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The problem of two phase bio-convective nano fluid flow past a vertical stretching flat surface in presence of volume fraction of the dust particles has been investigated. Governing partial differential equations of the problem for both fluid and dust phases are transformed into ordinary differential equations using suitable similarity transformations. The resulting non-linear differential equations are solved numerically using MATLAB built-in bvp4c solver scheme. The velocity, temperature of fluid motion and concentration of micro-organisms are presented graphically for various flow parameters. Coefficient of local skin friction, local Nusselt number and local density number of the micro-organisms are calculated and presented in tables for various parameters.

This paper considers a new bi-directional cascaded system of a fractional ordinary differential equation (FODE) and a fractional partial differential equation (FPDE) which interacts at an intermediate point. The space-dependent coefficients, interaction between the FODE and FPDE at an intermediate point and the presence of fractional calculus makes the FODE-FPDE cascaded system, representative. In this note, we first apply an invertible integral transformation to convert the system into a FODE-FPDE coupled system, as the target system, which is Mittag–Leffler stable. Using the backstepping method and under some assumptions of the space-dependent coefficients, we work out the kernel functions in the transformation and we design a boundary controller. Also, by the invertibility of the transformation, we show the Mittag–Leffler stability of the closed-loop system via the Lyapunov approach. Second, we propose an observer for which we prove that it can well estimate the original cascaded system. Then, we design an output feedback boundary control law and show that the closed-loop system is Mittag–Leffler stable under the designed output feedback control law. Finally, we present some numerical illustrations to show the correctness of the theoretical results.

This work aims to present nonlinear models that arise in ocean engineering. There are many models of ocean waves that are present in nature. In shallow water, the linearization of the equations requires critical conditions on wave capacity than it make in deep water, and the strong nonlinear belongings are spotted. We use Lie symmetry analysis to obtain different types of soliton solutions like one, two, and three-soliton solutions in a (2+1) dimensional variable-coefficient Bogoyavlensky Konopelchenko (VCBK) equation that describes the interaction of a Riemann wave reproducing along the y-axis and a long wave reproducing along the x-axis in engineering and science. We use the Lie symmetry analysis then the integrating factor method to obtain new solutions of the VCBK equation. To demonstrate the physical meaning of the solutions obtained by the presented techniques, the graphical performance has been demonstrated with some values. The presented equation has fewer dimensions and is reduced to ordinary differential equations using the Lie symmetry technique.

The depth of the deep neural network (DNN) refers to the number of hidden layers between the input and output layers of an artificial neural network. It usually indicates a certain degree of complexity of the computational cost (parameters and floating point operations per second) and expressiveness once the network structure is settled. In this study, we experimentally investigate the effectiveness of using neural ordinary differential equations (NODEs) as a component to provide further depth to relatively shallower networks rather than stacked layers (depth), which achieved an improvement with fewer parameters. Experiments are conducted on classic DNNs, including residual networks. Moreover, we construct infinite deep neural networks with flexible complexity based on NODEs, enabling the system to adjust its complexity during training. On a better hidden-space provided by adaptive step DNNs, adaptive step ResNet with NODE (ResODE) is managed to achieve better performances in terms of convergence and accuracy ( Fig. 1 ) than standard networks, and the improvements are widely observed in popular benchmarks.

Information diffusion in social Internet of Things (SIoT) brings convenience to life, but it also faces challenges. The explosive growth of information exceeds the computing capability of the network, making it difficult to control. Motivated by the challenge, a novel control strategy for information diffusion is proposed in SIoT, which is based on the advantages of edge computing with low latency, high energy efficiency and timely feedback. Furthermore, graph neural networks (GNNs) is combined with ordinary differential equation systems (ODEs) to model edge-aided control dynamics, describing high-dimensionality and complicated interactions. Theoretical analysis and comparative simulation results show that the proposed edge-aided control strategy can dominate information diffusion within the target scope at the lowest cost.

This paper presents a study of the performance of the Tau method using orthogonal polynomials as basis functions for solving fourth-order partial differential equations with boundary conditions. Because of the good numerical stability properties of integral operators in compare to differential operator, we first convert this PDE problem to Volterra-Fredholm integral equation and then apply the numerical Tau method to solve the obtained integral equation. Applying the Tau method yields a system of the ordinary differential equation such that this system is solved by piecewise polynomial collocation methods. Convergence analysis and error estimation of the Tau method are discussed. The advantages of converting PDE to integral equation are shown by the numerical examples. For this aim, we consider two cases to solve the proposed examples. In case1, we apply the Tau method to solve the converted problem (Volterra-Fredholm integral equation) and in case 2, we solve PDE problem directly by Tau method. Comparing the numerical results, we observe that the obtained errors in case 1 are less than the errors in case 2.

Asymptotic integration theory gives a collection of results which provide a thorough description of the asymptotic growth and zero distribution of solutions of (*) f ′ ′ + P ( z ) f = 0 , where P ( z ) is a polynomial. These results have been used by several authors to find interesting properties of solutions of (*). That said, many people have remarked that the proofs and discussion concerning asymptotic integration theory that are, for example, in E. Hille’s 1969 book Lectures on Ordinary Differential Equations are difficult to follow. The main purpose of this paper is to make this theory more understandable and accessible by giving complete explanations of the reasoning used to prove the theory and by writing full and clear statements of the results. A considerable part of the presentation and explanation of the material is different from that in Hille’s book.

