Search

In this article, we propose novel coupled nonlinear singular boundary value problems arising in epitaxial growth theory. The coupled equations are nonlinear, non‐self‐adjoint, and singular and have no exact solutions. We derive some qualitative properties of the coupled solutions, which depend on the size of parameters that occur in the coupled system. To prove the existence of the coupled solutions, we apply the monotone iterative technique on equivalent coupled integral equations in the presence of upper and lower solutions. We also compute the bounds for the parameters, which indicates the existence of coupled solutions for small values of the parameters, and nonexistence for large positive values of these parameters. To illustrate the theory developed, we consider test cases and compute the sequence of upper and lower solutions. To verify the bounds on the parameters, we have used a type of Adomian decomposition method. [ABSTRACT FROM AUTHOR]

We formulate a Neumann variational problem, which characterizes the capillary phenomena, and disLaplacian‐like operator defined bycuss the existence of nontrivial weak solutions. We get the results using sufficient hypotheses on the reaction term. [ABSTRACT FROM AUTHOR]

In this paper, we study a free‐boundary problem modeling the growth of spherically symmetric tumors with angiogenesis and a necrotic core, where the Robin boundary condition is imposed for the nutrient concentration. The existence of a global solution is established by first reducing the free‐boundary problem into an equivalent initial boundary value problem for a nonlinear strongly singular parabolic equation on a fixed domain, then proving that an approximation problem admits a unique solution by the Schauder fixed point theorem combined with the Lp$$ {L}^p $$ estimates for parabolic equations, and finally taking the limit. Compared with the Dirichlet boundary value condition problem, the Robin condition causes some new difficulties in making rigorous analysis of the model, particularly on the uniqueness of solutions to the approximation problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of transonic shocks in a cylindrical nozzle. More precisely, given the compatible outer normal momentum at the exit of a cylindrical nozzle, there exists a transonic shock solution such that the incoming supersonic flow is compressed to some subsonic flow. The shock position is uniquely determined by the exit normal momentum when we fix one point of the shock. This problem can be reduced to solve a nonlinear free boundary value problem for a hyperbolic–elliptic coupled system and the key ingredient is to solve a second‐order quasilinear elliptic equation with a nonlocal term by the method of separation of variables. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonlinear second‐order functional differential equations with piecewise constant arguments with applications to a thermostat that is controlled by the introduction of functional terms in the temperature and the speed of change of the temperature at some fixed instants. We first prove some comparison results for boundary value problems associated to linear delay differential equations that allow to give a priori bounds for the derivative of the solutions, so that we can control not only the values of the solutions but also their rate of change. Then, we develop the method of upper and lower solutions and the monotone iterative technique in order to deduce the existence of solutions in a certain region (and find their approximations) for a class of boundary value problems, which include the periodic case. In the approximation process, since the sequences of the derivatives for the approximate solutions are, in general, not monotonic, we also give some estimates for these derivatives. We complete the paper with some examples and conclusions. [ABSTRACT FROM AUTHOR]

The novelty and main aim of the present numerical analysis is to explore the magneto‐thermo‐fluid characteristic features of incompressible, time‐dependent electrically conducting second‐grade fluid flow about a micro‐cantilever sensor sheet suspended in a squeezed regime between two parallel plates under the influence of Lorentz force field and viscous dissipation effects. Pertaining to this physical situation, the primitive forms of produced coupled nonlinear partial differential equations are rendered to non‐dimensional ordinary differential equations through appropriate scaling transformations. The resultant coupled nonlinear boundary value physical problem is solved by deploying a robust BVP4C Matlab function. The key features of the emerged different fluid flow parameters are documented in terms of extensive graphical visualization and tabular discussion. The current numerical investigation showed that, the increasing second‐grade fluid parameter significantly decreases the flow field and enhances the thermal and mass diffusion fields. Magnifying permeable velocity number decreases the velocity and increases the heat and concentration transport process. The thermal distribution over a sensor region is boosted with increasing Eckert number through viscous dissipation and magnetic Ohmic heating. Further, the elevating squeezing number decreases the skin friction and enhancing second‐grade fluid number boosted the skin‐friction coefficient. In addition to this, the increasing Schmidt and squeezing parameters decreases the local Sherwood number. Finally, a reasonable agreement between the present numerical solutions with the former results is presented. [ABSTRACT FROM AUTHOR]

