Integral Conditions Research Articles

Painlevé analysis of the Sasa–Satsuma equation

The Sasa-Satsuma equation is considered. The Painlevé test for partial differential equations with application of the Kruskal variable is used to investigate the necessary condition of integrability of the equation. It is shown that the equation does not pass the Painlevé test in the general case. However, there exist constraints on parameters of the equation, when all Fuchs indices are integers. In these cases the Laurent series expansions of the solution exit and, consequently, there exist analytical solutions of this partial differential equation. We confirm that there are two integrable cases, at which the Sasa–Satsuma equation passes the Painlevé test. We also obtain that there is another set of parameter values at which the equation can be an integrable.

(k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional Langevin differential equation having multipoint boundary conditions

The primary objective of this manuscript is to investigate the existence and uniqueness of solutions for the Langevin (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional differential equation of different orders with multipoint nonlocal fractional integral boundary conditions. We consider the generalized version of the Hilfer fractional diferential equation called as (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional differential equation. We provide some significant outcomes about (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional Langevin differential equation that requires deriving equivalent fractional integral equation to (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer Langevin fractional differential equation. The existence result is established using the Krasnoselskii’s fixed-point theorem, while the uniqueness is addressed with the help of Banach contraction principle. Additionally, we investigate the different forms of Ulam stability for the solution of the mentioned problem, under specific conditions. To validate our main outcomes, we present a detailed example at the end of the manuscript.

Spline approximation methods for second order singularly perturbed convection-diffusion equation with integral boundary condition

Spline approximation methods for second order singularly perturbed convection-diffusion equation with integral boundary condition

Solvability of a Hadamard fractional boundary value problem with multi‐term integral and Hadamard fractional derivative boundary conditions

In the present paper, we construct the existence of nontrivial solutions to a new kind of Hadamard fractional boundary value problem on an unbounded domain. With the contribution of some fixed point theorems in cone and the corresponding Green function, we ensure sufficient conditions for the Hadamard fractional boundary value problem. Also, the paper concludes with two examples to demonstrate our results.

SCREEN PSEUDO-SLANT LIGHTLIKE SUBMERSIONS FROM INDEFINITE SASAKIAN MANIFOLDS ONTO LIGHTLIKE MANIFOLDS

As a generalization of screen slant lightlike submersions, we introduce the notion of screen pseudo-slant lightlike submersions from indefinite Sasakian manifolds onto lightlike manifolds. We give examples and prove a characterization theorem for the existence of such lightlike submersions. We also obtain integrability conditions of distributions involved in the definition of this class of lightlike submersions. Further, we find necessary and sufficient conditions for foliations determined by these distributions to be totally geodesic.

Construction of pairwise orthogonal Parseval frames generated by filters on LCA groups

The generalized translation invariant (GTI) systems unify the discrete frame theory of generalized shift-invariant systems with its continuous version, such as wavelets, shearlets, Gabor transforms, and others. This article provides sufficient conditions to construct pairwise orthogonal Parseval GTI frames in L2(G) satisfying the local integrability condition (LIC) and having the Calderón sum one, where G is a second countable locally compact abelian group. The pairwise orthogonality plays a crucial role in multiple access communications, hiding data, synthesizing superframes and frames, etc. Further, we provide a result for constructing N numbers of GTI Parseval frames, which are pairwise orthogonal. Consequently, we obtain an explicit construction of pairwise orthogonal Parseval frames in L2(R) and L2(G), using B-splines as a generating function. In the end, the results are particularly discussed for wavelet systems.

Inverse problem for wave equation of memory type with acoustic boundary conditions

In this article, for the direct problem, which consists of finding the velocity potential and the displacement of boundary points in the wave equation of memory type with initial and boundary acoustic conditions, the inverse problem of determining the memory kernel by the integral overdetermination condition is studied. By introducing a new function, the problem is reduced to a problem with homogeneous boundary conditions. Using the technique of estimating integral equations and the contraction mappings principle in Sobolev spaces, the existence and uniqueness theorem for the inverse problem is proved.

Biomaterials in Medical Applications

Abstract: Biomaterials, a fascinating and highly interdisciplinary field, have become integral to improving modern man's conditions and quality of life. It is done by many health-related problems arising from many sources. The first batch of biomaterials was produced as implants and medical equipment in the 1960s and 1970s. Biomaterials are primarily used in medicine and may be directly or indirectly exposed to biological systems. For instance, we could use them in cultures and mediums for cell development, plasma protein testing, biomolecular processing cultures, diagnostic gene chips, and packaging materials primarily for medical items. Biomaterials should have certain qualities for human-related problems, like being non-carcinogenic, not being pyrogenic or toxic, completely plasma compatible, and anti-inflammatory. This paper introduces the history, classification, and ideal parameters of biomaterials and where they are used in the current scenarios in the medical field, providing a brief outlook on the future.