Based on the integral transform theorem and the extended precise integration method (EPIM), this paper investigates the axisymmetric and in-plane transient dynamic responses of multilayered transversely isotropic (TI) poroelastic media. First, the governing equations are transformed into ordinary differential equations by virtue of the integral transform technique. It can be found that the transformed equations of the axisymmetric problem are similar to those of the in-plane problem. Then, we introduce the boundary conditions and apply the EPIM to obtain the transient dynamic solution in the transformed domain. The actual solution is further gotten by the numerical transform inversion. Various suddenly applied load forms are considered in this work, including concentrated point loads, line loads, uniform strip loads and circular loads. Several numerical examples are provided to verify the proposed theory and to investigate the differences of the transient problems in cylindrical coordinates and Cartesian coordinates. The influences of transverse isotropy and load forms on transient behaviors are also analyzed.

The viscoelastic fluid flow over a non linear stretching porous sheet is considered in this analysis. The case of suction and injection is also discuused. The effects of thermophoresis and brownain motion with viscous dissipation is taken into account. Under the flow assumptions, boundary layer approximation applied on the mathematical model and developed the partial differential equations. The similarity variable is applied on the partial differential equations which converted into ordinary differential equations. The dimensionless system further solved through numerical technique bvp4c method. The results of the current problem achieved after solving the problem which presented through graphs and table. Influence of numerous physical parameters on velocity function, concentration function and temperature function are manipulated through graphs. The tabular results are schemed to display skin friction and local Nusselt number. The higher values of N b which enhanced the velocity profile but declines velocity profile due to higher values of Ec . The thermal thickness enhances due larger values of γ and Pr but thermal thickness declines due to higher values of Ec and N t . The concentration profile enhances due to enhancing the values of Ec and Pr but concentration profile declines due to increasing the values of N t . Nusselt number enhaced due to higher values of Ec , S c and N t but declines of higher values of Pr .

In this paper we develop a numerical method for pricing American options under regime-switching model, whose solutions are option buyer indifference prices. The problem is formulated as a nonlinear complementarity problem. We apply interior penalty method to approximate the differential complementarity problem, which results in a system of one degenerate parabolic equation and one ordinary differential equation, weakly coupled by nonlinear exponential term. We formulate the problems in variational form in appropriate spaces and prove comparison principle. Then, as a consequence, we derive estimates about the convergence rate of the interior penalty method. We discretize the penalized problem by fully implicit finite difference scheme and prove that the early exercise constraint is strictly satisfied. We establish comparison principle, uniqueness and boundedness of the numerical solution. To solve the nonlinear system of algebraic equations we use Newton method and partial Newton method. We present numerical results that confirm the theoretical statements. To improve the computational efficiency, we unfold the two-grid idea, combining both iteration processes – Newton method and partial Newton method. We illustrate the efficiency of this approach.

The present paper is concerned with the solutions of first-order linear fuzzy differential equations where differentiability of fuzzy functions is defined under an assumption of linear correlation between fuzzy numbers. Motivated by the fact that the structure of the space of linearly correlated fuzzy numbers strongly depends on the symmetry of the basic fuzzy number, here we address first-order linear fuzzy differential equations by distinguishing whether the basic fuzzy number is symmetric or not. In the non-symmetric case, a first-order linear fuzzy differential equation may be transformed into an equivalent system of ordinary differential equations related to the representation functions of the linearly correlated fuzzy number-valued function. In the symmetric case, according to the monotonicity of the diameter of the fuzzy solution, a first-order linear fuzzy differential equation may be transformed into a system of ordinary differential equations associated with the representation functions of the canonical form of the linearly correlated fuzzy number-valued function. In addition, using our extension method one may obtain solutions with either increasing or decreasing diameters. Several examples are provided in order to illustrate the proposed method.

We investigate the dynamics of a delay differential equation obtained by perturbing a vector field f : R n → R n , admitting a stable periodic orbit, by using a delayed feedback control term η g ( x , x ( t − τ ) ) of same regularity, where η is small and τ large so that ητ be bounded but non small. We prove that trajectories starting in a neighborhood (of size independent of the parameters η , τ ) of this original periodic orbit enter asymptotically a periodic regime, and that the number of such distinct periodic regimes increases (almost) linearly when ητ increases infinitely. Our result is based on the construction of an invariant manifold via a process inspired by the Lyapunov-Perron method for integral operators associated to ordinary differential equations, and on the persistence of normally hyperbolic invariant manifolds for semi-flows on Banach spaces. The statement we provide here complements already known results on periodic orbits of delay differential equations.

A parameter uniform numerical method is presented for solving singularly perturbed time-dependent differential–difference equations with small shifts. To approximate the terms with the shifts, Taylor’s series expansion is used. The resulting singularly perturbed parabolic partial differential equation is solved using an implicit Euler method in temporal direction and cubic B-spline collocation method for the resulting system of ordinary differential equations in spatial direction, and an artificial viscosity is introduced in the scheme using the theory of singular perturbations. The proposed method is shown to be accurate of order O Δ t + h 2 by preserving ɛ -uniform convergence, where h and Δ t denote spatial and temporal step sizes, respectively. Several test examples are solved to demonstrate the effectiveness of the proposed method. The computed numerical results show that the proposed method provides more accurate results than some methods exist in the literature and suitable for solving such problems with little computational effort.