Sufficient conditions of blow up in finite time of solutions to initial boundary value problems for nonlinear systems of equations of thermoelasticity type are obtained. It is shown that solutions even with large enough initial energies of the considered problems may blow up in a finite time. [ABSTRACT FROM AUTHOR]

In a paper from 1960, Felix Browder established a theorem concerning the continuation of the fixed points of a family of continuous functions ft:X→X$f_t:X\rightarrow X$ depending continuously on a parameter t∈[0,1]$t\in [0,1]$, where X$X$ is a convex and compact subset of Rn$\mathbb {R}^n$. Here, the result is presented for a compact mapping f:A×X→X$f:A\times X\rightarrow X$ where X$X$ is a convex, closed, and bounded subset of an arbitrary normed space and A$A$ is an arcwise connected topological space. Applications to nonlinear boundary value problems are given; specifically, we shall present new viewpoints of known results, introduce some novel results, and exhibit some open problems. [ABSTRACT FROM AUTHOR]

The article considers a nonlinear boundary value problem for a linear delay differential equation. To solve the problem, the idea of parametrization method, namely, the interval at which the problem is being considered, is divided into subintervals whose lengths do not exceed the values of the constant delay; constant parameters are introduced at the left ends of these intervals; a new unknown function is introduced at each subinterval. Thus, the problem under consideration is reduced to an equivalent multipoint boundary value problem for differential equations with delay containing parameters. Auxiliary Cauchy problems without delay with zero initial conditions at the left ends of the subintervals are consistently considered on each of the subintervals. Using an analog of the Cauchy formula to represent the solution of a system of linear differential equations on each of the subintervals and given nonlinear boundary conditions, an algebraic system with respect to unknown parameters is obtained. The article proposes an algorithm for finding a solution to a multipoint boundary value problem for differential equations with a delay containing parameters. At each step of the algorithm, a system of nonlinear algebraic equations is solved to determine the values of the parameters, and an analog of the Cauchy formula is used to obtain solutions to auxiliary Cauchy problems. The results obtained are demonstrated on a test problem. [ABSTRACT FROM AUTHOR]

The solution is obtained and validated by an existence and uniqueness theorem for the following nonlinear boundary value problem ddx1+δy+γy2ndydx+2xdydx=0,x>0,y(0)=0,y(∞)=1,$$ \frac{d}{dx}\left[{\left(1+\delta y+\gamma {y}^2\right)}^n\frac{dy}{dx}\right]+2x\frac{dy}{dx}=0,x>0,y(0)=0,y\left(\infty \right)=1, $$which was proposed in 1974 by Cho and Sunderland to represent a Stefan problem with a nonlinear temperature‐dependent thermal conductivity on the semi‐infinite line (0,∞)$$ \left(0,\infty \right) $$. The modified error function of two parameters φδ,γ$$ {\varphi}_{\delta, \gamma } $$ is introduced to represent the solution of the problem above, and some properties of the function are established. This generalizes the results obtained in earlier studies. [ABSTRACT FROM AUTHOR]

In the present paper, we consider an important problem from the point of view of application in sciences and engineering, namely, Riemann–Liouville nonlinear fractional boundary value problem. Under new minimal conditions on the parameters 0 ≤ s, τ ≤ 1, it is shown that, based on the upper and lower solutions method using Schauder fixed point theorem, the positive solutions in a Sobolev spaces exist. Moreover, our results are illustrated by a numerical example. [ABSTRACT FROM AUTHOR]

