Special Issue "Nonlinear Equations: Theory, Methods, and Applications"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (28 February 2021).

Special Issue Editors

Prof. Dr. Ravi P. Agarwal
grade E-Mail Website
Guest Editor
Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA
Interests: nonlinear analysis; differential and difference equations; fixed point theory; general inequalities
Special Issues and Collections in MDPI journals
Prof. Dr. Bashir Ahmad
E-Mail Website
Guest Editor
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
Interests: differential equations; boundary value problems; nonlinear analysis; applications
Special Issues and Collections in MDPI journals

Special Issue Information

Dear Colleagues

This Special Issue will cover:

Ordinary differential equations;

Delay differential equations;

Functional equations;

Equations on time scales;

Partial differential equations;

Fractional differential equations;

Stochastic differential equations;

Integral equations.

Prof. Dr. Ravi P. Agarwal
Prof. Dr. Bashir Ahmad
Guest Editors

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Published Papers (27 papers)

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Research

Article
Mathematical Analysis of Oxygen Uptake Rate in Continuous Process under Caputo Derivative
Mathematics 2021, 9(6), 675; https://doi.org/10.3390/math9060675 - 22 Mar 2021
Cited by 6 | Viewed by 441
Abstract
In this paper, the wastewater treatment model is investigated by means of one of the most robust fractional derivatives, namely, the Caputo fractional derivative. The growth rate is assumed to obey the Contois model, which is often used to model the growth of [...] Read more.
In this paper, the wastewater treatment model is investigated by means of one of the most robust fractional derivatives, namely, the Caputo fractional derivative. The growth rate is assumed to obey the Contois model, which is often used to model the growth of biomass in wastewaters. The characteristics of the model under consideration are derived and evaluated, such as equilibrium, stability analysis, and steady-state solutions. Further, important characteristics of the fractional wastewater model allow us to understand the dynamics of the model in detail. To this end, we discuss several important analyses of the fractional variant of the model under consideration. To observe the efficiency of the non-local fractional differential operator of Caputo over its counter-classical version, we perform numerical simulations. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
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Article
Calculation of Two Types of Quaternion Step Derivatives of Elementary Functions
Mathematics 2021, 9(6), 668; https://doi.org/10.3390/math9060668 - 21 Mar 2021
Cited by 1 | Viewed by 455
Abstract
We aim to get the step derivative of a complex function, as it derives the step derivative in the imaginary direction of a real function. Given that the step derivative of a complex function cannot be derived using i, which is used [...] Read more.
We aim to get the step derivative of a complex function, as it derives the step derivative in the imaginary direction of a real function. Given that the step derivative of a complex function cannot be derived using i, which is used to derive the step derivative of a real function, we intend to derive the complex function using the base direction of the quaternion. Because many analytical studies on quaternions have been conducted, various examples can be presented using the expression of the elementary function of a quaternion. In a previous study, the base direction of the quaternion was regarded as the base separate from the basis of the complex number. However, considering the properties of the quaternion, we propose two types of step derivatives in this study. The step derivative is first defined in the j direction, which includes a quaternion. Furthermore, the step derivative in the j+k2 direction is determined using the rule between bases i, j, and k defined in the quaternion. We present examples in which the definition of the j-step derivative and (j,k)-step derivative are applied to elementary functions ez, sinz, and cosz. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
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Article
Models for COVID-19 Daily Confirmed Cases in Different Countries
Mathematics 2021, 9(6), 659; https://doi.org/10.3390/math9060659 - 19 Mar 2021
Cited by 5 | Viewed by 767
Abstract
In this paper, daily confirmed cases of COVID-19 in different countries are modelled using different mathematical regression models. The curve fitting is used as a prediction tool for modeling both past and upcoming coronavirus waves. According to virus spreading and average annual temperatures, [...] Read more.