A mathematical model for the two-dimensional flow of tangent hyperbolic nanofluid over a Riga plate in the existence of gyrotactic microorganisms is discussed. For velocity, temperature, and concentration, Wu's velocity slip, thermal convection, and zero nanoparticle flux conditions are imposed. Nanofluids gained tremendous notability due to their wide spread industrial, technological, and chemical applications. A set of pertinent transformations has been suggested to transform the governing non-linear partial differential equations into a system of non-linear ordinary differential equations (ODEs). The methodology involves solving the dimensionless equations using the bvp4c technique for computing the numerical solution. Additionally, the numerical results are constructed for these obtained equations by using the bvp4c tool in MATLAB. The effects of numerous prominent parameters on concentration, velocity, temperature, and motile microorganism profiles are executed graphically. From the obtained results it is noted that the velocity of the tangent hyperbolic fluid is boosted up by growing the variations of mixed convection parameters. The temperature of nanofluid behavior is reduced for a larger Prandtl number. Concentration of nano particles is enhanced by growing activation energy parameters. Microorganisms profile is improved with first-order velocity slip parameter while decaying for bio-convection Lewis number.

In the presented research article, the intelligence based numerical computation of artificial neural network backpropagated with Levenberg-Marquardt algorithm has been developed to analyze the novel ferrofluid flow model in the presence of magnetic dipole. Heat transfer effects are also incorporated along the horizontal. The designed fluid flow model initially represented by system of partial differential equations are converted into system of non-linear ordinary differential equations through suitable similarity transformations. The reference dataset of the possible outcomes is obtained from Adam numerical solver for the different scenarios of flow model by variation of co-efficient of the thermal expansion, Eckert number, suction parameter, magnetization and radiation parameter. The approximated solutions are interpreted for designed model by testing, training and validation process of backpropagated neural networks. Furthermore, the comparative studies and performance analysis of used algorithm is validated through regression analysis, histogram studies, correlation index and results of mean square error.

This paper is mainly concerned with the stability of inertial delayed neural networks (IDNNs) with stochastic delayed impulses. In contrast with previous related works, the stochastic impulsive intensity and density, and impulsive delays are considered simultaneously. Firstly, by appropriate variable substitution, the considered IDNNs model is converted to a first order differential equation with stochastic delayed impulsive effect. Secondly, by using the matrix measure approach and some stochastic analysis techniques, the stability criteria of IDNNs with stochastic delayed impulses are presented. Finally, numerical examples are provided to illustrate the obtained theoretical results.

The stability and inertia of sign pattern matrices with entries in { + , − , 0 } associated with dynamical systems of second-order ordinary differential equations x ¨ = A x ˙ + B x are studied, where A and B are real matrices of order n. An equivalent system of first-order differential equations has coefficient matrix C = [ A B I O ] of order 2n, and eigenvalue properties are considered for sign patterns C = [ A B D O ] of order 2n, where A , B are the sign patterns of A , B respectively, and D is a positive diagonal sign pattern. For given sign patterns A and B where one of them is a negative diagonal sign pattern, results are determined concerning the potential stability and sign stability of C , as well as the refined inertia of C . Applications include the stability of such dynamical systems in which only the signs rather than the magnitudes of entries of the matrices A and B are known.

Linear first order Riemann-Liouville fractional differential equations are studied. These new equations unify and generalize the Riemann-Liouville, modified Caputo and Caputo fractional differential equations. The equivalences between the fractional differential equations and the corresponding perturbed Abel's integral equations are obtained. These results are useful not only to study the initial or boundary value problems for nonlinear first order Riemann-Liouville fractional differential equations but also to study the solutions of the perturbed Abel's integral equations arising in a problem of mechanics and many other physical problems. The well-known Tonelli's result on solvability of the Abel's integral equation is generalized. We exhibit that there are nonconstant equilibria for some first order Caputo fractional equations. This is different from nonlinear first order ordinary differential equations which have only constant equilibria. The equivalence results are applied to generalize the classical Mean Value Theorem to the first order Riemann-Liouville fractional derivatives.

This paper continues our research devoted to an accurate nonlinear Bayesian filters' design. Our solution implies numerical methods for solving ordinary differential equations (ODE) when propagating the mean and error covariance of the dynamic state. The key idea is that an accurate implementation strategy implies the methods with a discretization error control involved. This means that the filters' moment differential equations are to be solved accurately, i.e. with negligible error. In this paper, we explore the continuous-discrete extended-cubature Kalman filter that is a hybrid method between Extended and Cubature Kalman filters (CKF). Motivated by recent results obtained for the continuous-discrete CKF in Bayesian filtering realm, we propose the numerically stable (to roundoff) square-root approach within a singular value decomposition (SVD) for the hybrid filter. The new method is extensively tested on a few application examples including stiff systems.

Given an autonomous first order algebraic ordinary differential equation F ( y , y ′ ) = 0 , we prove that every formal Puiseux series solution of F ( y , y ′ ) = 0 , expanded around any finite point or at infinity, is convergent. The proof is constructive and we provide an algorithm to describe all such Puiseux series solutions. Moreover, we show that for any point in the complex plane there exists a solution of the differential equation which defines an analytic curve passing through this point.

