Abdolali Neamaty

University of Mazandaran | UMZ · Faculty of Mathematical Sciences

In this research, we consider the fractional semilinear problem in a sequentially compact Banach space $X$: $x^{\alpha}(t)=A(t)x(t)+f(t,x(t))$, $t\in \mathbb R^{+} $, with the initial condition $x(0)=x_{0}$, $ x_{0} \in X $, where $A$ is the generator of an evolution system $({U(t,s)})_{t\leq s \leq {0}}$ and $f$ is a given function satisfying some...

In this work, we investigate the vibrations of embankments by the singular Sturm-Liouville equations. At first, we create the mathematical form of the vibrations by the shear beam (SB) model (see [21]) and transform this given form to the Sturm-Liouville form with a singularity. Finally, we discuss the numerical solution to the considered problem u...

Introduction In this study, we consider the differential equation with aftereffect under the separated boundary conditions on a finite interval. In fact, we consider the Sturm-Liouville operator disorganized by a Volterra integral operator. We obtain the numerical solution for the special case of the inverse aftereffect problem by applying Chebyshe...

In this research, we intend to show that the nonlocal fractional Cauchy problem Dαu(t) = A(t)u(t) + f(t, u(t)), t ∈ J = [0, 1] with integral initial condition u(0) = ∫¹0g(s, u(s))ds, in the Banach space X, where A is a generator of α-resolvent operator function {T(t)}t≥0 and f, g are given functions satisfying some assumptions, has an almost period...

This paper deals with the unique solvability of the inverse problem for second-order differential operators having a singular potential. It is shown that the coefficients of the differential operator can be determined from the spectral data. Then, we construct the theoretical bases and provide a numerical algorithm for solving the inverse problem o...

This work suggests a model for a population dynamic caused by an enemy attack to a domain of residential areas. With the help of a local non-integer order rate of change and a new structure induced on the real line, we derive a spatial discrete diffusion equation of fractional order. Then making use of the d'Alembert's change of variable we obtain...

In this paper, we give new versions of Hermite–Hadamard inequality via pseudo-fractional integral of order α>0 in two classes of semiring ([a,b],⊕,⊙). We show that if pseudo operations are defined by an increasing and continuous function g, then we can replace the convexity of g with a weaker condition. Also, for the second semiring ([a,b],sup,⊙),...

In this study, we consider Sturm–Liouville equation having a symmetric potential function under the separated boundary conditions on a finite interval. Then, we use Chebyshev polynomials of the first kind for calculating the approximate solution of the inverse Sturm–Liouville problem. Finally, we present the numerical results by providing some exam...

An inverse nodal problem has first been studied for the Sturm-Liouville equation with one turning point. The asymptotic representation of the corresponding eigenfunctions of the eigenvalues has been investigated and an asymptotic of the nodal points is obtained. For this problem, we give a reconstruction formula for the potential function. Furtherm...

In the present paper, we discuss the pseudo-fractional calculus, including two fields of fractional calculus and pseudo-analysis. We also provide pseudo-fractional integral/derivative operators on a semiring ([a,b],⊕,⊙)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsb...

This paper deals with a non-integer order calculus on time scales in a local sense. Taking advantage of a new field structure over real and complex numbers, the researchers have normalized the proposed derivative. Non-integer order mean value theorem on time scales is stated. In addition to a hypothesis on the existence of a Holder continuous funct...

In this article, we present some results about the Sturm–Liouville equation with turning points and singularities and transform them to each other. By applying a change of a variable, we can transform the differential equation with a turning point to the differential equation with a singularity. Also we will prove that a differential equation with...

In this paper, the researchers have studied asymptotic solution for fractional Airy differential equation (FADE) in the conformable sense with steepest descent method.

In this work, inverse problems for singular Sturm-Liouville operators at the finite interval are studied. In this study, we show the spectral characteristics and prove the uniqueness theorem for the solution of the inverse problem. Finally, we give an applied example and use the numerical technique to obtain the approximate solution of the problem.

