ at different values of

$r$ &

$\alpha$ for problem 1.

DownLoad: Full-Size Img PowerPoint

Figure 2. Solution plot of OHAM and Exact for

$\underline u (x, r)$ &

$\bar u(x, r)$ at different values of

$\alpha$ taking

$r = 0.75$ for problem 1.

DownLoad: Full-Size Img PowerPoint

Figure 3. Solution plot of OHAM and Exact for

$\underline u (x, r)$ &

$\bar u(x, r)$ at different values of

$\alpha$ taking

$r = 0.50$ for problem 1.

DownLoad: Full-Size Img PowerPoint

Figure 4. Solution plot of OHAM for

$\underline u (x, r)$ &

$\bar u(x, r)$ at different values of

$r$ &

$\alpha$ for problem 2.

DownLoad: Full-Size Img PowerPoint

Figure 5. Solution plot of OHAM and Exact for

$\underline u (x, r)$ &

$\bar u(x, r)$ at different values of

$\alpha$ taking

$r = 0.5$ for problem 2.

DownLoad: Full-Size Img PowerPoint

Figure 6. Solution plot of OHAM and Exact for

$\underline u (x, r)$ &

$\bar u(x, r)$ at different values of

$r$ taking

$\alpha = 0.5$ for problem 2.

DownLoad: Full-Size Img PowerPoint

6. Conclusions

In the research paper, a powerful technique known as Optimal Homotopy Asymptotic Method (OHAM) has been extended to the solution of system of fuzzy integro differential equations of fractional order. The obtained results are quite interesting and are in good agreement with the exact solution. Two numerical equations are taken as test examples which show the behavior and reliability of the proposed method. The extension of OHAM to system of fuzzy integro differential equations of fractional order is more accurate and as a result this technique will more appealing for the researchers for finding out optimum solutions of system of fuzzy integro differential equations of fractional order.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	H. Brunner, Collocation methods for Volterra integral and related functional equations, Cambridge: Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511543234
[2]	K. Ikeda, Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system, Opt. Commun., 30 (1979), 257–261. https://doi.org/10.1016/0030-4018(79)90090-7 doi: 10.1016/0030-4018(79)90090-7
[3]	K. Ikeda, H. Daido, O. Akimoto, Optical turbulence: Chaotic behavior of transmitted light from a ring cavity, Phys. Rev. Lett., 45 (1980), 709. https://doi.org/10.1103/PhysRevLett.45.709 doi: 10.1103/PhysRevLett.45.709
[4]	M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, T. Erneux, Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators, Phys. Rev. E., 79 (2009), 026208. https://doi.org/10.1103/PhysRevE.79.026208 doi: 10.1103/PhysRevE.79.026208
[5]	L. Weicker, G. Friart, T. Erneux, Two distinct bifurcation routes for delayed optoelectronic oscillators, Phys. Rev. E., 96 (2017), 032206. https://doi.org/10.1103/PhysRevE.96.032206 doi: 10.1103/PhysRevE.96.032206
[6]	J. Belair, Population models with state-dependent delays, In: Mathematical population dynamics, New York: Marcel Dekker, 1991,165–176. https://doi.org/10.1201/9781003072706-13
[7]	K. L. Cooke, An epidemic equation with immigration, Math. Biosci., 29 (1976), 135–158. https://doi.org/10.1016/0025-5564(76)90033-X doi: 10.1016/0025-5564(76)90033-X
[8]	H. W. Hethcote, P. van den Driesschew, Two SIS epidemiologic models with delays, J. Math. Biol., 40 (2000), 3–26. https://doi.org/10.1007/s002850050003 doi: 10.1007/s002850050003
[9]	A. Bellour, M. Bousselsal, A Taylor collocation method for solving delay integral equations, Numer. Algor., 65 (2014), 843–857. https://doi.org/10.1007/s11075-013-9717-8 doi: 10.1007/s11075-013-9717-8
[10]	H. Brunner, Iterated collocation methods for Volterra integral equations with delay arguments, Math. Comp., 62 (1994), 581–599. https://doi.org/10.2307/2153525 doi: 10.2307/2153525
[11]	H. Brunner, Y. Yatsenko, Spline collocation methods for nonlinear Volterra integral equations with unknown delay, J. Comput. Appl. Math., 71 (1996), 67–81. https://doi.org/10.1016/0377-0427(95)00228-6 doi: 10.1016/0377-0427(95)00228-6
[12]	F. Calio, E. Marchetti, R. Pavani, About the deficient spline collocation method for particular differential and integral equations with delay, Rend. Sem. Mat. Univ. Pol. Torino, 61 (2003), 287–300.
[13]	I. Ali, H. Brunner, T. Tang, Spectral methods for pantograph-type differential and integral equations with multiple delays, Front. Math. China, 4 (2009), 49–61. https://doi.org/10.1007/s11464-009-0010-z doi: 10.1007/s11464-009-0010-z
[14]	V. Horvat, On collocation methods for Volterra integral equations with delay arguments, Math. Commun., 4 (1999), 93–109.
[15]	Q. Y. Hu, Multilevel correction for discrete collocation solutions of Volterra integral equations with delay arguments, Appl. Numer. Math., 31 (1999), 159–171. https://doi.org/10.1016/S0168-9274(98)00127-5 doi: 10.1016/S0168-9274(98)00127-5
[16]	M. Khasi, F. Ghoreishi, M. Hadizadeh, Numerical analysis of a high order method for state-dependent delay integral equations, Numer. Algor., 66 (2014), 177–201. https://doi.org/10.1007/s11075-013-9729-4 doi: 10.1007/s11075-013-9729-4
[17]	P. K. Lamm, Full convergence of sequential local regularization methods for Volterra inverse problems, Inverse Probl., 21 (2005), 785. https://doi.org/10.1088/0266-5611/21/3/001 doi: 10.1088/0266-5611/21/3/001
[18]	T. T. Zhang, H. Liang, Multistep collocation approximations to solutions of first-kind Volterra integral equations, Appl. Numer. Math., 130 (2018), 171–183. https://doi.org/10.1016/j.apnum.2018.04.005 doi: 10.1016/j.apnum.2018.04.005
[19]	S. N. Elaydi, An introduction to difference equations, New York: Springer, 2005. https://doi.org/10.1007/0-387-27602-5
[20]	E. Hairer, C. Lubich, S. P. Nørset, Order of convergence of one-step methods for Volterra integral equations of the second kind, SIAM J. Numer. Anal., 20 (1983), 569–579. https://doi.org/10.1137/0720037 doi: 10.1137/0720037

