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In this paper, we investigate the method of solving partial differential equations (PDEs) using artificial neural network (ANN) structures, which have been actively applied in artificial intelligence models. The ANN model for solving PDEs offers the advantage of providing explicit and continuous solutions. However, the ANN model for solving PDEs cannot construct a conventionally solvable linear system with known matrix solvers; thus, computational speed could be a significant concern. We study the implementation of the multigrid method, developing a general concept for a coarse-grid correction method to be integrated into the ANN-PDE architecture, with the goal of enhancing computational efficiency. By developing a reduced form of the multigrid method for ANN, we demonstrate that it can be interpreted as an equivalent representation of the classic multigrid expansion. We validated the applicability of the proposed method through rigorous experiments, which included analyzing loss decay and the number of iterations along with improvements in terms of accuracy, speed, and complexity. We accomplished this by employing the gradient descent method and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method to update the gradients while solving the given ANN systems of PDEs. [ABSTRACT FROM AUTHOR]

In this paper, an attempt has been made to establish the sufficient conditions for the existence and multiplicity of positive solutions by using the fixed point index theory and the Avery–Peterson fixed point theorem respectively for the following class of nonlinear singular fractional differential equation D 0 + β ⁢ ( ϕ p ⁢ ( D 0 + α ⁢ u ⁢ ( t ) ) ) + f ⁢ ( t , u ⁢ ( t ) , … , u ( n - 2 ) ⁢ ( t ) ) = 0 , t ∈ ( 0 , 1 ) , D_{0^{+}}^{\beta}(\phi_{p}(D_{0^{+}}^{\alpha}u(t)))+f(t,u(t),\dots,u^{(n-2)}(t% ))=0,\quad t\in(0,1), with the boundary conditions u ( k ) ⁢ ( 0 ) = 0 , 0 ≤ k ≤ n - 2 , D 0 + α ⁢ u ⁢ ( 0 ) = 0 , D 0 + α - 1 ⁢ u ⁢ ( 1 ) = ∫ 0 1 D 0 + α - 1 ⁢ u ⁢ ( t ) ⁢ 푑 A ⁢ ( t ) , u^{(k)}(0)=0,\quad 0\leq k\leq n-2,\qquad D_{0^{+}}^{\alpha}u(0)=0,\qquad D_{0% ^{+}}^{\alpha-1}u(1)=\int_{0}^{1}D_{0^{+}}^{\alpha-1}u(t)\,dA(t), where 0 < β ≤ 1 {0<\beta\leq 1} , n - 1 < α ≤ n {n-1<\alpha\leq n} , n ≥ 3 {n\geq 3} , A : [ 0 , 1 ] → ℝ {A:[0,1]\to\mathbb{R}} is a function of bounded variation, ϕ p ⁢ ( s ) = | s | p - 2 ⁢ s {\phi_{p}(s)=|s|^{p-2}s} for p > 1 {p>1} and ϕ q ⁢ ( s ) {\phi_{q}(s)} is the inverse of ϕ p ⁢ ( s ) {\phi_{p}(s)} , where p , q {p,q} satisfies the relation 1 p + 1 q = 1 {\frac{1}{p}+\frac{1}{q}=1} , f may have singularity at t = 0 {t=0} . [ABSTRACT FROM AUTHOR]

We study the motion of an elastic rigid rod which is radially free-falling towards a Schwarzschild black hole. This is accomplished by reducing the corresponding free-boundary partial differential equation problem to a sequence of ODEs, which we integrate numerically. Starting with a rod at rest, we show that it is possible to choose its initial compression profile so that its midpoint falls substantially faster, or slower, than a free-falling particle with the same initial conditions. This seems to be a purely kinematic effect, since on average there is no net transfer of elastic energy to mechanical energy. [ABSTRACT FROM AUTHOR]