The present paper is concerned with nonlinear vibration of a simply supported functionally graded graphene reinforced composite (GRC) cylindrical panel subjected to the transverse excitation. Functionally graded structure is proposed to distribute more graphene platelets (GPLs) to the most needed layer for the optimal performance. The modified Halpin-Tsai model and the rule of mixture are carried out to predict the effective material properties of the GRC cylindrical panel, such as Young's modulus, Poisson's ratio and mass density. Nonlinear partial differential equations of motion are established within the framework of Hamilton principle in conjunction with first-order shear deformation theory (FSDT) and von Karman geometric nonlinearity. Navier approach is adopted to calculate natural frequencies of the GRC cylindrical panel. Then, considering the external excitation, Galerkin method is applied to realize the model reduction, which is expressed by the ordinary differential equations. Backward Differentiation Formula (BDF) is used to perform nonlinear transient responses of the system subjected to the pulse excitation (step load or triangular load). Finally, Runge-Kutta method is employed to carry out bifurcations and chaotic dynamics of the system subjected to the steady-state excitation. Results are presented in the form of natural frequencies and nonlinear responses to perform the parametric studies, such as distribution types and weight fractions of GPLs, geometry parameters and excitation parameters of the GRC cylindrical panel.

We extend the theory of lower and upper solutions to planar systems of ordinary differential equations with separated boundary conditions, both in the well-ordered and in the non-well-ordered cases. We are able to deal with general Sturm–Liouville boundary conditions in the well-ordered case, and we analyze the Dirichlet problem in the non-well-ordered case. Our results apply in particular to scalar second order differential equations, including those driven by the mean curvature operator. Higher dimensional systems are also treated, with the same approach.

The present investigation illustrates the parametric instability of damped functionally graded carbon nanotube (CNT) reinforced composite cylindrical panel subjected to periodic in-plane excitations and thermal environment. The Rayleigh damping is taken into consideration to incorporate the damping in the present formulation. The shell panel model accounts for the shear deformation theory of higher-order and geometric nonlinearity of the von-Karman type. The functional properties of the material for carbon nanotube reinforced composite (CNTRC) panels are considered to be temperature-dependent. The governing partial differential equations (PDE) are transformed into ordinary differential equations (ODE) through Galerkin’s approach. The dynamic instability regions are obtained for the periods T and 2T as suggested by Bolotin, and the time history responses are evaluated following Newmark’s integration technique. Numerical investigation examined the influence of damping on the parametric instability of CNTRC shell panels. In addition to this, the present analysis also explores the effect of CNT distribution, volume fraction, curvature effect, thermal environment, and static pre-loading on the instability responses of the shell panels.

The current work aims at studying a constructed mathematical model with an examination of heat transfer in water-based nanofluids containing ferromagnetic nanoparticles flowing between parallel stretchable spinning discs with variable viscosity influence and variable conductivity. The nonlinear coupling of the ordinary differential equations of the momentum and energy equation with the partial differential equations based on the Navier-Stokes equation employing some influential similarity transformations. The transformed system of ordinary differential equations has been solved through the Chebyshev spectral collocation procedure (CSCP). The numerical results for the velocity and temperature distributions are shown in terms of graphical presentations. The existing available literature was utilized to test for validation of the numerical findings. The outcomes demonstrate that the stretching of the lower and upper disks and spinning parameters strengthens the impetus boundary layer and diminished the temperature boundary layer, whilst the variable thermal conductivity improved the convective and conductive strength of the ferromagnetic nanoparticles considered, and the Fe3O4 nanofluid displays a higher thermal conductivity strength than the Mn-ZnFe2O4 nanofluid.

The current research highlights the impact of thermal stratification on the magnetohydrodynamic flow of F e 3 O 4 hybrid carbon nanotubes (CNTs) scattered in the mixture of carrier-based fluid such as ethylene glycol and water (50–50%) over a slender stretching sheet. Single-walled and multi-walled carbon nanotubes were considered in the current study. The problem was initially constructed in the form of partial differential equations (PDEs) and then transformed to ordinary differential equations (ODEs) by using a suitable set of transformations. These transformed ODEs were solved using a built-in code accessible in MATLAB software called the boundary value problem of fourth-order (bvp4c). The emerging dimensionless constraints were achieved after utilizing the transformations and then later numerically computed. The outcomes indicated that the velocities and temperature augment due to the hybrid nanoparticles, while the velocities initially enhance and then decline in the case of decelerated flow ( A ≤ 0 ) . However, the reverse impact was noted for the profiles of temperature distribution. The magnetic field reduces the velocities but augments the temperature distribution. Moreover, the thermal stratification elevates the temperature as well as the velocities.

Here, continuous two-dimensional boundary layer flow of a Fe2O3 CuO/water hybrid nanofluid over a porous stretching/shrinking wedge with radiation and MHD effects has been investigated semi–analytically. The dual solutions of the problem and also its stability analysis are taken into account. Moreover, the mass-based procedure for hybrid nanofluid modeling has been used. According to this procedure, the volume fraction of first and second nanoparticles is written in terms of both nanoparticles as well as base fluid masses. The Tiwari-Das model joined with mass-based hybrid nanofluid procedure is applied to find the governing partial differential equations which are then transferred to a set of dimensionless ordinary differential equations with help of the similarity transform method. The numerical method of solution has been chosen based on a famous finite difference scheme (bvp4c) from MATLAB software. The results demonstrate that dual solutions exist for a certain domain of the wedge stretching/shrinking parameter. Also it is observed that always the first solutions have thinner boundary layer thickness than the second ones. Moreover, the critical value in which the solution is in existence increase with the enhancement of magnetic parameter and suction parameter. The magnetic parameter and the mass suction at the wedge's surface also leads to the increase in the local Nusselt number. Further, it is inferred that the second nanoparticle's mass enhancement results in the amplification of the skin friction coefficient and the local Nusselt number for the first solutions. In addition, the rise in radiation parameter tends to enhance the thermal boundary layer thickness.

