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In [19] , Smoczyk showed that parabolic expansion of convex curves and hypersurfaces by the reciprocal of the harmonic mean curvature gives rise to a linear second order equation for the evolution of the support function, with corresponding representation formulae for solutions. In this article we obtain representation formula for the corresponding linear hyperbolic evolution equation for the support function of locally convex curves of general nonzero rotation number, finding some natural similarities and differences in behaviour of solutions as compared with the parabolic case. As applications, we consider linear hyperbolic flows that preserve length or generalised enclosed area of the evolving curve, flow of curves with Neumann boundary conditions inside cones and hyperbolic approaches to the Yau problem of flowing one curve to another. We also consider radial graphs evolving by similar linear equations and linear higher order hyperbolic equations. Our results may be compared with the corresponding parabolic cases consider by the first author, Schrader and Wheeler, in [17]. [ABSTRACT FROM AUTHOR]

This paper deals with the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments of advanced type. First of all, we obtain the expression of analytic solution by the method of separation variable, then the sufficient conditions for stability are obtained. Semidiscrete and fully discrete schemes are derived by Galerkin finite element method, and their convergence are both analyzed in L²-norm. Moreover, the stability of the two schemes are investigated. The semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are derived under which the analytic solution is asymptotically stable. Finally, some numerical experiments are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Throughout this article, generalizations of some Grónwall–Bellman integral inequalities for two real-valued unknown functions in n independent variables are introduced. We are looking at some novel explicit bounds of a particular class of Young and Pachpatte integral inequalities. The results in this paper can be utilized as a useful way to investigate the uniqueness, boundedness, continuousness, dependence and stability of nonlinear hyperbolic partial integro-differential equations. To highlight our research advantages, several implementations of these findings will be presented. Young's method, which depends on a Riemann method, will follow to prove the key results. Symmetry plays an essential role in determining the correct methods for solving dynamic inequalities. [ABSTRACT FROM AUTHOR]

Recently, many authors studied the numerical solution of the classical Darboux problem in its integral form via two‐dimensional nonlinear Volterra–Fredholm integral equation. In the present article, a numerical technique based on the Chebyshev wavelet is proposed to solve the Darboux problem directly without converting into a nonlinear Volterra–Fredholm integral equation. The proposed technique is different from the techniques discussed in the literature. The proposed approach produces higher accuracy than its counterpart techniques. The proposed scheme is illustrated with suitable examples to show the advantages in terms of its accuracy with a lesser grid size. [ABSTRACT FROM AUTHOR]

Abstract This article establishes a theorem that guarantees the occurrence of chaotic oscillations in a system governed by a linear hyperbolic partial differential equation (PDE) with a nonlinear boundary condition (NBC). Compared with the NBCs in all previous related literature, such an NBC is more general. Both the left end and the right end of the system parameter interval for the occurrence of chaotic oscillations are precisely characterized. The chaotic results obtained are further applied to two specific NBCs and the telegraph equation. Finally, numerical examples verify the effectiveness of theoretical prediction. [ABSTRACT FROM AUTHOR]

The meshless element-free Galerkin method is developed for numerical analysis of hyperbolic initial-boundary value problems. In this method, only scattered nodes are required in the domain. Computational formulae of the method are analyzed in detail. Error estimates and convergence are also derived theoretically and verified numerically. Numerical examples validate the performance and efficiency of the method. [ABSTRACT FROM AUTHOR]

This paper proposes a novel, low dimensional and accurate approximate model for the distributed parabolic solar collector, based on a modified Gaussian interpolation along the spatial domain. The proposed reduced model, written in a form of a low dimensional bilinear state representation, enables the reproduction of the heat transfer dynamics along the collector tube for system analysis. Moreover, presented as a reduced order bilinear state space model, the well established control theory for this class of systems can be applied. The approximation efficiency has been proven by several simulation tests, which have been performed considering parameters of the Acurex field with real external working conditions. Model accuracy has been evaluated by comparison to the analytical solution of the hyperbolic distributed model and its semi discretized approximation highlighting the benefits of using the proposed numerical scheme. Furthermore, model sensitivity to the different parameters of the Gaussian interpolation has been studied. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider the expansion problem of a wedge of gas into vacuum for the two-dimensional Euler equations in isothermal flow. By the bootstrapping argument, they prove the global existence of the smooth solution through the direct method in the case $$0 < \theta \leqslant \bar \theta = \arctan \tfrac{1} {{\sqrt {2 + \sqrt 5 } }}$$, where θ is the half angle of the wedge. Furthermore, they get the uniform C estimates of the solution to the expansion problem. [ABSTRACT FROM AUTHOR]

This note presents a weak generalization of a time-delayed partial differential equation which, in turn, generalizes the well-known Burger–Fisher and Burgers–Huxley models. In this work, we provide a full discretization which is consistent with the integro-differential equation under consideration. The main analytical result of this note establishes that the discrete temporal rate of change of the discretization yields a consistent approximation to the differential form of the integro-differential equation investigated. Some numerical examples are provided in order to assess the efficiency and effectiveness of our methodology. [ABSTRACT FROM AUTHOR]