This research study presents a novel highly accurate matrix approach for numerical treatments of a strongly nonlinear boundary value problem occurring in the modeling of the corneal shape of the human eye. Using the technique of quasilinearization, the nonlinear model is reduced into a sequence of linearized problems. Then, a spectral collocation procedure based on novel shifted Vieta‐Fibonacci (SVF) is applied to transform each subproblem into a linear algebraic system of equations. An upper bound for the error is provided, and convergence analysis of the SVF series solution in the weighted L2$$ {L}_2 $$ norm is discussed. The technique of error correction is used to improve the SVF polynomial solutions by means of the residual error function. Various numerical tests are provided to demonstrate the accuracy and efficiency of the presented collocation algorithm. The validation of the proposed approach is shown by comparison with available existing numerical solutions. [ABSTRACT FROM AUTHOR]

In this study, an effective approach is presented to obtain a numerical solution of linear and nonlinear singular boundary value problems. The proposed method is constructed by combining reproducing kernel and Legendre polynomials. Legendre basis functions are used to get the kernel function, and then the approximate solution is obtained as a finite series sum. Comparison of numerical results is made with the results obtained by other methods available in the literature. Furthermore, efficiency and accuracy of the method are demonstrated in tabulated results and plotted graphs. The numerical outcomes demonstrate that our method is very effective, applicable, and convenient. [ABSTRACT FROM AUTHOR]

In this paper a thorough analysis is carried out of the type of order reduction that Lawson methods exhibit when used to integrate nonlinear initial boundary value problems. In particular, we focus on nonlinear reaction‐diffusion problems, and therefore, this study is important in a large number of practical applications modeled by this type of nonlinear equations. A theoretical study of the local and global error of the total discretization of the problem is carried out, taking into account both, the error coming from the space discretization and that due to the integration in time. These results are also corroborated by the numerical experiments performed in this paper. [ABSTRACT FROM AUTHOR]

In this article, we explore the monotone iterative technique (MI‐technique) to study the existence of solutions for a class of nonlinear Neumann four‐point, boundary value problems (BVPs) defined as, −ϕ(2)(z)=G(z,ϕ,ϕ(1)),00$$ \left(\partial G/\partial \phi \right)>0 $$. To demonstrate that the proposed technique is effective, we compute the solution of the nonlinear multi‐point BVPs. We don't require sign restriction on G$$ \boldsymbol{G} $$ which is very common and strict condition. [ABSTRACT FROM AUTHOR]

In this paper, numerical results for a fourth‐order nonlinear wave equation of Kirchhoff type with a viscoelastic term are established. First, the existence and uniqueness of a solution is proved by applying the Faedo–Galerkin method and the weak compact method. Next, an algorithm to the initial boundary value problem for a nonlinear evolution equation of fourth order is studied where the spatial‐variable discretization and time‐variable discretization are constructed to approximate derivatives by the difference schemes. Finally, the numerical results are shown to present the errors of proposed algorithm. Simulations of the approximate solution and the exact solution are also visualized. [ABSTRACT FROM AUTHOR]

This paper is concerned with the initial boundary value problem for a nonlinear parabolic equation with a small parameter. The existence of a boundary layer as the parameter goes to zero is obtained together with the estimation on the thickness of the boundary layer. The main result extends an earlier work of Frid and Shelukhin (1999). [ABSTRACT FROM AUTHOR]

We analyze the classical problem of finding the shape of the column that optimizes certain criteria. The new formulation proposed here may be stated as: given the critical buckling load F of the column and the length L, find cross‐sectional area A, such that the volume W of the column attains minimal value. This is a classical Clausen problem. However, in this work we shall use the generalized constitutive equations of the column that allows for shear deformation and axis compressibility. This, as well as the novel use of the first integral, are the main novelties of our work. We will formulate a nonlinear boundary value problem for post critical deformation of optimally shaped rod. Finally we show that optimally shaped rod exhibits pitchfork supercritical bifurcation at critical buckling load. [ABSTRACT FROM AUTHOR]