Analytical solutions of two-dimensional statics problems of elasticity are presented in bipolar coordinates for homogeneous isotropic bodies bounded by the coordinate lines of a bipolar coordinate system. In particular, various boundary problems for an eccentric circular ring, a half-plane with circular holes, and others are considered. The equilibrium equations and Hooke's law are expressed using bipolar coordinates. This paper does not address the static equilibrium requirement of the external load at each circular boundary of the study area. This requirement, which significantly limits the range of problems that can be solved, typically appears in papers dealing with the aforementioned problems. In addition, the proposed method for obtaining an exact (analytical) solution is much simpler compared to the traditional approach. The exact solutions are derived using the method of separation of variables. By utilizing MATLAB software, the numerical results of some boundary value problems for an eccentric semi-ring are obtained, and the corresponding diagrams are presented. [ABSTRACT FROM AUTHOR]

The solution of a nonlinear boundary value problem in ordinary differential equations of the form $ L(x(t))+f(x(t)) = 0 $ can be obtained by solving the associated system of nonlinear equations. In this paper, a three-step iterative method with ninth order convergence is proposed to solve the system of nonlinear equations. The scheme is kept free from second and higher order Frechet derivatives to make it computationally efficient. The nonlinear boundary value problem is discretized to produce a system of nonlinear equations which is then solved using the proposed method. Later, the ninth order method is generalized to obtain a m-step scheme with $ 3m $ order of convergence. Standard problems like Bratu [3] one-dimensional problem and Frank-Kamenetzki [8] problem are solved using the new method. The performance of the new method is compared with an existing method [18] to establish its computational superiority. [ABSTRACT FROM AUTHOR]

In the present paper, we generalized some of the operators defined in (p , q) -calculus with respect to another function. More precisely, the generalized (p , q) - ϕ -derivatives and (p , q) - ϕ -integrals were introduced with respect to the strictly increasing function ϕ with the help of different orders of the q-shifting, p-shifting, and (q / p) -shifting operators. Then, after proving some related properties, and as an application, we considered a generalized (p , q) - ϕ -difference problem and studied the existence property for its unique solutions with the help of the Banach contraction mapping principle. [ABSTRACT FROM AUTHOR]

Integral formulas represent a methodological basis for the determination of gravitational fields generated by planetary bodies. In particular, spherical integral transformations are preferred for their symmetrical properties with the integration domain being the entire surface of the sphere. However, global coverage of boundary values is rarely guaranteed. In practical calculations, we therefore split the spherical surface into a near zone and a far zone, for convenience, by a spherical cap. While the gravitational effect in the near zone can be evaluated by numerical integration over available boundary values, the contribution of the far zone has to be precisely quantified by other means. Far-zone effects for the isotropic integral transformations and those depending on the direct azimuth have adequately been discussed. On the other hand, this subject has only marginally been addressed for the spherical integral formulas that are, except for other variables, also functions of the backward azimuth. In this article, we significantly advance the existing geodetic methodology by deriving the far-zone effects for the two classes of spherical integral transformations: (1) the analytical solutions of the horizontal, horizontal–horizontal, and horizontal–horizontal–horizontal BVPs including their generalisations with arbitrary-order vertical derivative of respective boundary conditions and (2) spatial (vertical, horizontal, or mixed) derivatives of these generalised analytical solutions up to the third order. The integral and spectral forms of the far-zone effects are implemented in MATLAB software package, and their consistency is tested in closed-loop simulations. The presented methodology can be employed in upward/downward continuation of potential field observables or for a quantification of error propagation through spherical integral transformations. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study a kind of boundary value problem with an integral boundary condition including Hadamard-type fractional differential equations. To do this, upper and lower solutions are used to guarantee their existence, and Schauder’s fixed point theorem is used to prove the uniqueness of the positive solutions to this problem. An illustrated example is presented to explain the theorems that have been proved. [ABSTRACT FROM AUTHOR]

