Fredholm and Volterra Integral Equations
الكلمات المفتاحية:
Fredholm integral equations, Volterra integral equations, Fredholm integral-differential equation, Integro-differential equations
الملخص
We will focus on Fredholm and Volterra integral equations. We have examined the development of integral equations that has significant applicability in physical problems. A multitude of initial and boundary value issues can be converted into integral equations. Mathematical physics problems are typically regulated by integral equations. There are several categories of integral equations. Singular integral equations are highly beneficial in numerous physical issues, including elasticity, fluid mechanics, and electromagnetic theory.
المراجع
1. A domain, G.Solving Frontier Problem: The Decomposition Method, Kluwer:
Dordrecht, 1994.
2. A Domain, G. Nonlinear Stochastic Operator Equations, Academic Press:
3. San Diego, CA, 1986.
4. A domain, G. A review of the decomposition method and some recent results
5. for nonlinear equation, Math. Compute. Modelling,13(7), pp. 17–43, 1992.
6. A Domain, G. & Rach, R. Noise terms in decomposition series solution,
7. Computers Math. Appl.,24(11), pp. 61–64, 1992.
8. 5. Abramowitz, M. & Stegun, I.E., Handbook of Mathematical Functions,
9. U.S. Department of Commerce, National Bureau of Standards, Applied Mathematics Series-55, 1970.
10. Rahman, M., Mathematical Methods with Applications, WIT Press:
11. Southampton, UK, pp. 456, 2000.
12. Gripenberg, G., Londen, S.-O., & Staffans, O. (1990). Volterra Integral and Functional Equations. Cambridge University Press.
13. A comprehensive resource covering theoretical aspects, stability, and applications of Volterra equations.
14. Lakshmikantham, V., & Vatsala, A. S. (1995). Theory of Integro-Differential Equations. World Scientific.
15. Focuses on existence, uniqueness, and stability theories for integro-differential equations.
16. Brunner, H. (2004). Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press.
17. Discusses numerical methods for solving Volterra integro-differential equations.
18. Diethelm, K. (2010). The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer.
19. Covers fractional-order Volterra equations and their applications.
20. Corduneanu, C. (2008). Integral Equations and Applications. Cambridge University Press.
21. Provides insights into Volterra integral and integro-differential equations.
22. Travis, C. C., & Webb, G. F. (1974). Existence and Stability for Volterra Integro-Differential Equations in Banach Spaces. Transactions of the American Mathematical Society, 192(1), 185-202.
23. Discusses the existence, uniqueness, and stability of solutions in Banach spaces.
24. Grünwald, A., & Letnikov, S. (1867). On Fractional Differentiation. Journal of Applied Mathematics.
25. One of the foundational papers introducing fractional calculus, relevant for fractional Volterra equations.
26. Podlubny, I. (1999). Fractional Differential Equations. Mathematics in Science and Engineering, 198.
27. A key reference for fractional Volterra equations, stability, and numerical solutions.
28. Brunner, H., & van der Houwen, P. J. (1986). The Numerical Solution of Volterra Equations. North-Holland Mathematics Studies.
29. An essential guide to numerical techniques for solving Volterra integro-differential equations.
30. Grönwall, T. H. (1919). Note on the Derivatives with Respect to a Parameter of the Solutions of a System of Differential Equations. Annals of Mathematics, 20(4), 292-296.
31. Introduces Grönwall’s inequality, a fundamental tool for analyzing Volterra equations.
Dordrecht, 1994.
2. A Domain, G. Nonlinear Stochastic Operator Equations, Academic Press:
3. San Diego, CA, 1986.
4. A domain, G. A review of the decomposition method and some recent results
5. for nonlinear equation, Math. Compute. Modelling,13(7), pp. 17–43, 1992.
6. A Domain, G. & Rach, R. Noise terms in decomposition series solution,
7. Computers Math. Appl.,24(11), pp. 61–64, 1992.
8. 5. Abramowitz, M. & Stegun, I.E., Handbook of Mathematical Functions,
9. U.S. Department of Commerce, National Bureau of Standards, Applied Mathematics Series-55, 1970.
10. Rahman, M., Mathematical Methods with Applications, WIT Press:
11. Southampton, UK, pp. 456, 2000.
12. Gripenberg, G., Londen, S.-O., & Staffans, O. (1990). Volterra Integral and Functional Equations. Cambridge University Press.
13. A comprehensive resource covering theoretical aspects, stability, and applications of Volterra equations.
14. Lakshmikantham, V., & Vatsala, A. S. (1995). Theory of Integro-Differential Equations. World Scientific.
15. Focuses on existence, uniqueness, and stability theories for integro-differential equations.
16. Brunner, H. (2004). Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press.
17. Discusses numerical methods for solving Volterra integro-differential equations.
18. Diethelm, K. (2010). The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer.
19. Covers fractional-order Volterra equations and their applications.
20. Corduneanu, C. (2008). Integral Equations and Applications. Cambridge University Press.
21. Provides insights into Volterra integral and integro-differential equations.
22. Travis, C. C., & Webb, G. F. (1974). Existence and Stability for Volterra Integro-Differential Equations in Banach Spaces. Transactions of the American Mathematical Society, 192(1), 185-202.
23. Discusses the existence, uniqueness, and stability of solutions in Banach spaces.
24. Grünwald, A., & Letnikov, S. (1867). On Fractional Differentiation. Journal of Applied Mathematics.
25. One of the foundational papers introducing fractional calculus, relevant for fractional Volterra equations.
26. Podlubny, I. (1999). Fractional Differential Equations. Mathematics in Science and Engineering, 198.
27. A key reference for fractional Volterra equations, stability, and numerical solutions.
28. Brunner, H., & van der Houwen, P. J. (1986). The Numerical Solution of Volterra Equations. North-Holland Mathematics Studies.
29. An essential guide to numerical techniques for solving Volterra integro-differential equations.
30. Grönwall, T. H. (1919). Note on the Derivatives with Respect to a Parameter of the Solutions of a System of Differential Equations. Annals of Mathematics, 20(4), 292-296.
31. Introduces Grönwall’s inequality, a fundamental tool for analyzing Volterra equations.
منشور
2025-03-23
كيفية الاقتباس
Dr. Raga Hassan Ali Shiekh. (2025). Fredholm and Volterra Integral Equations. Journal of Arts, Literature, Humanities and Social Sciences, (118), 383-392. https://doi.org/10.33193/JALHSS.118.2025.1386
إصدار
القسم
المقالات
الحقوق الفكرية (c) 2025 Dr. Raga Hassan Ali Shiekh
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