Neural networks are vulnerable to adversarial input perturbations imperceptible to human, which calls for robust machine learning for safety-critical applications. In this paper, we propose a new neural ODE layer which is inspired by Hopfield-type neural networks. We prove that the proposed ODE layer has global asymptotic stability on the projected space, which implies the existence and uniqueness of its steady state. We further show that the proposed layer satisfies the local stability condition such that the output is Lipschitz continuous in the ODE layer input, guaranteeing that the norm of perturbation on the hidden state does not grow over time. By experiments we show that an appropriate level of stability constraints imposed on the proposed ODE layer can improve the adversarial robustness of ODE layers, and present a heuristic method for finding good hyperparameters for stability constraints.

Hybrid nanofluid has become one of the major interest topics among researchers nowadays due to its significant impact in myriad applications. This paper modeled and explored the properties of flow and heat transfer of mixed convection stagnation point flow for hybrid nanofluid (alumina-copper/water) on a vertical plate with slips and suction. By using the method of similarity transformation, the governing set of partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs). These derived equations are then evaluated numerically by adopting the bvp4c function that is available in Matlab software. In the presence of slips together with the opposing flow of mixed convection, the local Nusselt number (heat transfer rate) can be augmented by increasing the solid volume fraction of copper. The surge of 0.5% of copper volume fraction can accelerate the heat transfer rate by approximately 1.3% in this present analysis within the specified value of slips, mixed convection, and suction. The stability analysis is conducted due to the existence of dual solutions and only the first solution is stable for practical application.

In present paper, a new technique is developed named Uniform Algebraic Trigonometric (UAT) tension B-spline based DQM, for the better numerical approximation of coupled 1D and coupled 2D Burgers’ equation. The formula of UAT tension B-spline of order 4 is developed in this paper, by using above-mentioned basis function. Strong stability preserving SSP-RK 43 method is implemented to solve obtained ordinary differential equations. To check the effectiveness and accuracy of the proposed scheme, L 2 and L ∞ error norms are used. Four numerical experiments are discussed to put light upon applicability of scheme. By means of tables and figures, results obtained are given. Obtained results are in a good agreement with existing ones. Stability of proposed scheme is also checked by implementing the Matrix stability analysis method. This novel scheme will surely open some new dimensions in the field of numerical approximation, for other researchers in their future work.

We extend to the multidimensional case a Wong–Zakai-type theorem proved by Hu and Oksendal (1996) for scalar quasi-linear Ito stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Ito’s SDEs, we consider for any finite partition of the time interval [ 0 , T ] a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional case this type of equations can be reduced, by means of a transformation related to the method of characteristics, to the study of a random ordinary differential equation. Here, instead, one is naturally led to the investigation of a semilinear hyperbolic system of partial differential equations that we utilize for constructing a solution of the Wong–Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker–Planck equation and that the sequence converges to the solution of the Ito equation, as the mesh of the partition tends to zero.

This paper addresses the stabilization issue of linear time delay system with input saturation and distinct input delays via predictor feedback boundary control algorithm by employing transport partial differential equations (PDEs). First, the addressed ordinary differential equation (ODE) system with input delay is equivalently represented as a cascade of an ODE and transport PDEs. Second, by employing the backstepping Volterra integral transformation technique, the equivalent cascade system is transformed into a stable target system, whose kernels are solved by the constraints satisfying transport PDEs. Third, based on the boundary conditions of the obtained invertible transformation, the proposed feedback control law can be formulated. Fourth, by applying semigroup operator theory, the well-posedness of the resulting system is proved and consequently, novel exponential stability conditions of the addressed system are established. Then, the domain of attraction region under the given actuator saturation constraints is estimated with the help of the solution of obtained stability conditions. Finally, a demonstrative simulation example is offered to verify the feasibility and usefulness of the results.

In the recent years, rapid developments in nanotechnology have developed a great prospects for researchers to check up. Thermal performance of nanofluid is well scrutinized by global scientists. Keeping aforesaid pragmatism of nanofluids, we have considered Sutterby liquid under thermophoretic and Brownian movement's aspects. Radiation and stratification phenomenon for thermal analysis of Sutterby nanoliquid are considered. By invoking some appropriate transformations and boundary conditions the given partial differential equations are converted into coupled system of ODEs (ordinary differential equations). Then by using Bvp4c algorithm, we resolved these ODEs numerically. Graphs are designed to check the behavior of appropriate parameters on velocity, temperature, and concentration distribution. Consequences extracts that increase in Hartmann number decays the velocity profile whereas opposite behavior is accounted for temperature distribution. It is also observed that for larger values of thermal stratification phenomenon deteriorates the transportation rate of heat. Furthermore, Sutterby fluid model calculates the features of pseudoplastic plus dilatant solutions. This study is very suitable for extraordinary polymer resolutions plus polymer melts.

