On New Picard-Mann Iterative Approximations with Mixed Errors for Implicit Midpoint Rule and Applications by Teng-fei Li, Heng-you Lan

“…In order to solve (partial) differential equations, implicit midpoint rules are often employed as a powerful numerical method. The purpose of this paper is to introduce and study a class of new Picard-Mann iteration processes with mixed errors for the implicit midpoint rules, which is different from existing methods in the literature, and to analyze the convergence and stability of the proposed method. …”
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Analysis of stochastic pantograph differential equations with generalized derivative of arbitrary order by Devaraj Vivek, Elsayed M. Elsayed, Kuppusamy Kanagarajan

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Approximation of the Fixed Point of Multivalued Quasi-Nonexpansive Mappings via a Faster Iterative Process with Applications by Godwin Amechi Okeke, Mujahid Abbas, Manuel de la Sen

“…In this paper, we approximate the fixed points of multivalued quasi-nonexpansive mappings via a faster iterative process and propose a faster fixed-point iterative method for finding the solution of two-point boundary value problems. …”
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Euler Operational Matrix of Integration and Collocation Method for Solving Functional Integral Equations by Sohrab Bazm, Fatemeh Pahlevani

“…First, some conditions that ensure the existence and uniqueness of the solutions to these equations within the space of square-integrable functions are established and then the Euler operational matrix of integration is constructed and applied within the collocation method for approximating the solutions. This approach transforms the integral equation into a set of nonlinear algebraic equations, which can be efficiently solved by employing standard numerical methods like Newton's method or Picard iteration. …”
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Newtons method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion by Henryk Leszczyński, Monika Wrzosek

“…The existence of solutions is proved by means of Picard iterations. Next we apply Newton's method. …”
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On the Existence and Uniqueness Results for Fuzzy Linear and Semilinear Fractional Evolution Equations Involving Caputo Fractional Derivative by Ali El Mfadel, Said Melliani, M’hamed Elomari

“…The existence theorems are proved by using fuzzy fractional calculus, Picard’s iteration method, and Banach contraction principle. …”
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