Skip to main content

An efficient method for solving neutrosophic Fredholm integral equations of second kind

Granular Computing volume 8, pages 1–22 (2023)Cite this article

Abstract

In this article, a numerical method for the solution of neutrosophic Fredholm integral equation has been investigated. In addition, the neutrosophic Fredholm integral equation has been presented in the sense of \((\alpha ,\beta ,\gamma )-\)cut using Riemann integration approach. Some basic properties of neutrosophic calculus such as neutrosophic integral, neutrosophic continuity have been introduced. An iterative method has been modified in neutrosophic environment to find the numerical solution of Fredholm integral equation of second kind. The convergence of the iterative method in neutrosophic environment has been demonstrated in terms of some theorems. In the iterative method, trapezoidal rule has been used to evaluate the integral and find the approximate solution of the equation. In addition, the convergence of the trapezoidal approximations has been provided in terms of theorem. The algorithm of the proposed method has been given in the numerical method section, which briefly helps to understand the proposed method. A comparison of our method with other existing methods has been discussed to show the efficiency and reliability of our proposed method. In addition, a brief discussion about the advantages and limitations of our method has been provided. Some numerical examples have been examined to show the validation and effectiveness of the proposed method. In addition, different types of error analysis have been investigated by providing different types of tables and figures.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

References

  • Agarwal P, Ramadan M, Osheba HS, Chu YM (2020) Study of hybrid orthonormal functions method for solving second kind fuzzy fredholm integral equations. Adv in Differ Equ 1:1–14

    MathSciNet  MATH  Google Scholar 

  • Agboola A, Akinleye S (2014) Neutrosophic vector spaces. Neutros Sets Syst 4:9–18

    Google Scholar 

  • Agboola AAA, Akwu AD, Oyebo YT (2012) Neutrosophic groups and subgroups. Math Combin 3:1–9

    Google Scholar 

  • Alhasan YA (2021) The neutrosophic integrals and integration methods. Neutros Sets Syst 43:290–301

    Google Scholar 

  • Bertone AM, Jafelice RM, de Barros LC, Gomide F (2018) Granular approximation of solutions of partial differential equations with fuzzy parameter. Gran Comp 3(1):1–7

    Article  Google Scholar 

  • Biswas S, Roy TK (2018) Adomian decomposition method for solving initial value problem for fuzzy integro-differential equation with an application in volterra’s population model. J Fuzzy Math 26(1):69–88

    MATH  Google Scholar 

  • Biswas S, Roy TK (2018) Generalization of seikkala derivative and differential transform method for fuzzy volterra integro-differential equations. J Intel & Fuzzy Syst 34(4):2795–2806

    Article  Google Scholar 

  • Biswas S, Roy TK (2019) A semianalytical method for fuzzy integro-differential equations under generalized seikkala derivative. Soft Comp 23(17):7959–7975

    Article  MATH  Google Scholar 

  • Biswas S, Moi S, Pal S (2021a) Study of interval type-2 fuzzy singular integro-differential equation by using collocation method in weighted space. New Mathematics and Natural Computation

  • Biswas S, Moi S, Sarkar SP (2021) Neutrosophic Riemann integration and its properties. Soft Comp 25(22):13987–13999

    Article  Google Scholar 

  • Biswas S, Moi S, Sarkar SP (2021) Numerical solution of fuzzy fredholm integro-differential equations by polynomial collocation method. Comp Appl Math 40(7):1–33

    Article  MathSciNet  MATH  Google Scholar 

  • Chen SM (1997) Interval-valued fuzzy hypergraph and fuzzy partition. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) 27(4):725–733

  • Chen SM, Hong JA (2014) Fuzzy multiple attributes group decision-making based on ranking interval type-2 fuzzy sets and the topics method. IEEE Trans Syst Man Cybern Syst 44(12):1665–1673

