Search Results (73)

In this article, we introduce and study a novel class of polynomial $\phi$-contractions, which simultaneously generalizes classical polynomial contractions and $\phi$-contractions within a unified framework. We establish generalized fixed point theorems that encompass some results in the existing literature. Furthermore, we derive explicit error estimates and convergence rates for the associated Picard iteration, providing practical insights into the speed of convergence. Several illustrative examples, including higher-degree polynomial contractions, demonstrate the scope and applicability of our results. As an application, we prove existence and uniqueness results for solutions of a class of fractional logistic growth equations, highlighting the relevance of our theoretical contributions to nonlinear analysis and applied mathematics. Full article

This work aims to propose an iterative method for approximating solutions of two-dimensional weakly singular Fredholm integral Equation (2D-WSFIE) by incorporating the product integration technique, an appropriate cubature formula, and the Picard algorithm. This iterative approach is utilized to approximate the solution of the 2D-WSFIE that arises in some contact problems in linear elasticity. Under some sufficient conditions, the existence and uniqueness of the solution are established, an error bound for the approximate solution is estimated, and the order of convergence of the proposed algorithm is discussed. The effectiveness of the proposed method is illustrated through its application to some contact problems involving weakly singular kernels. Full article

In this paper, we are studying a class of nonlinear fractional difference equations with time-varying delays in Banach space. By means of mathematical induction and the Picard iteration method, we first obtain the existence result of this fractional difference system. Under some new criteria along with the Schauder’s fixed point theorem, we then derive the attractivity conclusions. Subsequently, with the aid of Grönwall’s inequality, we prove that the system is globally attractive. Finally, we give two examples to prove the validity of our theorems. Full article

In this paper, we introduce a fast iterative scheme and establish its convergence under a contractive condition. This new scheme can be viewed as an extension and generalization of existing iterative schemes such as Picard–Noor and UO iterative schemes for solving nonlinear equations. We demonstrate theoretically and numerically that the new scheme converges faster than several existing iterative schemes with the fastest known convergence rates for contractive mappings. We also analyze the stability of the new scheme and provide numerical computations to validate the analytic results. Finally, we implement the new scheme in MATLAB R2023b to simulate the dynamics of the Ebola virus disease. Full article

This paper investigates a class of distributed fractional-order stochastic differential equations driven by fractional Brownian motion with a Hurst parameter $1/2<H<1$. By employing the Picard iteration method, we rigorously prove the existence and uniqueness of solutions with Lipschitz conditions. Furthermore, leveraging the Girsanov transformation argument within the $L^2$ metric framework, we derive quadratic transportation inequalities for the law of the strong solution to the considered equations. These results provide a deeper understanding of the regularity and probabilistic properties of the solutions in this framework. Full article

This paper introduces a novel Picard-type iterative algorithm for solving general variational inequalities in real Hilbert spaces. The proposed algorithm enhances both the theoretical framework and practical applicability of iterative algorithms by relaxing restrictive conditions on parametric sequences, thereby expanding their scope of use. We establish convergence results, including a convergence equivalence with a previous algorithm, highlighting the theoretical relationship while demonstrating the increased flexibility and efficiency of the new approach. The paper also addresses gaps in the existing literature by offering new theoretical insights into the transformations associated with variational inequalities and the continuity of their solutions, thus paving the way for future research. The theoretical advancements are complemented by practical applications, such as the adaptation of the algorithm to convex optimization problems and its use in real-world contexts like machine learning. Numerical experiments confirm the proposed algorithm’s versatility and efficiency, showing superior performance and faster convergence compared to an existing method. Full article

This paper aims to establish several fixed-point theorems within the framework of Banach spaces endowed with a binary relation. By utilizing enriched contraction principles involving two classes of altering-distance functions, the study encompasses various types of contractive mappings, including theoretic-order contractions, Picard–Banach contractions, weak contractions, and non-expansive contractions. A suitable Krasnoselskij iterative scheme is employed to derive the results. Many well-known fixed-point theorems (FPTs) can be obtained as special cases of these findings by assigning specific control functions in the main definitions or selecting an appropriate binary relation. To validate the theoretical results, numerous illustrative examples are provided. Furthermore, the paper demonstrates the applicability of the findings through applications to ordinary differential equations. Full article