This work deals with concepts of non-differentiability and a non-integer order differential on timescales. Through an investigation of a local non-integer order derivative on timescales, a mean value theorem (a fractional analog of the mean value theorem on timescales) is presented. Then, by illustrating a vanishing property of this derivative, its...

In this work, a non-integer order Airy equation involving Liouville differential operator is considered. Proposing an undetermined integral solution to the left fractional Airy differential equation, we utilize some basic fractional calculus tools to clarify the closed form. A similar suggestion to the right FADE, converts it into an equation in th...

In this study, we consider Sturm-Liouville (SL) equation under the separated boundary con- ditions on a finite interval. We get the approximate solutions of inverse SL problem by using different input data as eigenvalues and nodes (zeros of eigenfunctions), separately, and calculate the computed errors related to the obtained approximate solutions....

In this study, we consider Sturm-Liouville problem in two cases: the first case having no singularity and the second case having a singularity at zero. Then, we calculate the eigenvalues and the nodal points and present the uniqueness theorem for the solution of the inverse problem by using a dense subset of the nodal points in two given cases. Als...

In this paper, we first introduce the concept of fractional quantum integral with general kernels, which generalizes several types of fractional integrals known from the literature. Then we give more general versions of some integral inequalities for this operator, thus generalizing some previous results obtained by many researchers. 2,8,25,29,30,3...

In this paper we introduce a new type of quantum calculus, the p-calculus involving two concepts of p-derivative and p-integral. After familiarity with them some results are given.

In this paper, we establish the fundamental system of solutions (FSS) for differential pencils (1.1) by the Birkhoff’s method. We obtain the Weyl function which plays an important role in studying inverse problems. Then we give a formulation of the inverse problem and prove the uniqueness theorem.

In this study, we consider Sturm–Liouville problem with a boundary condition depending on spectral parameter. We apply the nodal points as input data and calculate the approximate solution of the inverse nodal problem using Chebyshev polynomials of the first kind. Finally, we illustrate the numerical results by providing several examples.

These notes are about the fractional chain rule on time scales. Under quite general conditions, some modification is made to the existing chain rule theorem due to its applicability in dynamics.

In this paper, we have established the \( \left( G^{\prime }/G\right) \)-expansion method to find exact solutions for time fractional fifth-order Caudrey–Dodd–Gibbon equation (FCDG5). This method is an effective method in finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. The present approach ha...

In this paper, variational homotopy perturbation iteration method (VHPIM) has been applied along with Caputo derivative to solve high-order fractional Volterra integro-differential equations (FVIDEs). The "VHPIM" is present in all two steps. In order to indicate the efficiency and simplicity of the proposed method, we have presented some examples....

This paper deals with a newly born fractional derivative and integral on time scales. A chain rule is derived and the given indefinite integral is being discussed. Also an application to the traffic flow problem with a fractional Burger's equation is presented.

In the present article, we investigate a fractional boundary value problem (FBVP) of complex order θ = m+iα, where 1 < m ≤ 2 and α ∈ R⁺ is studied. By applying two fixed point theorems Banach and Schauder, we achieved some new existence and uniqueness conclusions of complex solutions. We present an example to express our results.

In this article, we verify the existence and uniqueness of solution for a class of nonlinear boundary value problem for fractional differential equation of order(Formula presented.), with the standard fractional derivative in the sense of Riemann–Liouville. By using a variety of tools including the Schauder fixed-point theorem, the Krasnoseĺskii fi...

In this work, we consider a boundary value problem with aftereffect on a finite interval and study the asymptotic behavior of the solutions, eigenvalues, the nodal points and the associated nodal length and calculate the numerical values of the nodal points and the nodal length. Introduction In this paper, we consider the equation

In this paper we have established the simplest equation method to find approximate solutions for total Burgurs equations with time fractional derivative. This method is effective in finding approximate traveling wave solutions of nonlinear, fractional evolution equations (NLEEs) in mathematical physics. The effectiveness of this manageable method h...