We study unbendable rational curves, i.e., nonsingular rational curves in a complex manifold of dimension $n$ with normal bundles isomorphic to $\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus p} \oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus (n-1-p)}$ for some nonnegative integer $p$. Well-known examples arise from algebraic geometry as general minimal rational curves of uniruled projective manifolds. After describing the relations between the differential geometric properties of the natural distributions on the deformation spaces of unbendable rational curves and the projective geometric properties of their varieties of minimal rational tangents, we concentrate on the case of $p=1$ and $n \leq 5$, which is the simplest nontrivial situation. In this case, the families of unbendable rational curves fall essentially into two classes: Goursat type or Cartan type. Those of Goursat type arise from ordinary differential equations and those of Cartan type have special features related to contact geometry. We show that the family of lines on any nonsingular cubic 4-fold is of Goursat type, whereas the family of lines on a general quartic 5-fold is of Cartan type, in the proof of which the projective geometry of varieties of minimal rational tangents plays a key role., Comment: To appear in Journal de Math\'ematiques Pures et Appliqu\'ees

Boundary layer flow in the industrial applications such as extrusion processes is attributable to impulsive movement of an extensible moving surface. However, the velocity of the extruded surface may not necessarily be linear. Hence, this study proposes a unique model to investigate the unsteady stagnation point flow in hybrid nanofluid (Al2O3-Cu/H2O) induced by exponentially permeable stretching/shrinking sheet with the magnetic field. The system of ordinary differential equations is attained through a simplification of governing partial differential equations by employing appropriate similarity transformation. The numerical computation is determined by utilising the bvp4c procedure in the MATLAB program, which is proficient in generating non-uniqueness solutions. The noteworthy findings revealed that the skin friction coefficient intensifies in conjunction with the local Nusselt number by enhancing the suction/injection parameter and the nanoparticles volume fraction past an exponentially stretching/shrinking sheet. The results also signified that the increment in the unsteady parameter conclusively increases the skin friction coefficient, and the heat transfer rate shows an upsurge attribution resulting from the inclusion of the magnetic parameter towards an exponentially shrinking sheet. Also, the results are evidenced by having dual solutions, which ultimately lead to an investigation of the analysis of the solution stability, thereby validating the feasibility of the first solution. This crucial contribution will help to enhance industrial growth, especially in the manufacturing and processing sectors.

Fault-tolerant control (FTC) strategy based on adaptive integral sliding mode (AISM) is proposed to stabilize a flexible satellite, subject to inertia uncertainties, external disturbances, and actuator faults. Using Hamilton’s principle, the satellite dynamics is presented as a coupled Ordinary Differential Equation (ODE) and Partial Differential Equation (PDE). The control scheme is based on the infinite-dimensional model of the flexible satellite with no discretization, so the spillover instability phenomenon is eliminated. This is the most important advantage of the proposed control scheme over the previous FTC schemes that have been used for the flexible satellite. Stabilization and vibration suppression are performed using control torque that is applied to the rigid center, and there is no need to implement in-domain actuators on panels to stabilize their vibration. First, a novel nominal controller based on the infinite-dimensional model of the satellite is designed for a healthy system. Then, an integral sliding surface, including angular velocities, internal reaction torques, and nominal control, is proposed. Finally, an AISM controller with an adaptive estimator is designed to accommodate actuator faults and other uncertainties. System stability is guaranteed for small changes in a neighborhood around the sliding surface with simultaneous vibration damping. Numerical simulations illustrate the effectiveness of the proposed control strategy.

Second derivative general linear methods (SGLMs) have been already implemented in a variable stepsize environment using Nordsieck technique. In this paper, we introduce variable stepsize SGLMs directly on nonuniform grid. By deriving the order conditions of the proposed methods of order p and stage order q = p , some explicit examples of these methods up to order four are given. By some numerical experiments, we show the efficiency of the proposed methods in solving nonstiff problems and confirm the theoretical order of convergence.

In this paper, we are interested in investigating stability results for generalized ordinary differential equations (generalized ODEs in short), and their applications to measure differential equations and dynamic equations on time scales. First, we establish stability, asymptotic and exponential stability for the trivial solution of generalized ODEs. Secondly, we use the well known correspondence between solutions of generalized ODEs and solutions of measure differential equations, obtaining analogues results for the last equations. Finally, we apply some of these results for dynamic equations on time scales.

This article studies the implications of bio-convection and activation energy on Reiner–Rivlin nanofluid transportation over a disk in rotation with partial slips. A system of nonlinear differential equations in coupled form is acquired through the fundamental relations of Reiner–Rivlin fluids. A wide range of non-Newtonian fluid frameworks and coefficients of slip are notified for the useful numerical analysis to explain semblance conditions. Partial differential equations are transformed into ordinary differential equations with the help of Runge–Kutta order 4 method with the shooting approach. The impact of various parameters is discussed and drawn physically with the help of graphs. The influence of increasing values of Reiner–Rivlin fluid parameter K on velocity profile emerged to demonstrate a decrement in it. For greater the values of coefficients of wall slip, we consummated to accumulate the rotatory disk is the uniformly greater value of torque is required.