The purpose of this paper is to construct a new fixed-point iterative scheme, called the Picard-like iterative scheme, for approximating the fixed point of a mapping that satisfies condition $(B_{\gamma ,\mu })$ in the setting of a uniformly convex Banach space. We prove that this novel iterative scheme converges faster than some existing iterative schemes in the literature. Moreover, $\mathcal{G}$-stability and almost $\mathcal{G}$-stability results are proven. Furthermore, we apply our results for approximating the solution of an integral equation that models the spread of some infectious diseases. Similarly, we also applied the results for approximating the solution of the boundary value problem by embedding Green’s function in our novel method. Our results extend and generalize other existing results in the literature. Full article

The boundary shape function method (BSFM) and the variational iteration method (VIM) are merged together to seek the analytic solutions of nonlinear boundary value problems. The boundary shape function method transforms the boundary value problem to an initial value problem (IVP) for a new variable. Then, a modified variational iteration method (MVIM) is created by applying the VIM to the resultant IVP, which can achieve a good approximate solution to automatically satisfy the prescribed mixed-boundary conditions. By using the Picard iteration method, the existence of a solution is proven with the assumption of the Lipschitz condition. The MVIM is equivalent to the Picard iteration method by a back substitution. Either by solving the nonlinear equations or by minimizing the error of the solution or the governing equation, we can determine the unknown values of the parameters in the MVIM. A nonlocal BSFM is developed, which then uses the MVIM to find the analytic solution of a nonlocal nonlinear boundary value problem. In the second part of this paper, a new splitting–linearizing method is developed to expand the analytic solution in powers of a dummy parameter. After adopting the Liapunov method, linearized differential equations are solved sequentially to derive an analytic solution. Accurate analytical solutions are attainable through a few computations, and some examples involving two boundary layer problems confirm the efficiency of the proposed methods. Full article

In this work, we are devoted to discussing a system of fractional stochastic differential variational inequalities with Lévy jumps (SFSDVI with Lévy jumps), that comprises both parts, that is, a system of stochastic variational inequalities (SSVI) and a system of fractional stochastic differential equations(SFSDE) with Lévy jumps. Here it is noteworthy that the SFSDVI with Lévy jumps consists of both sections that possess a mutual symmetry structure. Invoking Picard’s successive iteration process and projection technique, we obtain the existence of only a solution to the SFSDVI with Lévy jumps via some appropriate restrictions. In addition, the major outcomes are invoked to deduce that there is only a solution to the spatial-price equilibria system in stochastic circumstances. The main contributions of the article are listed as follows: (a) putting forward the SFSDVI with Lévy jumps that could be applied for handling different real matters arising from varied domains; (b) deriving the unique existence of solutions to the SFSDVI with Lévy jumps under a few mild assumptions; (c) providing an applicable instance for spatial-price equilibria system in stochastic circumstances affected with Lévy jumps and memory. Full article

In the paper, two new analytic methods using the decomposition and linearization technique on nonlinear differential/integral equations are developed, namely, the decomposition–linearization–sequential method (DLSM) and the linearized homotopy perturbation method (LHPM). The DLSM is realized by an integrating factor and the integral of certain function obtained at the previous step for obtaining a sequential analytic solution of nonlinear differential equation, which provides quite accurate analytic solution. Some first- and second-order nonlinear differential equations display the fast convergence and accuracy of the DLSM. An analytic approximation for the Volterra differential–integral equation model of the population growth of a species is obtained by using the LHPM. In addition, the LHPM is also applied to the first-, second-, and third-order nonlinear ordinary differential equations. To reduce the cost of computation of He’s homotopy perturbation method and enhance the accuracy for solving cubically nonlinear jerk equations, the LHPM is implemented by invoking a linearization technique in advance is developed. A generalization of the LHPM to the nth-order nonlinear differential equation is involved, which can greatly simplify the work to find an analytic solution by solving a set of second-order linear differential equations. A remarkable feature of those new analytic methods is that just a few steps and lower-order approximations are sufficient for producing reasonably accurate analytic solutions. For all examples, the second-order analytic solution $x_2\left(t\right)$ is found to be a good approximation of the real solution. The accuracy of the obtained approximate solutions are identified by the fourth-order Runge–Kutta method. The major objection is to unify the analytic solution methods of different nonlinear differential equations by simply solving a set of first-order or second-order linear differential equations. It is clear that the new technique considerably saves computational costs and converges faster than other analytical solution techniques existing in the literature, including the Picard iteration method. Moreover, the accuracy of the obtained analytic solution is raised. Full article