    Article  Google Scholar 

  • Chen SM, Hsiao WH (2000) Bidirectional approximate reasoning for rule-based systems using interval-valued fuzzy sets. Fuzzy Sets Syst 113(2):185–203

    Article  MathSciNet  MATH  Google Scholar 

  • Chen SM, Hsiao WH, Jong WT (1997) Bidirectional approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets Syst 91(3):339–353

    Article  MathSciNet  MATH  Google Scholar 

  • Chen SM, Chang YC, Pan JS (2012) Fuzzy rules interpolation for sparse fuzzy rule-based systems based on interval type-2 gaussian fuzzy sets and genetic algorithms. IEEE Trans Fuzzy Syst 21(3):412–425

    Article  Google Scholar 

  • Dey A, Broumi S, Son LH, Bakali A, Talea M, Smarandache F (2019) A new algorithm for finding minimum spanning trees with undirected neutrosophic graphs. Gran Comp 4(1):63–69

    Article  Google Scholar 

  • Fahmi A, Aslam M, Riaz M (2020) New approach of triangular neutrosophic cubic linguistic hesitant fuzzy aggregation operators. Gran Comp 5(4):527–543

    Article  Google Scholar 

  • Friedman M, Ming M, Kandel A (1999) Solutions to fuzzy integral equations with arbitrary kernels. Int J Approx Reas 20(3):249–262

    Article  MathSciNet  MATH  Google Scholar 

  • Ishii K, Sugeno M (1985) A model of human evaluation process using fuzzy measure. Int J Man-Machine Stud 22(1):19–38

    Article  MATH  Google Scholar 

  • Liu J, Chen Z, Chen Y, Zhang Y, Li C (2021) Multiattribute group decision making based on interval-valued neutrosophic N-soft sets. Gran Comp 6(4):1009–1023

    Article  Google Scholar 

  • Ma M, Friedman M, Kandel A (1999) Numerical solutions of fuzzy differential equations. Fuzzy Sets Syst 105(1):133–138

    Article  MathSciNet  MATH  Google Scholar 

  • Mendel JM, John RI, Liu F (2006) Interval type-2 fuzzy logic systems made simple. IEEE Trans Fuzzy Syst 14(6):808–821

    Article  Google Scholar 

  • Moi S, Biswas S, Pal S (2021a) Neutrosophic linear differential equation with a new concept of neutrosophic derivative. In: Neutrosophic Operational Research, Springer, pp 395–410

  • Moi S, Biswas S, Pal S (2021b) Second-order neutrosophic boundary-value problem. Complex Intel Syst 7(2):1079–1098

    Article  Google Scholar 

  • Mondal SP, Goswami A, Kumar De S (2019) Nonlinear triangular intuitionistic fuzzy number and its application in linear integral equation. Adv Fuzzy Syst

  • Nouriani H, Ezzati R (2020) Application of simpson quadrature rule and iterative method for solving nonlinear fuzzy delay integral equations. Fuzzy Sets Syst 400:147–161

    Article  MathSciNet  MATH  Google Scholar 

  • Nouriani H, Ezzati R, Gholam AM (2021) Numerical solution of two-dimensional nonlinear fuzzy delay integral equations via iterative method and trapezoidal quadrature rule. Gran Comp 6(4):829–851

    Article  Google Scholar 

  • Rashidinia J, Maleknejad K, Jalilian H (2020) Convergence analysis of non-polynomial spline functions for the fredholm integral equation. Int J Computer Math 97(6):1197–1211

    Article  MathSciNet  MATH  Google Scholar 

  • Şahin M, Uluçay V, Menekşe M (2018) Some New Operations of (\(\alpha\), \(\beta\), \(\gamma\)) Interval Cut Set of Interval Valued Neutrosophic Sets. J Math Fundam Sci 50(2)

  • Salama A, Alblowi S (2012) Neutrosophic set and neutrosophic topological spaces. IOSR J Math 3(4):31–35