In this paper, the Sinc-collocation method is applied to solve a system of coupled nonlinear differential equations that report the chemical reaction of carbon dioxide CO2 and phenyl glycidyl ether in solution. The model has Dirichlet and Neumann boundary conditions. The given scheme has transformed this problem into some algebraic equations. The approach is quite simple to handle and the new numerical solutions are compared with some known solutions, which shows that the new technique is accurate and efficient. [ABSTRACT FROM AUTHOR]

This paper aims to study the existence of solutions for Hadamard fractional nonlocal boundary value problems with mean curvature operator at resonance. Based on the coincidence degree theory, some new results are established. Moreover, an example is given to verify our main results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish the existence of multiple positive solutions for a coupled system of second-order undamped Sturm-Liouville boundary value problems. The technique is based on the Six functionals fixed point theorem. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the rotational periodic boundary value problem for a fractional‐order nonlinear differential equation. Applying topology‐degree theory and the Leray‐Schauder fixed‐point theorem, we prove the existence and uniqueness of solution for the fractional‐order differential system. Furthermore, the existence of solution for a nonlinear differential system with a multivalued perturbation term is investigated by using set‐valued theory and techniques of functional analysis. Two examples of applications are given at the end. [ABSTRACT FROM AUTHOR]

In this paper, we study boundary value problem of system of generalized Sturm–Liouville and Langevin Hadamard fractional differential equations. Existence and uniqueness results are proved via Banach contraction principle and Leray–Schauder fixed point theorem. Besides, the Ulam–Hyers and Ulam–Hyers–Rassias stability results are addressed for the proposed problem. An example illustrating the effectiveness of the theoretical results is presented. [ABSTRACT FROM AUTHOR]

In a Hilbert space, a linear-quadratic optimal control problem with a fixed left end and a moving right end is considered on a fixed time interval. The objective functional is the sum of the integral and terminal components of quadratic type. Each component seeks its minimum on its admissible set independently. At the right end of the time interval, we have a linear programming problem. The solution to this problem implicitly determines the terminal condition for controlled dynamics. A saddle point approach is proposed for solving the problem, which reduces to calculating the saddle point of the Lagrange function. The approach is based on saddle point inequalities in both groups of variables: primal and dual. These inequalities are sufficient optimality conditions. A method for calculating the saddle point of the Lagrange function is formulated. Convergence in primal and dual variables is proved, namely, weak convergence in controls and strong convergence in phase and adjoint trajectories, as well as in terminal variables of the boundary value problem. Based on the saddle point approach, the control synthesis is constructed, i.e., feedback in the presence of constraints on controls in the form of a convex closed set. This is a new result, since, in the classical case in the theory of a linear regulator, a similar statement is proved in the absence of constraints on controls, which makes it possible to use the matrix Riccati equation. In the presence of constraints on control, these arguments no longer work. Therefore, the result obtained is based on the concept of a support plane to the control set. [ABSTRACT FROM AUTHOR]

This paper outlines a numerical method called the Bernstein operational matrix of derivative (BOMD) of order two and order three with the approach of the Chebyshev collocation technique to solve boundary value problems (BVP). BOMD with suitable collocation points is implemented to solve the BVP using the linear combination of Bernstein polynomials with unknown coefficients to approximate the solutions. The derivatives featured in the problem sets will be approximated by utilizing the matrix. The subsequent examination involves a mathematical analysis of the proposed method, including evaluating its order, absolute error metrics and comparative assessments with alternative methodologies. Four problems involving linear and non-linear equations and systems, along with practical real-world problems, are addressed to assess the reliability of the proposed method. [ABSTRACT FROM AUTHOR]

This paper communicates with the entropy generation analysis of Eyring Powell fluid flow through a microchannel in the thermal heat source coefficients. The dimensionless equations of the physical features along with the boundary conditions which are solved by using Homotopy analysis method. An expression of dimensionless velocity and dimensionless temperature are used to determination of the entropy generation number and bejan number explicitly. By using the analytical expression the thermal performances can be improved through the effects of magnetic field, viscous heating, joule heating and convective condition, futhermore it is explaining with graphically. [ABSTRACT FROM AUTHOR]