In this paper, daily confirmed cases of COVID-19 in different countries are modelled using different mathematical regression models. The curve fitting is used as a prediction tool for modeling both past and upcoming coronavirus waves. According to virus spreading and average annual temperatures, countries under study are classified into three main categories. First category, the first wave of the coronavirus takes about two-year seasons (about 180 days) to complete a viral cycle. Second category, the first wave of the coronavirus takes about one-year season (about 90 days) to complete the first viral cycle with higher virus spreading rate. These countries take stopping periods with low virus spreading rate. Third category, countries that take the highest virus spreading rate and the viral cycle complete without stopping periods. Finally, predictions of different upcoming scenarios are made and compared with actual current smoothed daily confirmed cases in these countries. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
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Article
Integral Comparison Criteria for Half-Linear Differential Equations Seen as a Perturbation
Mathematics 2021, 9(5), 502; https://doi.org/10.3390/math9050502 - 01 Mar 2021
Viewed by 312
Abstract
In this paper, we present further developed results on Hille–Wintner-type integral comparison theorems for second-order half-linear differential equations. Compared equations are seen as perturbations of a given non-oscillatory equation, which allows studying the equations on the borderline of oscillation and non-oscillation. We bring [...] Read more.
In this paper, we present further developed results on Hille–Wintner-type integral comparison theorems for second-order half-linear differential equations. Compared equations are seen as perturbations of a given non-oscillatory equation, which allows studying the equations on the borderline of oscillation and non-oscillation. We bring a new comparison theorem and apply it to the so-called generalized Riemann–Weber equation (also referred to as a Euler-type equation). Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
Stability Concepts of Riemann-Liouville Fractional-Order Delay Nonlinear Systems
Mathematics 2021, 9(4), 435; https://doi.org/10.3390/math9040435 - 22 Feb 2021
Cited by 1 | Viewed by 504
Abstract
First, we set up in an appropriate way the initial value problem for nonlinear delay differential equations with a Riemann-Liouville (RL) fractional derivative. We define stability in time and generalize Mittag-Leffler stability for RL fractional differential equations and we study stability properties by [...] Read more.
First, we set up in an appropriate way the initial value problem for nonlinear delay differential equations with a Riemann-Liouville (RL) fractional derivative. We define stability in time and generalize Mittag-Leffler stability for RL fractional differential equations and we study stability properties by an appropriate modification of the Razumikhin method. Two different types of derivatives of Lyapunov functions are studied: the RL fractional derivative when the argument of the Lyapunov function is any solution of the studied problem and a special type of Dini fractional derivative among the studied problem. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
Optimal Control for a Nonlocal Model of Non-Newtonian Fluid Flows
Mathematics 2021, 9(3), 275; https://doi.org/10.3390/math9030275 - 30 Jan 2021
Viewed by 426
Abstract
This paper deals with an optimal control problem for a nonlocal model of the steady-state flow of a differential type fluid of complexity 2 with variable viscosity. We assume that the fluid occupies a bounded three-dimensional (or two-dimensional) domain with the impermeable boundary. [...] Read more.
This paper deals with an optimal control problem for a nonlocal model of the steady-state flow of a differential type fluid of complexity 2 with variable viscosity. We assume that the fluid occupies a bounded three-dimensional (or two-dimensional) domain with the impermeable boundary. The control parameter is the external force. We discuss both strong and weak solutions. Using one result on the solvability of nonlinear operator equations with weak-to-weak and weak-to-strong continuous mappings in Sobolev spaces, we construct a weak solution that minimizes a given cost functional subject to natural conditions on the model data. Moreover, a necessary condition for the existence of strong solutions is derived. Simultaneously, we introduce the concept of the marginal function and study its properties. In particular, it is shown that the marginal function of this control system is lower semicontinuous with respect to the directed Hausdorff distance. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
New Series Solution of the Caputo Fractional Ambartsumian Delay Differential Equationation by Mittag-Leffler Functions
Mathematics 2021, 9(2), 157; https://doi.org/10.3390/math9020157 - 13 Jan 2021
Viewed by 513
Abstract
The fractional generalization of the Ambartsumian delay equation with Caputo’s fractional derivative is considered. The Ambartsumian delay equation is very difficult to be solved neither in the case of ordinary derivatives nor in the case of fractional derivatives. In this paper we combine [...] Read more.
The fractional generalization of the Ambartsumian delay equation with Caputo’s fractional derivative is considered. The Ambartsumian delay equation is very difficult to be solved neither in the case of ordinary derivatives nor in the case of fractional derivatives. In this paper we combine the Laplace transform with the Adomian decomposition method to solve the studied equation. The exact solution is obtained as a series which terms are expressed by the Mittag-Leffler functions. The advantage of the present approach over the known in the literature ones is discussed. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