Math Meth Appl Sci. 2021; 1 - 14. doi: 10.1002/mma.7549 3 Cai R-Y, Kou C-H. Mittag-Leffler stabilization for coupled fractional reaction-diffusion neural networks subject to boundary matched disturbance. Math Meth Appl Sci. 2021; 1 - 19. doi: 10.1002/mma.8000 5 Wu M, Zhang J, Huang Z, Li X, Dong Y. Numerical solutions of wavelet neural networks for fractional differential equations. This special issue is devoted to recent advances in neural network methods for FDE and its applications in different areas. [Extracted from the article]

We investigate the existence and multiplicity of positive solutions for a system of Riemann‐Liouville fractional differential equations with sequential derivatives, positive parameters and sign‐changing singular nonlinearities, subject to nonlocal uncoupled boundary conditions which contain various fractional derivatives and Riemann‐Stieltjes integrals. We apply the nonlinear alternative of Leray‐Schauder type and the Guo‐Krasnosel'skii fixed point theorem in the proof of our main existence results. [ABSTRACT FROM AUTHOR]

The paper is concerned with a class of nonlinear reaction–diffusion equations with a dissipating parameter. The problem is singularly perturbed from a mathematical perspective. Solutions of these problems are known to exhibit multiscale character. There are narrow regions in which the solution has a steep gradient. To approximate the multiscale solution, we present and analyze an iterative analytic method based on a Lagrange multiplier technique. We obtain closed‐form analytic approximation to nonlinear boundary value problems through iteration. The Lagrange multiplier is obtained optimally, in a general setting, using variational theory and Liouville–Green transforms. The idea of the paper is to overcome the well‐known difficulties associated with numerical methods. Two test examples are taken into account, and rigorous comparative analysis is presented. Moreover, we compare the proposed method with others found in the literature. [ABSTRACT FROM AUTHOR]

One of the variants of the nonlocal Ginzburg–Landau equation is considered. This equation arises in the mathematical modeling of physical phenomena, such as ferromagnetism. For the corresponding initial boundary value problem in the case of periodic boundary conditions, current problems of the theory of infinite‐dimensional dynamical systems are considered. The question of the existence of smooth global solutions and a global attractor is studied. Its dimension and structure are determined. All the solutions of the nonlinear initial boundary value problem are found in explicit form. In particular, it was established that all the solutions belonging to the global attractor will be periodic or quasiperiodic functions of the temporal variable. [ABSTRACT FROM AUTHOR]

This paper provides a new proof of the existence and uniqueness of the solution for a nonlinear boundary value problem (1+δy)y′′+2x(1+γy)y′=0,0

We consider a stationary boundary value problem describing a radiative‐conductive heat transfer in a system consisting of one absolutely black body and several semitransparent bodies. To describe the radiative transfer, the integro‐differential radiative transfer equation is used. We do not take into account the dependence of the radiation intensity and the properties of semitransparent materials on the radiation frequency. We proved at the first time the unique solvability of this problem. Besides, we proved the comparison theorems and established the results on improving the properties of solutions with increasing exponents of data summability. [ABSTRACT FROM AUTHOR]

The work reported here concerns the study of a generalized nonlinear fractional boundary value problem involving ϑ‐fractional derivative in the Riemann–Liouville sense. The existence and uniqueness of positive solutions to the problem at hand are proved. Our discussion relies on the properties of Green's function, the upper and lower solutions method, and the classical fixed point theorems in a cone. Moreover, building upper and lower control functions has an effective role in the analysis. Some examples are given to justify the validity of theoretical results. [ABSTRACT FROM AUTHOR]