This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy integral method, while the existence of the solution is equivalently reduced to solving Volterra integral equations of the second kind and a fractional differential equation. The study focuses on a mixed domain where the parabolic section aligns with the upper half-plane, and the hyperbolic section is bounded by two characteristics of the equation under consideration and a segment of the x-axis. By utilizing the solution representation of the fractional-order diffusion equation, a primary functional relationship is derived between the traces of the sought function on the x-axis segment from the parabolic part of the mixed domain. An explicit solution form for the modified Cauchy problem in the hyperbolic section of the mixed domain is presented. This solution, combined with the problem's boundary condition, yields a fundamental functional relationship between the traces of the unknown function, mapped to the interval of the equation's degeneration line. Through the conjugation condition of the problem, an equation with fractional derivatives is obtained by eliminating one unknown function from two functional relationships. The solution to this equation is explicitly formulated. For a specific solution of the proposed problem, visualizations are provided for various orders of the fractional derivative. The analysis demonstrates that the derivative order influences both the intensity of the diffusion (or subdiffusion) process and the shape of the wave front. [ABSTRACT FROM AUTHOR]

This research focuses on finding multiple solutions (MSs) to nonlinear fractional boundary value problems (BVPs) through a new development, namely the predictor Laplace fractional power series method. This method predicts the missing initial values by applying boundary or force conditions. This research provides a set of theorems necessary for deriving the recurrence relations to find the series terms. Several examples demonstrate the efficacy, convergence, and accuracy of the algorithm. Under Caputo's definition of the fractional derivative with symmetric order, the obtained results are visualized numerically and graphically. The behavior of the generated solutions indicates that altering the fractional derivative parameters within their domain symmetrically changes these solutions, ultimately aligning them with the standard derivative. The results are compared with the homotopy analysis method and are presented in various figures and tables. [ABSTRACT FROM AUTHOR]

It is known that the plane problems of the theory of elasticity and thermoelasticity are usually reduced to solving the biharmonic equation with respect to the Airy stress function. Solutions to spatial problems can also be defined using the Maxwell and Morera stress functions, etc. In this work, for solving a spatial thermoelastic problem, in the framework of the Beltrami-Michell equations, Poisson-type differential equations are found for each component of the stress tensor. In this case, the right-hand sides of these equations depend on temperature and elastic constants. The boundary conditions consist of the usual surface forces and "additional" boundary conditions derived from the equilibrium equations. The differential equations and boundary conditions of the plane problem of thermoelasticity in stresses are discussed in more detail. Grid equations for plane and spatial problems of thermoelasticity were compiled by the finite-difference method and solved by the iterative method. The problems of a thermoelastic rectangle and a parallelepiped located in a given temperature field are solved numerically. The validity of the formulated boundary value problems and the reliability of the results obtained are substantiated by comparing the numerical results with the solutions of the problems of a thermoelastic plate and a parallelepiped known in the literature. [ABSTRACT FROM AUTHOR]

In this paper we investigate the existence and uniqueness of solutions for a class of boundary value problem for fractional differential equations involving the Hadamard's fractional derivative, by applying a nonlinear alternative of Leray–Schauder due to Frigon and Granas for contraction maps in Fréchet spaces. [ABSTRACT FROM AUTHOR]