A free-boundary equilibrium solver for an axisymmetric tokamak geometry was developed based on the finite difference method and Picard iteration in a rectangular computational area. The solver can run either in forward mode, where external coil currents are prescribed until the converged magnetic flux function $\psi (R,Z)$ map is achieved, or in inverse mode, where the desired plasma boundary, with or without an X-point, is prescribed to determine the required coil currents. The equilibrium solutions are made consistent with prescribed plasma parameters, such as the total plasma current, poloidal beta, or safety factor at a specified flux surface. To verify the mathematical correctness and accuracy of the solver, the solution obtained using this numerical solver was compared with that from an analytic fixed-boundary equilibrium solver based on the EAST geometry. Additionally, the proposed solver was benchmarked against another numerical solver based on the finite-element and Newton-iteration methods in a triangular-based mesh. Finally, the proposed solver was compared with equilibrium reconstruction results in DIII-D experiments. Full article

Pore-scale flow velocity is an essential parameter in determining transport through porous media, but it is often miscalculated. Researchers use a static porosity value to relate volumetric or superficial velocities to pore-scale flow velocities. We know this modeling assumption to be an oversimplification. The variable fraction of porosity conducive to flow, what we define as hydrodynamic porosity, ${\theta }_{mobile}$, exhibits a quantifiable dependence on the Reynolds number (i.e., pore-scale flow velocity) in the Laminar flow regime. This fact remains largely unacknowledged in the literature. In this work, we quantify the dependence of ${\theta }_{mobile}$ on the Reynolds number via numerical flow simulation at the pore scale for rectangular pores of various aspect ratios, i.e., for highly idealized dead-end pore spaces. We demonstrate that, for the chosen cavity geometries, ${\theta }_{mobile}$ decreases by as much as 42% over the Laminar flow regime. Moreover, ${\theta }_{mobile}$ exhibits an exponential dependence on the Reynolds number, Re = $R$. The fit quality is effectively perfect, with a coefficient of determination (R²) of approximately 1 for each set of simulation data. Finally, we show that this exponential dependence can be easily fitted for pore-scale flow velocity through use of only a few Picard iterations, even with an initial guess that is 10 orders of magnitude off. Not only is this relationship a more accurate definition of pore-scale flow velocity, but it is also a necessary modeling improvement that can be easily implemented. In the companion paper (Part 2), we build upon the findings reported here and demonstrate their applicability to media with other pore geometries: rectangular and non-rectangular cavities (circular and triangular). Full article

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard’s iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two. Full article

In this paper, a large class of contractions is studied that contains Banach and Matkowski maps as particular cases. Sufficient conditions for the existence of fixed points are proposed in the framework of b-metric spaces. The convergence and stability of the Picard iterations are analyzed, giving error estimates for the fixed-point approximation. Afterwards, the iteration proposed by Kirk in 1971 is considered, studying its convergence, stability, and error estimates in the context of a quasi-normed space. The properties proved can be applied to other types of contractions, since the self-maps defined contain many others as particular cases. For instance, if the underlying set is a metric space, the contractions of type Kannan, Chatterjea, Zamfirescu, Ćirić, and Reich are included in the class of contractivities studied in this paper. These findings are applied to the construction of fractal surfaces on Banach algebras, and the definition of two-variable frames composed of fractal mappings with values in abstract Hilbert spaces. Full article