Mathematics is a common and widespread language that is not spoken very well in different parts of the world. Math Methods Appl Sci. 2020. https://doi.org/10.1002/mma.5967 30 Li C, Li M-M, Sun X. Finite difference/Galerkin finite element methods for a fractional heat conduction-transfer equation. Math Methods Appl Sci. 2020. https://doi.org/10.1002/mma.6011 33 Seba D, Habani S, Benaissa A, Rebai H. The Henstock-Kurzweil-Pettis integral and multiorders fractional differential equations with impulses and multipoint fractional integral boundary conditions in Banach spaces. However, it is possible to proffer a thematic special issue to bring together diverse fields of mathematics as well as a range of researchers from different parts of the world to represent a valued panel that addresses the different aspects and methods of mathematics, from pure, applied mathematics to fractional calculus. [Extracted from the article]

This paper is devoted to the existence of weak solutions for a multipoint fractional integral boundary value problem of an impulsive nonlinear differential equation involving multiorders fractional derivatives and deviating argument. We make use of an appropriate fixed point theorem combined with the technique of measures of weak noncompactness. Our investigation is considered in a Banach space. The applicability of the obtained results is illustrated by an example. [ABSTRACT FROM AUTHOR]

In this article, the first purpose is treating a coupled system of nonlinear boundary value problems (BVPs) of fractional‐order differential equations (FODEs) for existence of solutions. The corresponding fractional‐order derivative is taken in Riemann‐Liouville sense. The require results for iterative solutions are obtained by using monotone iterative techniques combine with the method of upper and lower solutions. In this regard, two sequences are established for upper and lower solutions, respectively, in which one is monotonically increasing and converges to upper solution, while other one is monotonically decreasing converges to lower solution of the considered problem. The second purpose is discussing different kinds Ulam stability results for the proposed problem. Some applications of our results are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions to abstract boundary value problems of fractional differential equations on R+. The choice of the Banach space C01(R+,E) allows the solutions to be unbounded. Our approach mainly depends on the technique of Hausdorff measure of noncompactness in conjunction with the criterion of compactness in C01(R+,E). Finally, an example in the Banach sequence space ℓ1 is given to illustrate our results. [ABSTRACT FROM AUTHOR]

The aims of this manuscript is to establish conditions for obtaining mild solutions to a coupled systems of multipoint boundary value problems (BVPs) of fractional order hybrid differential equations (FHDEs) with nonlinear perturbations of second type. In the concerned problem, we consider a proportional type delay that represent a famous class of differential equation called pantograph equations. Hence, we establish sufficient conditions for at least one mild solution to the coupled system of fractional hybrid pantograph differential equations (FHPDEs) by using Burton and couple‐type fixed‐point theorems. We also provide examples to show the applicability of our results. [ABSTRACT FROM AUTHOR]

In this article, we consider the fourth order non‐self‐adjoint singular boundary value problem 1rr1rrϕ′′′′=ϕ′ϕ″r+λ,with λ as a parameter measures the speed at which new particles are deposited. This differential equation is non‐self‐adjoint, so finding its solutions is not easy by usual methods. Also, this does not have a unique solution; therefore, finite differences and discrete methods may not be applicable. Here, to find approximate solutions, we propose to use the Adomian decomposition method (ADM) and compute the approximate solutions which are fully dependent on the size of the parameter λ. Here, we prove theoretically that the approximate solutions converge to the exact solutions. For small positive values of parameter λ, the above equation has two solutions. No solutions could be found for large positive values of the parameter. The uniqueness of the solution may not be guaranteed for a fixed positive value of the parameter. For negative values of parameter existences of the solution are always guaranteed. [ABSTRACT FROM AUTHOR]

This paper aims to study the sufficient conditions for the existence and uniqueness of solutions to the multipoint coupled boundary value problem of nonlinear Caputo–Hadamard fractional differential equations associating with nonlocal integral boundary conditions. The required conditions are obtained by using classical results of functional analysis and fixed point theory. Furthermore, the Hyers–Ulam stability of solutions is discussed, and sufficient conditions for the stability are developed. Obtained results are supported by examples and illustrated in the last section. [ABSTRACT FROM AUTHOR]