This article has been cited by:

1.	Tareq Manzoor, S. Iqbal, Mohd Asif Shah, A note on the slip effects of an Oldroyd 6-constant fluid: Optimal homotopy asymptotic method, 2022, 10, 2296-424X, 10.3389/fphy.2022.1003000
2.	Laiq Zada, Rashid Nawaz, Wasim Jamshed, Rabha W. Ibrahim, El Sayed M. Tag El Din, Zehba Raizah, Ayesha Amjad, New optimum solutions of nonlinear fractional acoustic wave equations via optimal homotopy asymptotic method-2 (OHAM-2), 2022, 12, 2045-2322, 10.1038/s41598-022-23644-5
3.	HIMAYAT ULLAH JAN, HAKEEM ULLAH, MEHREEN FIZA, ILYAS KHAN, ABDULLAH MOHAMED, ABD ALLAH A. MOUSA, MODIFICATION OF OPTIMAL HOMOTOPY ASYMPTOTIC METHOD FOR MULTI-DIMENSIONAL TIME-FRACTIONAL MODEL OF NAVIER–STOKES EQUATION, 2023, 31, 0218-348X, 10.1142/S0218348X23400212
4.	RI ZHANG, NEHAD ALI SHAH, ESSAM R. EL-ZAHAR, ALI AKGÜL, JAE DONG CHUNG, NUMERICAL ANALYSIS OF FRACTIONAL-ORDER EMDEN–FOWLER EQUATIONS USING MODIFIED VARIATIONAL ITERATION METHOD, 2023, 31, 0218-348X, 10.1142/S0218348X23400285
5.	Nagwa Saeed, Deepak Pachpatte, Fuzzy Solutions of Fuzzy Fractional Parabolic Integro Differential Equations, 2025, 8, 2619-9653, 81, 10.32323/ujma.1631793

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(1011) PDF downloads(38) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(4) / Tables(6)

AIMS Mathematics

Study of numerical treatment of functional first-kind Volterra integral equations

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Application of OHAM

4. Application and accuracy

5. Result and discussion

6. Conclusions

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

Study of numerical treatment of functional first-kind Volterra integral equations

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Application of OHAM

4. Application and accuracy

5. Result and discussion

6. Conclusions

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog