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Browsing by Author "Mishra, Lakshmi Narayan"

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    An e ffective method for solving nonlinear integral equations involving the Riemann-Liouville fractional operator
    (2023) Baleanu, Dumitru; Mishra, Lakshmi Narayan; Mishra, Vishnu Narayan; Baleanu, Dumitru; 56389
    In this paper, under some conditions in the Banach space C([0; beta];R), we establish the existence and uniqueness of the solution for the nonlinear integral equations involving the Riemann-Liouville fractional operator (RLFO). To establish the requirements for the existence and uniqueness of solutions, we apply the Leray-Schauder alternative and Banach's fixed point theorem. We analyze Hyers-Ulam-Rassias (H-U-R) and Hyers-Ulam (H-U) stability for the considered integral equations involving the RLFO in the space C([0; beta];R). Also, we propose an e ffective and e fficient computational method based on Laguerre polynomials to get the approximate numerical solutions of integral equations involving the RLFO. Five examples are given to interpret the method.
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    An effective method for solving nonlinear integral equations involving the Riemann-Liouville fractional operator
    (2023) Baleanu, Dumitru; Mishra, Lakshmi Narayan; Mishra, Vishnu Narayan; Baleanu, Dumitru; 56389
    In this paper, under some conditions in the Banach space C([0, β], R), we establish the existence and uniqueness of the solution for the nonlinear integral equations involving the Riemann-Liouville fractional operator (RLFO). To establish the requirements for the existence and uniqueness of solutions, we apply the Leray-Schauder alternative and Banach’s fixed point theorem. We analyze Hyers-Ulam-Rassias (H-U-R) and Hyers-Ulam (H-U) stability for the considered integral equations involving the RLFO in the space C([0, β], R). Also, we propose an effective and efficient computational method based on Laguerre polynomials to get the approximate numerical solutions of integral equations involving the RLFO. Five examples are given to interpret the method.
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    Analysis of mixed type nonlinear Volterra–Fredholm integral equations involving the Erdélyi–Kober fractional operator
    (2023) Baleanu, Dumitru; Mishra, Lakshmi Narayan; Mishra, Vishnu Narayan; Baleanu, Dumitru; 56389
    This paper investigates the existence, uniqueness and stability of solutions to the nonlinear Volterra–Fredholm integral equations (NVFIE) involving the Erdélyi–Kober (E–K) fractional integral operator. We use the Leray–Schauder alternative and Banach's fixed point theorem to examine the existence and uniqueness of solutions, and we also explore Hyers–Ulam (H–U) and Hyers–Ulam–Rassias (H–U–R) stability in the space C([0,β],R). Furthermore, three solution sets Uσ,λ, Uθ,1 and U1,1 are constructed for σ>0, λ>0, and θ∈(0,1), and then we obtain local stability of the solutions with some ideal conditions and by using Schauder fixed point theorem on these three sets, respectively. Also, to achieve the goal, we choose the parameters for the NVFIE as δ∈( [Formula presented], 1), ρ∈(0,1), γ>0. Three examples are provided to clarify the results.
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    Approximation of solutions for nonlinear functional integral equations
    (2022) Baleanu, Dumitru; Pathak, Vijai Kumar; Baleanu, Dumitru; 56389
    In this article, we consider a class of nonlinear functional integral equations, motivated by an equation that offers increasing evidence to the extant literature through replication studies. We investigate the existence of solution for nonlinear functional integral equations on Banach space C[0, 1]. We use the technique of the generalized Darbo’s fixed-point theorem associated with the measure of noncompactness (MNC) to prove our existence result. Also, we have given two examples of the applicability of established existence result in the theory of functional integral equations. Further, we construct an efficient iterative algorithm to compute the solution of the first example, by employing the modified homotopy perturbation (MHP) method associated with Adomian decomposition. Moreover, the condition of convergence and an upper bound of errors are presented.
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    On the Solvability of Mixed-Type Fractional-Order Non-Linear Functional Integral Equations in the Banach Space C(I)
    (2022) Baleanu, Dumitru; Mishra, Lakshmi Narayan; Mishra, Vishnu Narayan; Baleanu, Dumitru; 56389
    This paper is concerned with the existence of the solution to mixed-type non-linear fractional functional integral equations involving generalized proportional ((Formula presented.))-Riemann–Liouville along with Erdélyi–Kober fractional operators on a Banach space (Formula presented.) arising in biological population dynamics. The key findings of the article are based on theoretical concepts pertaining to the fractional calculus and the Hausdorff measure of non-compactness (MNC). To obtain this goal, we employ Darbo’s fixed-point theorem (DFPT) in the Banach space. In addition, we provide two numerical examples to demonstrate the applicability of our findings to the theory of fractional integral equations