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Article
Quasilinear Dirichlet Problems with Degenerated p-Laplacian and Convection Term
Mathematics 2021, 9(2), 139; https://doi.org/10.3390/math9020139 - 11 Jan 2021
Viewed by 489
Abstract
The paper develops a sub-supersolution approach for quasilinear elliptic equations driven by degenerated p-Laplacian and containing a convection term. The presence of the degenerated operator forces a substantial change to the functional setting of previous works. The existence and location of solutions [...] Read more.
The paper develops a sub-supersolution approach for quasilinear elliptic equations driven by degenerated p-Laplacian and containing a convection term. The presence of the degenerated operator forces a substantial change to the functional setting of previous works. The existence and location of solutions through a sub-supersolution is established. The abstract result is applied to find nontrivial, nonnegative and bounded solutions. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
Boundary Value Problems for Hilfer Fractional Differential Inclusions with Nonlocal Integral Boundary Conditions
Mathematics 2020, 8(11), 1905; https://doi.org/10.3390/math8111905 - 31 Oct 2020
Cited by 4 | Viewed by 405
Abstract
In this paper, we study boundary value problems for differential inclusions, involving Hilfer fractional derivatives and nonlocal integral boundary conditions. New existence results are obtained by using standard fixed point theorems for multivalued analysis. Examples illustrating our results are also presented. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
Fractional-Order Integro-Differential Multivalued Problems with Fixed and Nonlocal Anti-Periodic Boundary Conditions
Mathematics 2020, 8(10), 1774; https://doi.org/10.3390/math8101774 - 14 Oct 2020
Cited by 2 | Viewed by 459
Abstract
This paper studies a new class of fractional differential inclusions involving two Caputo fractional derivatives of different orders and a Riemann–Liouville type integral nonlinearity, supplemented with a combination of fixed and nonlocal (dual) anti-periodic boundary conditions. The existence results for the given problem [...] Read more.
This paper studies a new class of fractional differential inclusions involving two Caputo fractional derivatives of different orders and a Riemann–Liouville type integral nonlinearity, supplemented with a combination of fixed and nonlocal (dual) anti-periodic boundary conditions. The existence results for the given problem are obtained for convex and non-convex cases of the multi-valued map by applying the standard tools of the fixed point theory. Examples illustrating the obtained results are presented. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method
Mathematics 2020, 8(10), 1675; https://doi.org/10.3390/math8101675 - 01 Oct 2020
Viewed by 721
Abstract
Differential equations of fractional order are believed to be more challenging to compute compared to the integer-order differential equations due to its arbitrary properties. This study proposes a multistep method to solve fractional differential equations. The method is derived based on the concept [...] Read more.
Differential equations of fractional order are believed to be more challenging to compute compared to the integer-order differential equations due to its arbitrary properties. This study proposes a multistep method to solve fractional differential equations. The method is derived based on the concept of a third-order Adam–Bashforth numerical scheme by implementing Lagrange interpolation for fractional case, where the fractional derivatives are defined in the Caputo sense. Furthermore, the study includes a discussion on stability and convergence analysis of the method. Several numerical examples are also provided in order to validate the reliability and efficiency of the proposed method. The examples in this study cover solving linear and nonlinear fractional differential equations for the case of both single order as α(0,1) and higher order, α1,2, where α denotes the order of fractional derivatives of Dαy(t). The comparison in terms of accuracy between the proposed method and other existing methods demonstrate that the proposed method gives competitive performance as the existing methods. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
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Article
Stability of Ulam–Hyers and Existence of Solutions for Impulsive Time-Delay Semi-Linear Systems with Non-Permutable Matrices
Mathematics 2020, 8(9), 1493; https://doi.org/10.3390/math8091493 - 03 Sep 2020
Cited by 3 | Viewed by 447
Abstract
In this paper, the stability of Ulam–Hyers and existence of solutions for semi-linear time-delay systems with linear impulsive conditions are studied. The linear parts of the impulsive systems are defined by non-permutable matrices. To obtain solution for linear impulsive delay systems with non-permutable [...] Read more.