In this work, we have applied the variational iteration method and He's polynomials to solve partial differential equation (PDEs) with time-fractional derivative. The variational homotopy perturbation iteration method (VHPIM) is presented in two steps. Some illustrative examples are given in order to show the ability and simplicity of the approach....

In this paper, we study second-order differential operators on the half-line having jump condition in an interior point. We obtain properties of the spectral characteristics, present a formulation of the inverse problem and prove the uniqueness theorem.

In this paper, we study a second-order differential equation on the halfline having a turning point and jump condition. We establish properties of the spectrum, obtain the formulation of the inverse problem and prove the uniqueness theorem for the solution of the inverse problem.

In this paper, we consider a boundary value problem with aftereffect on a finite interval with one turning point. First, we study the asymptotic behavior of the solution and eigenvalues. Then, we investigate infinite product representation of the solution of the problem.

In this paper, we consider a boundary value problem with aftereffect on a finite interval. Then, the asymptotic behavior of the solutions, eigenvalues, the nodal points and the associated nodal length are studied. We also calculate the numerical values of the nodal points and the nodal length. Finally, we prove the uniqueness theorem for the invers...

This work, dealt with the classical mean value theorem and took advantage of it in the fractional calculus. The concept of a fractional critical point is introduced. Some su?cient conditions for the existence of a critical point is studied and an illustrative example rele- vant to the concept of the time dilation e?ect is given. The present paper a...

The purpose of this paper is to investigate the inverse problem for a second order differential equation the so-called differential pencil on the finite interval 􁈾0,1􁈿 when the solutions are not smooth. We establish properties of the spectral characteristics, derive the Weyl function and prove the uniqueness theorem for this inverse problem.

In this paper, we study the inverse problem for Sturm–Liouville operator with discontinuity on the half line. Using Mochizuki and Trooshin’s method, we show that the potential function can be uniquely determined by a set of values of eigenfunctions at some interior points and one spectrum.

In this paper, we investigate a problem of seismic response of dam by inverse spectral method. In order to analyze the safety and stability of an earth dam during an earthquake, a boundary value problem is formulated and this model is created by shear beam model (SB model). By taking the Liouville transformation, we transform the ordinary different...

In this work, second-order differential equations on the half-line having turning points and discontinuities are studied. We establish spectral characteristics using fundamental system of solutions of a differential equation, calculate the eigenvalues and give the Weyl function.

In this paper, the asymptotic representation of the corresponding eigenfunctions of the eigenvalues has been investigated. Furthermore, we obtain the zeros of eigenfunctions.

In this paper, we use a method based on the operational matrices to the solution of the fractional partial differential equations. The main approach is based on the operational matrices of the Haar wavelets to obtain the algebraic equations. The fractional derivatives are described in Caputo sense. Some examples are included to demonstrate the vali...

We investigate the inverse spectral problem of recovering pencils of second-order differential operators on the half-line with turning point. Using the asymptotic distribution of the Weyl function, we give a formulation of the inverse problem and prove the uniqueness theorem for the solution of the inverse problem.

We apply the lower and upper solutions method and fixed-point theorems to prove the existence of positive solution to fractional boundary value problem D(0+)(alpha)u(t) + f(t, u(t)) = 0, 0 < t < 1, 2 < alpha <= 3, u(0) = u'(0) = 0, D-0(alpha-1),u(1) = beta u(xi), 0 < xi < 1, where D-0+(alpha) denotes Riemann-Liouville fractional derivative, beta is...

In this paper, we investigate the existence and uniqueness of solution for fractional boundary value problem for nonlinear fractional differential equation D-0+(alpha) u(t) = f(t,u(t)), 0 < t < 1, 2 < alpha <= 3, with the integral boundary conditions {u(0) - gamma(1) u(1) = lambda(1) integral(1)(0) g(1) (s, u(s))ds, u'(0) - gamma(2)u'(1) = lambda(2...

In this paper, we study the solution of the boundary value problem for secondorder differential operator on the halfline having jump point in an interior point. Using of the fundamental system of solutions, we investigate the asymptotic distribution of eigenvalues.