Copyright of Reviews of Geophysics & Planetary Physics is the property of Editorial Office of Reviews of Geophysics & Planetary Physics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper investigates the existence, uniqueness, and continuous dependence of solutions to fractional order nonlocal thermistor two-point boundary value problems on time scales. We employ the Schauder fixed point theorem to establish the existence of solutions, and the contraction principle to prove uniqueness. We also obtain a result on the continuous dependence of solutions. Finally, we present several examples to illustrate our findings. This work is the first to study a fractional model of thermistor on Department of Medical Research,time scales, and it makes a significant contribution to the field of modeling on time scales. The results of this paper can be used to develop new and improved mathematical models for thermistors, which can be used to design more efficient and reliable thermistor-based devices. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Guangdong University of Technology is the property of Journal of Guangdong University of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this article, we establish certain sufficient conditions to show the existence of solutions of boundary value problem for fractional differential equations on the half-line in a Fréchet space. The main result is based on Tykhonoff fixed point theorem combining with a suitable measure of non-compactness. An example is given to illustrate our approach. [ABSTRACT FROM AUTHOR]

A boundary value problem of the Bitsadze-Samarskii type is studied in the article for a fractionalorder diffusion equation and a degenerate hyperbolic equation with singular coefficients at lower terms in an unbounded domain. The article considers a mixed domain where the parabolic part of the domain under consideration coincides with the upper half-plane and the hyperbolic part is bounded by two characteristics of the equation under consideration and a segment of the abscissa axis. The uniqueness of the solution to the problem under consideration is proven by the method of energy integrals. The existence of a solution to the problem under consideration is reduced to the concept of solvability of a fractional-order differential equation. An explicit form of the solution to the modified Cauchy problem is given in the hyperbolic part of the mixed domain under consideration. Using this solution, due to the boundary condition of the problem, the main functional relationship between the traces of the unknown function brought to the interval of the degeneracy line of the equation is obtained. Further, using the representation of the solution of the diffusion equation of fractional order, the second main functional relationship between the traces of the sought-for function on the interval of the abscissa axis from the parabolic part of the considered mixed domain is obtained. Through the conjugation condition of the problem under study, an equation with fractional derivatives is obtained from two functional relationships by eliminating one unknown function; its solution is written out in explicit form. In the study of the boundary value problem, generalized fractional integro-differentiation operators with the Gauss hypergeometric function are employed. The properties of the Wright and Mittag-Leffler type functions are extensively utilized in the study.

This paper aims to study the existence of solutions for Hadamard fractional nonlocal boundary value problems with mean curvature operator at resonance. Based on the coincidence degree theory, some new results are established. Moreover, an example is given to verify our main results.

Modeling the gravitational field of irregular celestial bodies is a common problem currently faced by the researchers in the fields of Earth science and deep space exploration. This paper reviews the developmental history of gravitational field modeling theory in these two fields. It summarizes research progress in the last decade, focusing on multipole expansion and potential-flow simulation theories. The gravitational field multipole expansion theory introduces the complex algebraic function commonly used in electromagnetism and quantum mechanics to the gravitational field modeling research field, which can express the (internal and external) gravitational field of celestial bodies more concisely. The calculation rules and mathematical tools from the complex function theory make solving the gravitational dynamics problems in Earth and planetary science more convenient. The potential-flow simulation theory is proposed on top of the equivalence of the Earth's external gravitational vector field and the potential-flow velocity field. That is, both are source- and curl-free vector fields. It introduces the numerical algorithms of computational fluid dynamics to model the external gravitational fields of celestial bodies. Because the potential-flow simulation theory is not established based on the perturbation theory, it can overcome the divergence problem of conventional methods, especially when inverting for the gravitational field of an irregularly shaped celestial body. The two theories expand the theoretical research of gravitational field modeling by using seemingly irrelevant knowledge from other research fields, and such methodology is instructive for Earth or deep-space science researchers to make subsequent theoretical improvements. Moreover, we also systematically categorize, compare, and analyze the existing gravitational field modeling methods based on the mathematical definition of the gravitational field, as well as its variants, and summarize their advantages and disadvantages, helping to select proper gravitational field modeling methods for the Earth and deep-space science researchers.