In this paper, we develop a numerical scheme to approximate the solution of a general class of nonlinear singular boundary value problems (SBVPs) subject to Neumann and Robin boundary conditions. The original differential equation has a singularity at the point x=0, which is removed via L'Hospital's law with an assumption about the derivative of the solution at the point x=0. An exponential B‐spline collocation approach is then constructed to solve the resulting boundary value problem. Convergence analysis of the method is discussed. Numerical examples are provided to illustrate the applicability and efficiency of the method. Our results are compared with those obtained by other three numerical methods such as uniform mesh cubic B‐spline collocation (UCS) method, nonstandard finite difference method, and finite difference method based on Chawla's identity. [ABSTRACT FROM AUTHOR]

In this paper, we study a new boundary value problem of coupled nonlinear multi‐term fractional differential equations supplemented with integral multi‐strip coupled boundary conditions. The modern tools of fractional analysis are applied to derive the existence and uniqueness results for the given problem. We emphasize that our results are new and enrich the literature on the topic. [ABSTRACT FROM AUTHOR]

We consider a nonlinear traction boundary value problem for the Lamé equations in an unbounded periodically perforated domain. The edges lengths of the periodicity cell are proportional to a positive parameter δ, whereas the relative size of the holes is determined by a second positive parameter ε. Under suitable assumptions on the nonlinearity, there exists a family of solutions {u(ε,δ,·)}(ε,δ)∈]0,ε′[×]0,δ′[. We analyze the asymptotic behavior of two integral functionals associated to such a family of solutions when the perturbation parameter pair (ε, δ) is close to the degenerate value (0, 0). [ABSTRACT FROM AUTHOR]

In this article, we offer a comparison in terms of computational efficiency between two techniques to avoid order reduction when using Strang method to integrate nonlinear initial boundary value problems with time‐dependent boundary conditions. We see that it is important to consider an exponential method for the integration of the linear nonhomogeneous and stiff part in the technique by Einkemmer et al. so that the latter is comparable in efficiency with that suggested by Alonso et al. Some other advantages of the technique suggested by Alonso et al. are stated in the conclusions. [ABSTRACT FROM AUTHOR]

Summary: The main purpose is the study of optimal control problem in a domain with rough boundary for the mixed Dirichlet‐Neumann boundary value problem for the strongly nonlinear elliptic equation with exponential nonlinearity. A density of surface traction u acting on a part of rough boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable functional class in which we look for solutions, we prove the consistency of the original optimal control problem and show that it admits a unique optimal solution. Then we derive a first‐order optimality system assuming the optimal solution is slightly more regular. [ABSTRACT FROM AUTHOR]

Based on the pyroelectricity, piezoelectricity, and thermoelasticity in nonlinear constitutive equations, we report a thermal piezoelectric semiconductor model under a mechanical load, a Schottky contact boundary condition, and a temperature gradient. To solve this nonlinear boundary value problem, we propose a shooting method. In comparison with the results obtained by COMSOL, the shooting method is reliable. Furthermore, the influence of the temperature gradient on the extended displacements and stress fields is discussed. The numerical results show that a small change in the temperature has the remarkable effect on a mechanical electric field in thermal piezoelectric semiconductors. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the existence and convergence of solutions to periodic boundary value problems for one‐dimensional Kirchhoff equation. By employing analytical skills and the coincidence degree method, some new results are obtained, which enrich and generalize the previous results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a nonlinear fractional boundary value problem on a star graph. By using a transformation, the suggested problem is converted into an equivalent system of fractional boundary value problem. Schaefer's fixed point theorem and Banach's contraction principle is used to establish its existence and uniqueness results. Further, different kinds of Ulam's type stability results for the proposed problem have been discussed. Finally, two examples are presented to illustrate the application of the obtained results. [ABSTRACT FROM AUTHOR]