The uniqueness theorem is studied for boundary value problem with “aftereffect” on a finite interval with discontinuity conditions in an interior point. The oscillation of the eigenfunctions corresponding to large modulus eigenvalues is established and an asymptotic of the nodal points is obtained. By using these new spectral parameters, a uniquene...

In this paper we consider differential systems having a singularity and one turning point. First, by a replacement, we transform the system to a linear second-order equation of Sturm–Liouville type with a singularity. Using the infinite product representation of solutions provided in [8], we obtain the dual equation, then we investigate the uniquen...

In this paper, we are going to investigate the canonical property of solutions of systems of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm-Liouville equation with turning point. Using of the asymptotic estimates provided by Eberhard, Freiling, and Schneider...

In this paper, we derive Haar wavelet operational matrix of the fractional integration and use it to obtain eigenvalues of fractional Sturm-Liouville problem. The fractional derivative is described in the Caputo sense. The efficiency of the method is demonstrated by examples.

In this paper, we investigate the canonical property of solutions of a system of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm-Liouville equation with a turning point. Using the asymptotic estimates for a special fundamental system of solutions of Sturm-Lio...

The aim of this paper is to present an efficient and reliable treatment of the homotopy analysis method (HAM) for solving fractional Sturm-Liouville differential equation, which the second order derivatives is replaced by a fractional derivative. y(t) is positive and smooth function and D a that denotes the fractional differential operator of order...

We consider the differential equation where I contain three turning points, that is here, zeros of φ. Using of the asymptotic estimates provided in [5] for a special fundamental system of solutions of (i) in I, we study the infinite product representation of solutions of (i). Also, we use the infinite product representations of the solutions to der...

This paper deals with the boundary value problems for second-order differential equation on the half-line having a turning point and jump condition. By using the asymptotic estimates for a fundamental system of solutions of the differential equation on the half-line, we study the asymptotic distribution for eigenvalues. Then we establish the primar...

We consider the Sturm-Liouville equation on the bounded interval with two singularities in end points. This equation contains one turning point together with discontinuity conditions, moreover the turning point lies before the jump point. In this paper, by using the spectral characteristic function, we study eigenvalues.

Abstract. In this paper, we verify the solution around an α-ordinary point x0 ∈ [a, b] for fractional Sturm-Liouville equation (D 2αy)(x) + p(x)y(x) = λq(x)y(x), 1 2 < α < 1. (1) Also, the solutions around an α-singular point x0 ∈ [a, b] for fractional differential equation (x − x0) 2α(D 2αy)(x) + p(x)y(x) = (x − x0) 2αλq(x)y(x), 1 2 < α < 1, (2) i...

We applied the variational iteration method and the homotopy perturbation method to solve Sturm-Liouville eigenvalue and boundary value problems. The main advantage of these methods is the flexibility to give approximate and exact solutions to both linear and nonlinear problems without linearization or discretization. The results show that both met...

In this paper we consider the linear differential equation of the form $$ - y''(x) + q(x)y(x) = \lambda y(x), - \infty < a < x < b < \infty $$ where y satisfies Dirichlet boundary conditions and q is a real-valued function which has even number of singularities at c 1, …, c 2n ∈ (a, b). We will study the asymptotic eigenvalue near the singulari...

In this paper, we study the properties of solutions of systems of differential equations having singularities and turning points. First, by a replacement, we transform the system to the Sturm-Liouville equation with singularities. Using of the asymptotic estimates provided in [2] for a special fundamental system of solutions of Sturm-Liouville equa...

This paper presents the asymptotic eigenvalues of the Sturm-Liouville problems with Neumann condition. In this paper we apply the concept of turning points of second order differential equation. Note that the weight function in this equation has two zeros in domain. By making use of the solutions of equation we obtain the higher-order approximation...