Study of Caputo fractional derivative and Riemann–Liouville integral with different orders and its application in multi‐term differential equations

In this article, we initially provided the relationship between the RL fractional integral and the Caputo fractional derivative of different orders. Additionally, it is clear from the literature that studies into boundary value problems involving multi‐term operators have been conducted recently, and the aforementioned idea is used in the formulation of several novel models. We offer a unique coupled system of fractional delay differential equations with proper respect for the role that multi‐term operators play in the research of fractional differential equations, taking into account the newly established solution for fractional integral and derivative. We also made the assumptions that connected integral boundary conditions would be added on top of ‐fractional differential derivatives. The requirements for the existence and uniqueness of solutions are also developed using fixed‐point theorems. While analyzing various sorts of Ulam's stability results, the qualitative elements of the underlying model will also be examined. In the paper's final section, an example is given for purposes of demonstration.

Analysis of a parametric delay functional differential equation with nonlocal integral condition

Analysis of a parametric delay functional differential equation with nonlocal integral condition

Solutions of Second-Order Nonlinear Implicit ψ-Conformable Fractional Integro-Differential Equations with Nonlocal Fractional Integral Boundary Conditions in Banach Algebra

In this paper, we introduce and thoroughly examine new generalized ψ-conformable fractional integral and derivative operators associated with the auxiliary function ψ(t). We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit ψ-conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications.

Multiple Positive Symmetric Solutions for the Fourth-Order Iterative Differential Equations Involving $\mathrm{p}$-Laplacian with Integral Boundary Conditions

Multiple Positive Symmetric Solutions for the Fourth-Order Iterative Differential Equations Involving $\mathrm{p}$-Laplacian with Integral Boundary Conditions

Analysis of Caputo Sequential Fractional Differential Equations with Generalized Riemann–Liouville Boundary Conditions

This paper delves into a novel category of nonlocal boundary value problems concerning nonlinear sequential fractional differential equations, coupled with a unique form of generalized Riemann–Liouville fractional differential integral boundary conditions. For single-valued maps, we employ a transformation technique to convert the provided system into an equivalent fixed-point problem, which we then address using standard fixed-point theorems. Following this, we evaluate the stability of these solutions utilizing the Ulam–Hyres stability method. To elucidate the derived findings, we present constructed examples.

Solvability of a boundary value problem with integrable conditions for partial differential equations of fractional order

Differential equations have a remarkable ability to predict the world around us. They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. The study of existence and uniqueness, periodicity, asymptotic behavior, stability, and methods of analytic and numerical solutions of fractional differential equations have been studied extensively in a large cycle works; Especially, The study of existence and uniqueness of solution of fractional partial differential equations are then proved by mani methods as: the well-known Lax–Milgram theorem, Energy estimate and fixed point theorem... Among them, we only mention here the apriori estimate method (energy inequality method). The current piece of work concentrates on exploring the existence and uniqueness of a solution for a non-linear boundary value problem that has integrable conditions in the case of fractional partial differential equations. For this we split the proof into two sections: linear and non-linear problem; for the associated linear problem, we derive the a priori bound and demonstrate the density of the operator generated by the problem posed; we solve the non-linear problem by introducing a iterative process. The results show the efficiency of energy inequality method in the case of time fractional order differential equations with integrable conditions our results illustrate the existence and uniqueness of the continuous dependence of solution on fractional order. The work of the authors could be considered as a contribution to the development of the functional analysis method. And to conclude we argue that the research carried out in this article could serve as a contribution for possible studies in the field of applied mathematics.

On the Existence of a Positive Solution of a Boundary Value Problem for a Nonlinear Second-Order Functional-Differential Equation with Integral Boundary Conditions

On the Existence of a Positive Solution of a Boundary Value Problem for a Nonlinear Second-Order Functional-Differential Equation with Integral Boundary Conditions

A Fitted Approximate Method for Solving Singularly Perturbed Volterra–Fredholm Integrodifferential Equations with Integral Boundary Condition

A Fitted Approximate Method for Solving Singularly Perturbed Volterra–Fredholm Integrodifferential Equations with Integral Boundary Condition

Existence of solution of a system of non-linear differential inclusions with non-local, integral boundary conditions via fixed points of hybrid contractions

In the present article, we have introduced the notions of γ-admissibility for the pair of q-ROF set-valued maps and admissible hybrid q-ROF Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{Z}$\\end{document}-contraction. Notions introduced in the article generalizes the existing concepts in fuzzy literature. Common fixed point result for a pair of γ-admissible q-ROF mappings in b-metric spaces utilizing the introduced contraction is presented. A nontrivial example to support the obtained results is also included. As an application, we have discussed the existence of solution of system of non-linear n-th order differential inclusions with non-local and integral boundary conditions.

A numerical solver based on Haar wavelet to find the solution of fifth-order differential equations having simple, two-point and two-point integral conditions

A numerical solver based on Haar wavelet to find the solution of fifth-order differential equations having simple, two-point and two-point integral conditions

Enhanced resolution in solving first-order nonlinear differential equations with integral condition: a high-order wavelet approach

Enhanced resolution in solving first-order nonlinear differential equations with integral condition: a high-order wavelet approach

Painlevé analysis of the resonant third-order nonlinear Schrödinger equation

The resonant Schrödinger equation of the third order is studied. The Painlevé test for nonlinear partial differential equations is used to determine integrability of equation. It is shown that the necessary condition for integrability of partial differential equations by the inverse scattering transform is fulfilled at certain parameter restriction. Analytical solutions in the form of periodic and solitary wave are presented.