In this paper, the stability of Ulam–Hyers and existence of solutions for semi-linear time-delay systems with linear impulsive conditions are studied. The linear parts of the impulsive systems are defined by non-permutable matrices. To obtain solution for linear impulsive delay systems with non-permutable matrices in explicit form, a new concept of impulsive delayed matrix exponential is introduced. Using the representation formula and norm estimation of the impulsive delayed matrix exponential, sufficient conditions for stability of Ulam–Hyers and existence of solutions are obtained. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
Fixed Point Theorem Based Solvability of 2-Dimensional Dissipative Cubic Nonlinear Klein-Gordon Equation
Mathematics 2020, 8(7), 1103; https://doi.org/10.3390/math8071103 - 05 Jul 2020
Cited by 1 | Viewed by 574
Abstract
The purpose of this article is to establish the solvability of the 2-Dimensional dissipative cubic nonlinear Klein-Gordon equation (2DDCNLKGE) through periodic boundary value conditions (PBVCs). The analysis of this study is founded on the Galerkin’s method (GLK) and the Leray-Schauder’s fixed point theorem [...] Read more.
The purpose of this article is to establish the solvability of the 2-Dimensional dissipative cubic nonlinear Klein-Gordon equation (2DDCNLKGE) through periodic boundary value conditions (PBVCs). The analysis of this study is founded on the Galerkin’s method (GLK) and the Leray-Schauder’s fixed point theorem (LS). First, the GLK method is used to construct some uniform priori estimates of approximate solution to the corresponding equation of 2DDCNLKGE. Finally, the LS fixed point theorem is applied to obtain the efficient and straightforward existence and uniqueness criteria of time periodic solution to the 2DDCNLKGE. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
On a Class of Generalized Nonexpansive Mappings
Mathematics 2020, 8(7), 1085; https://doi.org/10.3390/math8071085 - 03 Jul 2020
Cited by 1 | Viewed by 602
Abstract
In our recent work we have introduced and studied a notion of a generalized nonexpansive mapping. In the definition of this notion the norm has been replaced by a general function satisfying certain conditions. For this new class of mappings, we have established [...] Read more.
In our recent work we have introduced and studied a notion of a generalized nonexpansive mapping. In the definition of this notion the norm has been replaced by a general function satisfying certain conditions. For this new class of mappings, we have established the existence of unique fixed points and the convergence of iterates. In the present paper we construct an example of a generalized nonexpansive self-mapping of a bounded, closed and convex set in a Hilbert space, which is not nonexpansive in the classical sense. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
On Hybrid Type Nonlinear Fractional Integrodifferential Equations
Mathematics 2020, 8(6), 984; https://doi.org/10.3390/math8060984 - 16 Jun 2020
Cited by 1 | Viewed by 508
Abstract
In this paper, we introduce and investigate a hybrid type of nonlinear Riemann Liouville fractional integro-differential equations. We develop and extend previous work on such non-fractional equations, using operator theoretical techniques, and find the approximate solutions. We prove the existence as well as [...] Read more.
In this paper, we introduce and investigate a hybrid type of nonlinear Riemann Liouville fractional integro-differential equations. We develop and extend previous work on such non-fractional equations, using operator theoretical techniques, and find the approximate solutions. We prove the existence as well as the uniqueness of the corresponding approximate solutions by using hybrid fixed point theorems and provide upper and lower bounds to these solutions. Furthermore, some examples are provided, in which the general claims in the main theorems are demonstrated. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
Multiparametric Contractions and Related Hardy-Roger Type Fixed Point Theorems
Mathematics 2020, 8(6), 957; https://doi.org/10.3390/math8060957 - 11 Jun 2020
Cited by 4 | Viewed by 628
Abstract
In this paper we present some novel fixed point theorems for a family of contractions depending on two functions (that are not defined on t = 0 ) and on some parameters that we have called multiparametric contractions. We develop our study in [...] Read more.
In this paper we present some novel fixed point theorems for a family of contractions depending on two functions (that are not defined on t = 0 ) and on some parameters that we have called multiparametric contractions. We develop our study in the setting of b-metric spaces because they allow to consider some families of functions endowed with b-metrics deriving from similarity measures that are more general than norms. Taking into account that the contractivity condition we will employ is very general (of Hardy-Rogers type), we will discuss the validation and usage of this novel condition. After that, we introduce the main results of this paper and, finally, we deduce some consequences of them which illustrates the wide applicability of the main results. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
A Regularity Criterion in Weak Spaces to Boussinesq Equations
Mathematics 2020, 8(6), 920; https://doi.org/10.3390/math8060920 - 05 Jun 2020
Cited by 33 | Viewed by 845