We consider the sliding mode control (SMC) problem for a diffuse interface tumor growth model coupling a Cahn–Hilliard equation with a reaction–diffusion equation perturbed by a maximal monotone nonlinearity. We prove existence and regularity of strong solutions and, under further assumptions, a uniqueness result. Then, we show that the chosen SMC law forces the system to reach within finite time a sliding manifold that we chose in order that the tumor phase remains constant in time. [ABSTRACT FROM AUTHOR]

In this study, sinc‐Galerkin and Sinc collocation methods coupled with double exponential transformation have been developed to obtain a numerical solution of nonlinear two‐point boundary value problem arising in steady state nonlinear reaction diffusion equation containing a nonlinear term related to Michaelis‐Menten of the enzymatic reaction. Application of these methods reduce the solution of the Michaelis‐Menten equation to the solution of nonlinear system of algebraic equations. The upper bound of the error was calculated. Two problems of this model, chosen from the literature, were considered and for several parameters, numerical solution was shown. Obtained numerical results and error of the methods show that these methods are very fast with high efficiency in convergence. [ABSTRACT FROM AUTHOR]

We prove the global existence and blow‐up of solutions of an initial boundary value problem for nonlinear nonlocal parabolic equation with nonlinear nonlocal boundary condition. Obtained results depend on the behavior of variable coefficients for large values of time. [ABSTRACT FROM AUTHOR]

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlocal boundary value problem of nonlinear impulsive Langevin equation with mixed derivatives. Some sufficient conditions are constructed to observe the existence, uniqueness, and generalized Ulam‐Hyers‐Rassias stability of our proposed model, with the help of Diaz‐Margolis' fixed‐point approach over generalized complete metric space. We give an example that supports our main result. [ABSTRACT FROM AUTHOR]

Inverse problems in partial deferential equations are severely ill posed in the sense of Hadamard. So the heat equation with a terminal condition problem is ill posed even in the sobolev space so regularization is needed. In this paper, we discuss about the convergence result of the approximation problem in the Sobolev space H2(Rn), which is well posed. By using a small parameter, we construct an approximation problem and use a quasi‐boundary value method to regularize nonlinear heat equation. Finally, we prove the approximation solution converges to the original solution whenever the parameter goes to zero in H2(Rn). [ABSTRACT FROM AUTHOR]

In this paper, the sinc‐collocation method (SCM) is investigated to obtain the solution of the nonlinear fractional order differential equations based on the relatively new defined fractional derivative, beta‐derivative. For this purpose, a theorem is proved for the approximate solution obtained from the SCM. Moreover, convergence analysis of the SCM is presented. To show the efficiency and the simplicity of the proposed method, some examples are solved, and the results are compared with the exact solutions of the considered equations. [ABSTRACT FROM AUTHOR]

The paper is concerned with the solvability for several nonlinear boundary value problems of fractional p‐Laplacian differential equation involving the right‐handed Riemann‐Liouville derivative. By applying monotone iterative technique, lower and upper solutions method and the Banach fixed point theorem, sufficient conditions for existence and uniqueness of extremal solutions are obtained and they extend existing results. At last, two examples are provided to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, an initial boundary value problem for nonlinear Klein‐Gordon equation is considered. Giving an additional condition, a time‐dependent coefficient multiplying nonlinear term is determined, and existence and uniqueness theorem for small times is proved. The finite difference method is proposed for solving the inverse problem. [ABSTRACT FROM AUTHOR]

Linear and nonlinear elliptic complex partial differential equations of higher‐order are considered under Schwarz conditions in the upper‐half plane. Firstly, using the integral representations for the solutions of the inhomogeneous polyanalytic equation with Schwarz conditions, a class of integral operators is introduced together with some of their properties. Then, these operators are used to transform the problem for linear equations into singular integral equations. In the case of nonlinear equations such a transformation yields a system of integro‐differential equations. Existence of the solutions of the relevant boundary value problems for linear and nonlinear equations are discussed via Fredholm theory and fixed point theorems, respectively. [ABSTRACT FROM AUTHOR]