    Article  Google Scholar 

  • Shabir M, Ali M, Naz M, Smarandache F (2013) Soft neutrosophic group. Neutros Sets Syst 1:13–25

    Google Scholar 

  • Shiri B, Perfilieva I, Alijani Z (2021) Classical approximation for fuzzy fredholm integral equation. Fuzzy Sets Syst 404:159–177

    Article  MathSciNet  MATH  Google Scholar 

  • Singh PK (2020) Multi-granular-based n-valued neutrosophic context analysis. Gran Comp 5(3):287–301

    Article  Google Scholar 

  • Smarandache F (2005) Neutrosophic set-a generalization of the intuitionistic fuzzy set. Int J Pure Appl Math 24(3):287

    MathSciNet  MATH  Google Scholar 

  • Smarandache F (2013) Introduction to neutrosophic measure, neutrosophic integral, and neutrosophic probability. Infinite Study

  • Smarandache F, Khalid HE (2018) Neutrosophic precalculus and neutrosophic calculus. Infinite Study

  • Son NTK, Dong NP, Long HV, Khastan A et al (2020) Linear quadratic regulator problem governed by granular neutrosophic fractional differential equations. ISA Trans 97:296–316

    Article  Google Scholar 

  • Sumathi I, Priya VM (2018) A new perspective on neutrosophic differential equation. Infinite Study

  • Sumathi I, Sweety CAC (2019) New approach on differential equation via trapezoidal neutrosophic number. Complex Intel Syst 5(4):417–424

    Article  Google Scholar 

  • Turksen IB (1986) Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst 20(2):191–210

    Article  MathSciNet  MATH  Google Scholar 

  • Ullah Z, Ullah A, Shah K, Baleanu D (2020) Computation of semi-analytical solutions of fuzzy nonlinear integral equations. Adv Differ Equ 1:1–11

    MathSciNet  MATH  Google Scholar 

  • Uluçay V (2021) Some concepts on interval-valued refined neutrosophic sets and their applications. J Amb Intell Hum Comp 12(7):7857–7872

    Article  Google Scholar 

  • Vasantha Kandasamy W, Smarandache F (2006) Neutrosophic rings. arXiv Mathematics e-prints pp math–0607765

  • Wang H, Smarandache F, Zhang Y, Sunderraman R (2010) Single valued neutrosophic sets, multispace and multistructure, 4

  • Yucesan M, Gul M (2021) Failure prioritization and control using the neutrosophic best and worst method. Gran Comp 6(2):435–449

    Article  Google Scholar 

  • Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning-i. Inform Sci 8(3):199–249

    Article  MathSciNet  MATH  Google Scholar 

  • Zulqarnain RM, Xin XL, Saqlain M, Saeed M, Smarandache F, Ahamad MI (2021) Some fundamental operations on interval valued neutrosophic hypersoft set with their properties. Neutros Sets Syst 40:134–148

    Google Scholar 

Download references

Acknowledgements

In this work, the study of Sandip Moi is funded by Council of Scientific and Industrial Research (CSIR), Government of India (File No.- 08/003(0135)/2019-EMR-I).

Author information

Authors and Affiliations

  1. Mathematics Discipline, School of Sciences, Indira Gandhi National Open University, New Delhi, 110068, India

    Suvankar Biswas

  2. Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, B. Garden, Howrah, 711103, India

    Sandip Moi & Smita Pal Sarkar

Authors
  1. Sandip Moi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suvankar Biswas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Smita Pal Sarkar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Suvankar Biswas.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest regarding the publication of this research article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Moi, S., Biswas, S. & Sarkar, S.P. An efficient method for solving neutrosophic Fredholm integral equations of second kind. Granul. Comput. 8, 1–22 (2023). https://doi.org/10.1007/s41066-021-00310-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41066-021-00310-1

Keywords

  • Neutrosophic integration
  • Neutrosophic integral equation
  • Fredholm integral equation
  • Trapezoidal rule
  • Numerical integration