Abstract
In this paper, we study the regularity of weak solutions to the incompressible Boussinesq equations in R 3 × ( 0 , T ) . The main goal is to establish the regularity criterion in terms of one velocity component and the gradient [...] Read more.
In this paper, we study the regularity of weak solutions to the incompressible Boussinesq equations in R 3 × ( 0 , T ) . The main goal is to establish the regularity criterion in terms of one velocity component and the gradient of temperature in Lorentz spaces. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
Averaging Methods for Second-Order Differential Equations and Their Application for Impact Systems
Mathematics 2020, 8(6), 916; https://doi.org/10.3390/math8060916 - 04 Jun 2020
Cited by 1 | Viewed by 680
Abstract
In this paper, we discuss the averaging method for periodic systems of second order and the behavior of solutions that intersect a hyperplane. We prove an averaging theorem for impact systems. This allows us to investigate the approximate dynamics of mechanical systems, such [...] Read more.
In this paper, we discuss the averaging method for periodic systems of second order and the behavior of solutions that intersect a hyperplane. We prove an averaging theorem for impact systems. This allows us to investigate the approximate dynamics of mechanical systems, such as the weakly nonlinear and weakly periodically forced Duffing’s equation of a hard spring with an impact wall, or a weakly nonlinear and weakly periodically forced inverted pendulum with double impacts. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
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Article
On Sequential Fractional q-Hahn Integrodifference Equations
Mathematics 2020, 8(5), 753; https://doi.org/10.3390/math8050753 - 09 May 2020
Cited by 3 | Viewed by 541
Abstract
In this paper, we prove existence and uniqueness results for a fractional sequential fractional q-Hahn integrodifference equation with nonlocal mixed fractional q and fractional Hahn integral boundary condition, which is a new idea that studies q and Hahn calculus simultaneously. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited
Mathematics 2020, 8(5), 743; https://doi.org/10.3390/math8050743 - 08 May 2020
Cited by 5 | Viewed by 827
Abstract
We consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we find an equivalent Volterra integral equation with two parameter Mittag–Leffler function in the kernel to the mentioned equation. We used [...] Read more.
We consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we find an equivalent Volterra integral equation with two parameter Mittag–Leffler function in the kernel to the mentioned equation. We used the contraction mapping theorem and Weissinger’s fixed point theorem to obtain existence and uniqueness of global solution in the spaces of Lebesgue integrable functions. The new representation formula of the general solution helps us to find the fixed point problem associated with the fractional Langevin equation which its contractivity constant is independent of the friction coefficient. Two examples are discussed to illustrate the feasibility of the main theorems. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
Extended Simulation Function via Rational Expressions
Mathematics 2020, 8(5), 710; https://doi.org/10.3390/math8050710 - 03 May 2020
Cited by 8 | Viewed by 753
Abstract
In this paper, we introduce some common fixed point theorems for two distinct self-mappings in the setting of metric spaces by using the notion of a simulation function introduced in 2015. The contractivity conditions have not to be verified for all pairs of [...] Read more.
In this paper, we introduce some common fixed point theorems for two distinct self-mappings in the setting of metric spaces by using the notion of a simulation function introduced in 2015. The contractivity conditions have not to be verified for all pairs of points of the space because it is endowed with an antecedent conditions. They are also of rational type because the involved terms in the contractivity upper bound are expressed, in some cases, as quotients. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
Exact Traveling and Nano-Solitons Wave Solitons of the Ionic Waves Propagating along Microtubules in Living Cells
Mathematics 2020, 8(5), 697; https://doi.org/10.3390/math8050697 - 02 May 2020
Cited by 14 | Viewed by 756
Abstract
In this paper, the weakly nonlinear shallow-water wave model is mathematically investigated by applying the modified Riccati-expansion method and Adomian decomposition method. This model is used to describe the propagation of waves in weakly nonlinear and dispersive media. We obtain exact and solitary [...] Read more.
In this paper, the weakly nonlinear shallow-water wave model is mathematically investigated by applying the modified Riccati-expansion method and Adomian decomposition method. This model is used to describe the propagation of waves in weakly nonlinear and dispersive media. We obtain exact and solitary wave solutions of this model by using the modified Riccati-expansion method then using these solutions to determine the boundary and initial conditions. These conditions are employed to evaluate the semi-analytical wave solutions and calculate the absolute value of error. The values of absolute error show the accuracy of the obtained solutions. Some solutions are sketched to show the perspective view of the solution of this model. Moreover, the novelty of the obtained solutions is illustrated by showing the similarity and differences between our and previous solutions of the model. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