In this paper, we consider the incompressible 3D Euler and Navier–Stokes equations in a smooth bounded domain. First, we study the 3D Euler equations endowed with slip boundary conditions and we prove the same criteria for energy conservation involving the gradient, already known for the Navier–Stokes equations. Subsequently, we utilize this finding, which is based on a proper approximation of the velocity (and does not require any estimate or additional assumption on the pressure), to explore energy conservation for Beltrami flows. Finally, we explore Beltrami solutions to the Navier–Stokes equations and demonstrate that conditions leading to energy conservation are significantly distinct from those implying regularity. This remains true even when making use of the bootstrap regularity improvement, stemming from the solution being a Beltrami vector field. [ABSTRACT FROM AUTHOR]

For bounded domains Ω with Lipschitz boundary Γ , we investigate boundary value problems for elliptic operators with variable coefficients of fourth-order subject to Wentzell (or dynamic) boundary conditions. Using form methods, we begin by showing general results for an even wider class of operators of type A = B ∗ B 0 - N b B γ , where B is associated to a quadratic form b and N b an abstractly defined co-normal Neumann trace. Even in this general setting, we prove generation of an analytic semigroup on the product space H : = L 2 (Ω) × L 2 (Γ) . Using recent results concerning weak co-normal traces, we apply our abstract theory to the elliptic fourth-order case and are able to fully characterize the domain in terms of Sobolev regularity for operators in divergence form B = - div Q ∇ with Q ∈ C 1 , 1 (Ω ¯ , R d × d) , also obtaining Hölder-regularity of solutions. Finally, we also discuss asymptotic behavior and (eventual) positivity. [ABSTRACT FROM AUTHOR]

In this paper the authors present three different Lyapunov-type inequalities for a higher-order Caputo fractional differential equation with identical boundary conditions marking the inaugural instance of such an approach in the existing literature. Their findings extend and complement certain prior results in the literature.

The aim of this paper is to estimate an interval of the existence of positive solutions for a boundary value problem for the system of three nonlinear second-order ordinary differential equations. Krasnosel'skiĭ–Precup fixed point theorem is used to determine this interval theoretically. For Dirichlet boundary conditions theoretical result is compared with result obtained numerically.

In this study, a multipoint boundary value problem for Volterra-Fredholm integro-differential equations is considered. The addition of a new function converts the system of Volterra-Fredholm integro-differential equations to a system of Fredholm integro-differential equations. In contrast to the original problem, the dimension of a Fredholm integro-differential equation is determined by the number of matrices in the degenerate kernel of the Volterra integral. A numerical algorithm of Dzhumabaev parameterization method for addressing a multipoint boundary value problem for Volterra-Fredholm integro-differential equations is proposed. The main advantage of the proposed method is splitting the problem into auxiliary Cauchy problems for ordinary differential equations and a system of algebraic equations with respect to the parameters. The conditions for the unique solvability of the multipoint boundary value problem for Fredholm integro-differential equations are established. Finally, various numerical examples are provided to demonstrate the efficiency and correctness of the suggested technique.

This research addresses the numerical solution of singularly perturbed convection-diffusion kind boundary value problem of second-order with a discontinuity term. Due to the perturbation parameter and discontinuity term, the problem solution has a boundary layer and an interior layer. A nonpolynomial cubic spline method is utilized to solve the boundary value problem. A specific set of parameters associated with nonpolynomial spline is used to tailor the method. A comprehensive analysis of the stability and convergence of the recommended method is presented which gives second-order convergence results. The suggested method is implemented on two examples, and the obtained results are contrasted with an existing method, highlighting the precision and efficacy of the proposed method, which would enhance the method's novelty.

