Existence and uniqueness of solutions for nonlinear Volterra‐Fredholm integro‐differential equation of fractional order with boundary conditions

Laadjal, Zaid; Ma, Qingwei

doi:10.1002/mma.5845

Cited by 10 publications

(6 citation statements)

References 14 publications

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“…The differential equations, specially a fractional and ordinary, have motivated a number of researchers to explore the topic's theoretical and practical aspects. For more details, see previous works, 21,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] and the reader can also look up the references cited in the papers.…”

Section: 𝜇([0 𝜍]) = 𝜇({0}) + ∫ [0𝜍]mentioning

confidence: 99%

Existence and uniqueness of solutions of differential equations with respect to non‐additive measures

Negi

Torra

2022

Math Methods in App Sciences

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Funding informationKempe foundation (Kempestiftelserna 891 80 Örnsköldsvik); Knut and Alice Wallenberg Foundation By taking Sugeno-derivative into account, first, we investigate the existence of solutions to the initial value problems (IVP) of first-order differential equations with respect to non-additive measure (more precisely, distorted Lebesgue measure). It particularly occurs in the mathematical modeling of biology. We begin by expressing the differential equation in terms of ordinary derivative and the derivative with respect to the distorted Lebesgue measure. Then, by using the fixed point theorem on cones, we construct an operator and prove the existence of positive non-decreasing solutions on cones in semi-order Banach spaces. In addition, we also use Picard-Lindelöf theorem to prove the existence and uniqueness of the solution of the equation.Second, we investigate the existence of a solution to the boundary value problem (BVP) on cones with integral boundary conditions of a mix-order differential equation with respect to non-additive measures. Moreover, the Krasnoselskii fixed point theorem is also applied to both BVP and IVP and obtains at least one positive non-decreasing solution. Examples with graphs are provided to validate the results.

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Section: 𝜇([0 𝜍]) = 𝜇({0}) + ∫ [0𝜍]mentioning

confidence: 99%

Existence and uniqueness of solutions of differential equations with respect to non‐additive measures

Negi

Torra

2022

Math Methods in App Sciences

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“…Te existence and uniqueness of the solution of such equations have been widely studied in many papers, as can be seen in [11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Theoretical and Numerical Study for Volterra−Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind

2023

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In this paper, we develop an efficient numerical method to approximate the solution of fractional integro-differential equations (FI-DEs) of mixed Volterra−Fredholm type using spectral collocation method with shifted Chebyshev polynomials of the third kind (S-Cheb-3). The fractional derivative is described in the Caputo sense. A Chebyshev−Gauss quadrature is involved to evaluate integrals for more precision. Two types of equations are studied to obtain algebraic systems solvable using the Gauss elimination method for linear equations and the Newton algorithm for nonlinear ones. In addition, an error analysis is carried out. Six numerical examples are evaluated using different error values (maximum absolute error, root mean square error, and relative error) to compare the approximate and the exact solutions of each example. The experimental rate of convergence is calculated as well. The results validate the numerical approach’s efficiency, applicability, and performance.

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“…Indeed, many techniques are always used to prove the existance of solutions for ordinary and fractional equations. The readers can be referred to [6,7,13,14], but here we interest on using the topological degree theory. This methode is an effective tool for the existance of solutions to boundary value problems (BVPs for short).…”

Section: Introductionmentioning

confidence: 99%

Untitled

2022

Adv. Math., Sci. J.

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Existence and uniqueness of solutions for nonlinear Volterra‐Fredholm integro‐differential equation of fractional order with boundary conditions

Cited by 10 publications

References 14 publications

Existence and uniqueness of solutions of differential equations with respect to non‐additive measures

Existence and uniqueness of solutions of differential equations with respect to non‐additive measures

Theoretical and Numerical Study for Volterra−Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind

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