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Article
Existence and Multiplicity Results for Nonlocal Boundary Value Problems with Strong Singularity
Mathematics 2020, 8(5), 680; https://doi.org/10.3390/math8050680 - 01 May 2020
Viewed by 708
Abstract
In this paper, we study singular φ -Laplacian nonlocal boundary value problems with a nonlinearity which does not satisfy the L 1 -Carathéodory condition. The existence, nonexistence and/or multiplicity results of positive solutions are established under two different asymptotic behaviors of the nonlinearity [...] Read more.
In this paper, we study singular φ -Laplacian nonlocal boundary value problems with a nonlinearity which does not satisfy the L 1 -Carathéodory condition. The existence, nonexistence and/or multiplicity results of positive solutions are established under two different asymptotic behaviors of the nonlinearity at ∞. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
Some Identities Involving Two-Variable Partially Degenerate Hermite Polynomials Induced from Differential Equations and Structure of Their Roots
Mathematics 2020, 8(4), 632; https://doi.org/10.3390/math8040632 - 20 Apr 2020
Cited by 2 | Viewed by 674
Abstract
In this paper, we introduce two-variable partially degenerate Hermite polynomials and get some new symmetric identities for two-variable partially degenerate Hermite polynomials. We study differential equations induced from the generating functions of two-variable partially degenerate Hermite polynomials to give identities for two-variable partially [...] Read more.
In this paper, we introduce two-variable partially degenerate Hermite polynomials and get some new symmetric identities for two-variable partially degenerate Hermite polynomials. We study differential equations induced from the generating functions of two-variable partially degenerate Hermite polynomials to give identities for two-variable partially degenerate Hermite polynomials. Finally, we study the symmetric properties of the structure of the roots of the two-variable partially degenerate Hermite equations. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
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Article
Existence and Integral Representation of Scalar Riemann-Liouville Fractional Differential Equations with Delays and Impulses
Mathematics 2020, 8(4), 607; https://doi.org/10.3390/math8040607 - 16 Apr 2020
Cited by 1 | Viewed by 649
Abstract
Nonlinear scalar Riemann-Liouville fractional differential equations with a constant delay and impulses are studied and initial conditions and impulsive conditions are set up in an appropriate way. The definitions of both conditions depend significantly on the type of fractional derivative and the presence [...] Read more.
Nonlinear scalar Riemann-Liouville fractional differential equations with a constant delay and impulses are studied and initial conditions and impulsive conditions are set up in an appropriate way. The definitions of both conditions depend significantly on the type of fractional derivative and the presence of the delay in the equation. We study the case of a fixed lower limit of the fractional derivative and the case of a changeable lower limit at each impulsive time. Integral representations of the solutions in all considered cases are obtained. Existence results on finite time intervals are proved using the Banach principle. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
Limiting Values and Functional and Difference Equations
Mathematics 2020, 8(3), 407; https://doi.org/10.3390/math8030407 - 12 Mar 2020
Cited by 2 | Viewed by 781
Abstract
Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used [...] Read more.
Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. As main results, this involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We establish these by the relation between bases of the Kubert space of functions. Then these expressions are equated with other expressions in terms of special functions introduced by some difference equations, giving rise to analogues of the Lerch-Chowla-Selberg formula. We also state Abelian results which not only yield asymptotic formulas for weighted summatory function from that for the original summatory function but assures the existence of the limit expression for Laurent coefficients. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Article
Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients
Mathematics 2020, 8(3), 374; https://doi.org/10.3390/math8030374 - 07 Mar 2020
Cited by 1 | Viewed by 978
Abstract
Motivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge–Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which [...] Read more.
Motivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge–Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with a periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
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