In this paper, we study the existence of positive solution to boundary value problem of fractional differential equations with p$$ p $$‐Laplacian operator. By using Avery–Peterson theorem, some new existence results of three increasing positive solutions are obtained. [ABSTRACT FROM AUTHOR]

A boundary value problem for a nonlinear second-order functional-differential equation on the interval with integral boundary conditions is considered. Sufficient conditions for the existence of at least one positive solution of the problem under consideration are found by using special topological tools. A nontrivial example in which the sufficient conditions for the unique solvability of the problem are satisfied is given. [ABSTRACT FROM AUTHOR]

We consider the p-Laplace boundary value problem with the Dirichlet boundary condition. A new lower estimate for positive solutions of the problem is obtained. As an application of this new lower estimate, some sufficient conditions for the existence and nonexistence of positive solutions for the p-Laplace problem are obtained. [ABSTRACT FROM AUTHOR]

The aim of this work is the study of a nonlinear boundary value problem which theoretically generalizes the Lamé system with disturbance in a thin 3D domain with friction and a generalized boundary condition. For the resolution of the considered problem and after the variational formulation, we construct an operator from the variational problem. Then we prove that this operator has certain properties which allows us to apply the theorem of existence and uniqueness of the solution of variational inequalities of the 2nd kind. Finally, using a change of scale, we transport the variational problem to an equivalent problem defined on a domain independent of the parameter ζ {{\zeta}} and subsequently we obtain the limit problem and the generalized weak equations of the initial problem. [ABSTRACT FROM AUTHOR]

In recent years of the literature, fixed point iteration schemes have been studied theoretically for many nonlinear problems, including differential and integral equations; however, numerically, there has been a little work on these iterative schemes for finding numerical solutions of differential equations. The main purpose of this paper is to propose a new numerically efficient iterative approach based on fixed point theory and Green's function. First, we compute a Green's function for a wider class fourth order BVPs and express the sought solution as a fixed point of a certain integral operator. We show by Banach Contraction Principle that this new integral operator admits a unique fixed point which is the solution for a given BVP. We embed the integral operator into our iterative scheme and obtain a generalized iterative scheme. We prove the convergence of our new scheme under possible mild assumptions. Instead of stability, we prove a weak w 2 -stability result for our scheme which is a natural and weak notion of the stability for any given iterative scheme. To support numerically the convergence of our scheme, we provide two different examples. Using these examples, we compare the high accuracy of our new scheme with the classical approaches of the literature. As an application of our scheme, we show that the scheme can be used for finding solutions for the Bratu's problem in a Banach space setting. At the end of the paper, we include some interesting discussion about the main results of the paper. [ABSTRACT FROM AUTHOR]

This work provides some information on the eigenvalues and eigenfunctions of a problem which is constructed by a system of equations and symmetric boundary conditions that includes the ordinary second‐order Sturm–Liouville boundary value problem. In particular, we show that the problem has an infinite number of discrete eigenvalues with a greatest lower bound and the corresponding eigenfunctions are complete in mean and energy. We introduce the results using the variational approach that enables us to consider only continuous pair functions instead of absolutely continuous pair functions. [ABSTRACT FROM AUTHOR]

This paper establishes the existence and uniqueness of a solution to the quasilinear fractional delay integro impulsive differential equations' boundary value problem using the Caputo derivative. We also took into account the system with integral and nonlocal boundary conditions. The Banach contraction principle is used to derive novel findings on the uniqueness and existence of solutions. We provided a numerical example with a graphical representation to help with comprehension and confirmation of the results. [ABSTRACT FROM AUTHOR]

This study is concerned with a new class of fractional differential equations with arbitrary order, coupled with multi-point boundary conditions. To solve this problem. we extract the formulation of an equivalent integral equation. We establish a continuous operator within a Banach space, leveraging cone theory and fixed point theorems. These tools allow us to establish the existence of multiple solutions for our fractional boundary value problem, contingent upon certain assumptions. To conclude, we offer illustrative examples to exemplify the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, existence of pairs of solutions is obtained for compact potential operators on Hilbert spaces. An application to a second-order boundary value problem is also given as an illustration of our results. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Dalian University of Technology / Dalian Ligong Daxue Xuebao is the property of Journal of Dalian University of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