Graphical abstract, Highlights • A new general integral transform which is covered all class of integral transform in the class of Laplace transform. • We investigated the application of this new transform for solving ODE with constant and variable coefficient. • This new transform can handle easily for fractional order integral equations and fractional order differential equations. • We have discussed the advantage and disadvantage of other integral transformed which is defined during last 2 decades. • We proved the related theorems for this new transform., Introduction Integral transforms are important to solve real problems. Appropriate choice of integral transforms helps to convert differential equations as well as integral equations into terms of an algebraic equation that can be solved easily. During last two decades many integral transforms in the class of Laplace transform are introduced such as Sumudu, Elzaki, Natural, Aboodh, Pourreza, Mohand, G_transform, Sawi and Kamal transforms. Objectives In this paper, we introduce a general integral transform in the class of Laplace transform. We study the properties of this transform. Then we compare it with few exiting integral transforms in the Laplace family such as Laplace, Sumudu, Elzaki and G\_transforms, Pourreza, Aboodh and etc. Methods A new integral transform is introduced. Then some properties of this integral transform are discussed. This integral transform is used to solve this new transform is used for solving higher order initial value problems, integral equations and fractional order integral equation. Results It is proved that those new transforms in the class of Laplace transform which are introduced during last few decades are a special case of this general transform. It is shown that there is no advantage between theses transforms unless for special problems. Conclusion It has shown that this new integral transform covers those exiting transforms such as Laplace, Elzaki and Sumudu transforms for different value of p(s) and q(s). We used this new transform for solving ODE, integral equations and fractional integral equations. Also, we can introduce new integral transforms by using this new general integral transform.

In this paper, a novel iterative algorithm is developed to solve second-order nonlinear singular boundary value problem, whose solution exactly satisfies the Robin boundary conditions specified on the boundaries of a unit interval. The boundary shape function is designed such that the boundary conditions can be fulfilled automatically, which renders a new algorithm with the solution playing the role of a boundary shape function. When the free function is viewed as a new variable, the original singular boundary value problem can be properly transformed to an initial value problem. For the new variable the initial values are given, whereas two unknown terminal values are determined iteratively by integrating the transformed ordinary differential equation to obtain the new terminal values until they are convergent. As a consequence, very accurate solutions for the nonlinear singular boundary value problems can be obtained through a few iterations. The present method is different from the traditional shooting method, which needs to guess initial values and solve nonlinear algebraic equations to approximate the missing initial values. As practical applications of the present method, we solve the Blasius equation for describing the boundary layer behavior of fluid flow over a flat plate, where the Crocco transformation is employed to transform the third-order differential equation to a second-order singular differential equation. We also solve a nonlinear singular differential equation of a pressurized spherical membrane with a strong singularity.

We present a Dual Non-Autonomous Deep Convolutional Neural Network , namely DNA-Net, and employ it to resolve the image denoising problem . The DNA-Net models the forward and backward processes of network training as two plain ordinary differential equations (ODEs), while the non-autonomous ODEs are employed in both the two processes for efficient feature learning and gradient propagation. The non-autonomy of the forward process is introduced by incorporating the network input to each layer, which allows low-level features to be well sustained in the entire network. The non-autonomy of the backward process is realized by connecting the network output to each layer, facilitating the gradient to be fluently propagated without vanishing. We prove that the forward process of DNA-Net can stably converge to an input dependent equilibrium in the sense of expectation. As a consequence, DNA-Net can be more efficiently trained than the existing deep networks such as the residual network . Experimental results on image denoising demonstrate the leading performance of the proposed DNA-Net.

This paper introduces a novel framework for the solution of (large-scale) Lyapunov and Sylvester equations derived from numerical integration methods. Suitable systems of ordinary differential equations are introduced. Low rank approximations of their solutions are produced by Runge-Kutta methods. Appropriate Runge-Kutta methods are identified following the idea of geometric numerical integration to preserve a geometric property, namely a low rank residual. For both types of equations we prove the equivalence of one particular instance of the resulting algorithm to the well known ADI iteration.

Social media has recently become one of the most used media in the world. This has resulted in a great hotbed for the growth of rumors, as anyone can spread knowledge and opinions without confirmation. Previous works on rumor detection focused on hand-extracted features and spent less effort on text representation. In this research, a novel method for rumor detection on social media is proposed, which integrates entity recognition, sentence reconfiguration and ordinary differential equation network under a unified framework called ESODE. An entity recognition method to enhance the semantic understanding of rumor texts is used. Then, a sentence reconfiguration to improve the frequency of important words is designed. The complete feature map is established by further collecting statistical features from three aspects: linguistic features on the content of rumors, characteristics of users involved in rumor propagating, and propagation network structures. Finally, the ordinary differential equation network (ODEnet) is applied to detect rumors. Experimental results on datasets from Twitter and Weibo show that the proposed method achieves better performance than previous ones.

The impact of gyrotactic microorganisms in the mixed convection stagnation point flow of Cu-Al2O3/water hybrid nanofluid towards an immovable plate is numerically examined. The boundary layer equation coupled with the energy, nanoparticle volume fraction and conservation equations for microorganisms are simplified into a set of ordinary differential equations by adopting similarity transformation. The validation results are in a good agreement with previous studies approximately by 0% difference relative error. The results imply that the dual solutions only exist in the opposing flow up to a critical value/turning point. Beyond this turning point, the solutions are unavailable which reflects the separation of flow. The hybrid nanofluid is essential in delaying the separation process where the results show that λ c , h n f = − 3.5804 > λ c , C u = − 3.3903 > λ c , A l = − 3.2016 . Moreover, the advantage of this hybrid Cu-Al2O3/water nanofluid is in the augmentation of heat and mass transfer rates including the density of microorganisms which is higher than the Cu- and Al2O3-water nanofluids.