In this paper, we introduce an iterative method for solving fractional differential equation Dα(p(x)y� (x)) + λq(x)y(x )= g(x), 0

We consider the following system of differential equations dy dt = iρ 1 R1(t) x, dx dt =( iρR2(t )+ q(t) iρR1(t) ) y (∗) on a finite interval I =( a, b). In this paper, we transform (∗) to the equation with poles and turning points of first order. Using of the asymptotic estimates provided in (2) for a special fundamental system of solutions of Stu...

In this work, we have studied the solution of the second order differential equation with two turning points case. It is well known that the solutions of the equation are obtained by the asymptotic solution. The aim of this article is to show the higher order distribution negative eigenvalues of Sturm-Liouville problem with Neumann boundary conditi...

‎‎We consider the following system of differential equations ‎ ‎‎\begin{eqnarray*}‎ \frac{dy_{2}}{dt}=i \rho \frac{1}{R_{1}(t)}y_{1} \ \ ‎, \‎ \‎ ‎‎ ‎\frac{dy_{1}}{dt}=( i\rho R_{2}(t) +\frac{q(t)}{ i\rho R_{1}(t)})y_{2} \ \ ‎\qquad‎‎\qquad ‎(*)‎ ‎\end{eqnarray*}‎ with initial conditions ‎$‎y_{1}(a,\rho)=0‎$, ‎ $‎y_{2}(a,\rho)=1‎$,‎ on a finite int...

In this work, the non-selfadjoint Sturm-Liouville operator on a finite in-terval with jump conditions is investigated. We establish properties of the solutions and find it's asymptotic forms , also we calculate the eigenvalues.

We consider the differential equation -y '' +q(x)y=ρ 2 φ 2 (x)yforx∈I:=[0,1],(*) where I contain three turning points (zeros of φ). Using asymptotic estimates for a special fundamental system of solutions of (*) in I, we study the infinite product representation of solutions of (*).

We consider the differential equation 0)) () ((' ' 2 2 = − + y x q x y φ ρ (*) on a finite interval ] 1 , 0 [ = I , where I contains three turning points, that is here, zeros of φ , under the assumption that one of the turning points is of even order while the others are of odd order. Using of the asymptotic estimates provided in [1] for a special...

In this paper, we apply the Olver's solution of second order differential equation with the two turning points case. Note that the weight function in this equation has two zeros in the domain, the so called "turning points". As a basic result we derive the distribution of the eigenvalues with the Neumann boundary conditions.

We consider the differential systems dy dt = iρ 1 R 1 (t) x , dx dt = iρR 2 (t) y (*) on a finite interval I = [t 0 , t 1 ]. In this paper, we transform (*) to the equation with singularity and turning point. As a basic result we derive asymptotic estimates for a special fundamental system of solutions of the corresponding differential equation and...

A differential systems having a finite number of arbitrary order sin-gularities and turning points is investigated. We establish properties of the spectral characteristics, find the asymptotic forms of the solution of the system (1) and we calculate the eigenvalues.

We consider eigenvalue problems for second-order differential equa-tion on a finite interval having a turning point. As a basic result we de-rive asymptotic estimates for a special fundamental system of solutions of the corresponding differential equation and determine the asymptotic distribution of the eigenvalues.

Boundary value problems for second-order differential equations on the half-line in which the eigenfunctions have a discontinuity in an inte-rior point are investigated. Using of the asymptotic estimates provided in [1] for a special fundamental system of solutions of equation, we study the behavior asymptotic solution and eigenvalues.

In this paper we study some properties of the eigenfunctions corresponding to the eigenvalues of Sturm-Liouville Problem in two turning points case. There is connection between the eigenvalues of Sturm-Liouville Problem and infinite product representation. Before representing the solution in the form of an infinite product, we make a slight digress...

Consider the Sturm - Liouville equation w " + (lambda(1 - zeta (2)) - psi (zeta))w(zeta) = 0, -infinity < a < zeta < 1 (1) with Dirichlet boundary conditions w(a) =w(zeta) = 0. Let the function psi(zeta) be continuous and -1 is an element of (a,zeta). For zeta is an element of (0,1), the Dirichlet boundary conditions for (1) have an infinite number...