A class of solved with respect to the th order derivative evolution linear equations in Banach spaces on the real axis is considered. The equation is not endowed by initial conditions. A theorem on criterion in terms of the Fourier transform for analytic in a containing bisectorial region functions with values in a Banach space is proved. On the basis of this criterion we define a class of bisectorial operators. For the equation with a bisectorial operator, it is proved a unique solution existence. Moreover, it is shown that the unique solution has the form of the convolution of the inverse Fourier transform for and of the right-hand side of the equation. The obtained general results are used in the study of a boundary value problem to a class of partial differential equations. [ABSTRACT FROM AUTHOR]

In this article, we study the problem of recovering an unknown function based on its known integrals over a specified family of curves, with consideration given to a weight function. The main objective is to establish a methodology for solving the integral geometry problem by transforming it into a boundary value problem involving a second-order partial differential equation of mixed type. The obtained results provide valuable insights into the solution of the finite difference problem, which encompasses both the boundary and a certain neighborhood in its vicinity, taking into account the presence of specific types of singularities. The estimation theorem for the solution is proven using a specialized factor-based technique. The implications of these findings extend to diverse applications, notably in fields such as seismic and medical imaging, as well as non-destructive material characterization. By presenting a comprehensive analysis and a detailed exploration of the problem at hand, this study contributes to the advancement of the theory and practical implementation of integral geometry in various scientific and technological domains. [ABSTRACT FROM AUTHOR]

Due to the importance of some physical systems, in this paper, we aim to investigate a generalized impulsive ρ -Caputo differential equation equipped with a p-Laplacian operator. In fact, our problem is a generalization of fractional differential equations equipped with the integral boundary conditions, impulsive forms and p-Laplacian operators under the Nemytskii operators. In this direction, we prove some theorems on the existence property along with the uniqueness of solutions under the Nemytskii operator. More precisely, we use the Schauder’s and Schaefer’s fixed point theorems, along with the Banach contraction principle. In the sequel, two examples are provided to show the validity of the obtained results in practical. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fourth-order compact discretization method for solving a secondorder boundary value problem governed by the nonlinear Fredholm integro-differential equations. Using an efficient approximate polynomial, a compact numerical integration method is first designed. Then by applying the derived numerical integration formulas, the original problem is converted into a nonlinear system of algebraic equations. It is shown that the proposed method is easy to implement and has the third order of accuracy in the infinity norm. Some numerical examples are presented to demonstrate its approximation accuracy and computational efficiency, as well as to compare the derived results with those obtained in the literature. [ABSTRACT FROM AUTHOR]

Under suitable continuity and uniqueness conditions, solutions of an α order Caputo fractional boundary value problem are differentiated with respect to boundary values and boundary points. This extends well-known results for nth order boundary value problems. The approach used applies a standard algorithm to achieve the result and makes heavy use of recent results for differentiation of solutions of Caputo fractional intial value problems with respect to initial conditions and continuous dependence for Caputo fractional boundary value problems. [ABSTRACT FROM AUTHOR]

For the differential equations of the barotropic dynamics of compressible viscous multifluids in a bounded three-dimensional domain with an immobile rigid boundary, a study of the solvability of the boundary value problem is made. Weak generalized solutions to the boundary value problem are shown to exist with weak constraints on the types of viscosity matrices and constitutive equations for pressure and momentum exchange. [ABSTRACT FROM AUTHOR]

This paper deals with the global approximation of the solutions of Boundary Value Problems (BVPs) of second order on the real line. We first reduce the BVP to an equivalent Fredholm integral equation of the second kind and then approximate its solution by a Nyström type method based on a suitable product quadrature rule. Such quadrature formula is based on a truncated interpolation process at the Hermite zeros. The stability and the convergence of the method as well as the well conditioning of the involved linear systems are studied in weighted spaces of continuous functions. Numerical tests confirming the theoretical error estimates are shown. [ABSTRACT FROM AUTHOR]