The aim of this paper is the investigation of heat transfer regarding the cases of both stretching and shrinking sheets with a sponge-like horizontal wall that allows for mass transpiration. The effects of Prandtl number, radiation and external magnetic field are extensively examined. The Navier-Stokes equations are reduced to partial differential equations, which are eventually become ordinary differential equations and solved analytically. Furthermore, the power-law wall temperature and heat flux boundary conditions are imposed on the boundary layer energy equation for obtaining exact analytical solutions. The results revealed that in both the stretching and shrinking sheet scenarios the thickness of the thermal boundary layer decreases with either increasing of transpiration as well as the Chandrasekhar and Prandtl number numbers or decreasing radiation number. Additionally, the characteristics of the heat transfer regarding a shrinking sheet and those of a stretching sheet are found not to be similar. In fact, a new solution branch appeared which indicates that multiple solutions may emerge under certain circumstances. Finally, by using the present analytical relationships, theoretical guidelines can be given for regulating the procedure.

The purpose of this work is to analyze the dynamic response of transversely isotropic saturated media under circular moving loads, in which the vertical and tangential loads can be simultaneously considered. Based on Biot's poroelastic theory and the Galilean transformation, the partial differential governing equations in a fixed Cartesian coordinate system are converted into those in a moving coordinate system. The ordinary differential equations are obtained in the transformed domain by using the Fourier transform. The extended precise integration method for layered media is adopted to solve the equations in the transformed domain by introducing the boundary conditions. The solutions in the physical domain are acquired by the Fourier transform inversion. After verifying the accuracy of the presented method, we perform several numerical examples to analyze the influences of the transverse isotropy and stratification of media, load shapes and the tangential moving load on the dynamic behaviors.

This paper analyses the existence and properties of solutions of the extended play-type model which was proposed in [16] to incorporate hysteresis in unsaturated flow through porous media. The model, when regularised, reduces to a nonlinear degenerate parabolic equation coupled to an ordinary differential equation. This PDE–ODE coupled system, also studied in the fields of bio-film and cellular biology, possesses an interesting mathematical structure which, to our knowledge, remains relatively unexplored. The existence of solutions for the non-degenerate version of the model is shown using the Rothe's method. Furthermore, an L ∞ bound is obtained for the solutions under certain restrictions. Existence of solutions for the degenerate case is then shown assuming that the initial condition is bounded away from the degenerate points. Finally, it is proven that if the solution for the unregularised case exists, then it is contained within physically consistent bounds.

In the issue of the environmental protection, the role of the widespread application of solar heating systems is very important. Well-usable mathematical models or their solutions can help to promote this aim. In the literature, analytical solutions for the models of solar heating systems are rather rare, despite of their advantages (fast and easy usability, low computational demand, low cost). In the present paper, we intend to (partly) make up for this shortage with respect to a two-dimensional ordinary differential equation model of complete collector-storage systems. The worked out new analytical system solution provides easy- and fast-to-use explicit formulas (for direct algebraic substitutions) to determine both the temperature of the collector and the temperature of the storage. The analytical solution is validated with the measured data of a real solar heating system. As a part of the validation of the (two-dimensional) system solution, the analytical solution of the (one-dimensional) collector sub-model (which is already known and successfully used in the literature) is also validated separately with the measured data of a real solar collector. Based on the validation containing a relatively long time period of 19 days, the error of the new system solution is 3.4% in case of the collector and 2.0% in case of the storage, which means an outstanding precision in the field.

This paper presents the neural dynamical network to compute the generalized and restricted singular value decompositions (GSVD/RSVD) in the regularization methods for ill-posed problems. The neural network model is defined by ordinary differential equations (ODE) which can be solved by many state-of-the-art techniques. The main purpose of the paper is to develop two neural network models for finding approximations of the GSVD and the RSVD. The globally asymptotic stability analyses are provided and numerical experiments illustrate our theory and methods. For small scale problems, the estimation can be as accurate as O ( 10 - 15 ) . For ill-posed problems, the truncation regularization method implementing the GSVD/RSVD algorithms also produces accurate results.

Solutions to nonlinear free vibration analysis of an arbitrarily laminated thin rectangular plate subjected to thermo-piezoelectric load is investigated. The governing equation of motion is based on von Karman’s deflection of thin plates to capture the effects of moderate geometric nonlinearity. Induced strain actuation assumption is confined to a linear model for applied low electric fields and varying temperature distribution along the thickness direction. The solution for lateral displacement field for simply supported and clamped condition edges are determined by displacement-stress; mixed formulation. The applied Galerkin’s projection reduces the governing nonlinear partial differential equation to a time dependent ordinary differential equation. Solutions are obtained using a predictor-corrector numerical integration scheme. The effects of randomly applied actuation potential and thermal loading on nonlinear frequency ratio are examined. The marginal values provided using various probabilistic methods are found to be in good agreement with each other.