Continuous sediment has a wide application in chemical engineering process, such as wastewater treatment, water reuse, mineral waste manage and processing. This paper provides analytical solutions of the ideal continuous settling model by using the method of characteristics. The analytical solutions are compared with experiment data to show that the solutions accurately predict the sediment height and concentration as a function of time and loading conditions. Additionally, three alternative methods, using finite differences are compared to the analytical solutions, and their accuracy and efficiency are evaluated. It is shown that all three methods are reliable for solving sedimentation problems but have varying efficiency. Method YRD is the most accurate but also has the greatest computation and implementation cost. Method SG is the least accurate but is the easiest to implement with lowest computation cost. Method G is a compromise between the two methods, providing acceptable accuracy and low computation cost. [ABSTRACT FROM AUTHOR]

Compared with the Monte Carlo method, the population density method is efficient for modeling collective dynamics of neuronal populations in human brain. In this method, a population density function describes the probabilistic distribution of states of all neurons in the population and it is governed by a hyperbolic partial differential equation. In the past, the problem was mainly solved by using the finite difference method. In a previous study, a continuous Galerkin finite element method was found better than the finite difference method for solving the hyperbolic partial differential equation; however, the population density function often has discontinuity and both methods suffer from a numerical stability problem. The goal of this study is to improve the numerical stability of the solution using discontinuous Galerkin finite element method. To test the performance of the new approach, interaction of a population of cortical pyramidal neurons and a population of thalamic neurons was simulated. The numerical results showed good agreement between results of discontinuous Galerkin finite element and Monte Carlo methods. The convergence and accuracy of the solutions are excellent. The numerical stability problem could be resolved using the discontinuous Galerkin finite element method which has total-variationdiminishing property. The efficient approach will be employed to simulate the electroencephalogram or dynamics of thalamocortical network which involves three populations, namely, thalamic reticular neurons, thalamocortical neurons and cortical pyramidal neurons. [ABSTRACT FROM AUTHOR]

Pattern formation in self-organized biological aggregation is a phenomenon that has been studied intensively over the past 20 years. In general, the studies on pattern formation focus mainly on identifying the biological mechanisms that generate these patterns. However, identifying the mathematical mechanisms behind these patterns is equally important, since it can offer information on the biological parameters that could contribute to the persistence of some patterns and the disappearance of other patterns. Also, it can offer information on the mechanisms that trigger transitions between different patterns (associated with different group behaviors). In this article, we focus on a class of nonlocal hyperbolic models for self-organized aggregations and show that these models are O(2)-equivariant. We then use group-theoretic methods, linear analysis, weakly nonlinear analysis, and numerical simulations to investigate the large variety of patterns that arise through O(2)-symmetric codimension-two bifurcations (i.e., Hopf/Hopf, steady-state/Hopf, and steady-state/steady-state mode interactions). We classify the bifurcating solutions according to their isotropy types (subgroups), and we determine the criticality and stability of primary branches of solutions. We numerically show the existence of these solutions and determine scenarios of secondary bifurcations. Also, we discuss the secondary bifurcating solutions from the biological perspective of transitions between different group behaviors. [ABSTRACT FROM AUTHOR]

We study mixed-moment models (full zeroth moment, half higher moments) for a Fokker-Planck equation in one space dimension. Mixed-moment minimum-entropy models are known to overcome the zero net-flux problem of full-moment minimum-entropy Mn models. A realizability theory for these mixed moments of arbitrary order is derived, as well as a new closure, which we refer to as Kershaw closure. They provide nonnegative distribution functions combined with an analytical closure. Numerical tests are performed with standard first-order finite volume schemes and compared with a finite difference Fokker-Planck scheme. [ABSTRACT FROM AUTHOR]

• Derives theoretical solution of convective heat transfer from a plate with time-dependent and space-dependent temperature. • Utilizes Karman-Pohlhausen polynomials to derive solution. • Results are shown to agree with several past papers for special cases. • Applications in several practical scenarios are discussed. • Results help improve our understanding of convective heat transfer. Convective heat transfer due to laminar, incompressible flow past a flat plate has been extensively researched due to applications in a variety of engineering systems. In several cases, such as thermal management of Li-ion cells and phase change based energy storage, the flat plate temperature may vary as a function of both time and space, and the resulting plate heat flux is of interest. This paper presents an integral method based analytical model for predicting the heat flux distribution from a flat plate with a time-varying and spatially-varying temperature. Third-order Karman-Pohlhausen polynomial forms of velocity and temperature distributions are used in the integral energy equation to derive an ordinary differential equation that predicts the plate heat flux distribution in response to an arbitrary time- and space-dependent plate temperature distribution. The generalized results derived here are shown to reduce to previously reported results for special cases of only time-dependent or only spatially varying or constant plate temperature. Results are also shown to be in good agreement with finite-element simulations. The analytical model developed here is used to predict the plate heat flux in response to realistic plate temperature distributions that may be encountered in practical applications. These results improve the fundamental understanding of an important convective heat transfer process, and may help in the design and optimization of a broad variety of engineering systems involving convective heat transfer. [ABSTRACT FROM AUTHOR]

Abstract: We study a semilinear mildly damped wave equation that contains the telegraph equation as a special case. We consider Neumann velocity boundary feedback and prove the exponential stability of the closed loop system. We show that for vanishing damping term in the partial differential equation, the decay rate of the system approaches the rate for the system governed by the wave equation without damping term. In particular, this implies that arbitrarily large decay rates can occur if the velocity damping in the partial differential equation is sufficiently small. [Copyright &y& Elsevier]

Solving the Saint-Venant equations by numerical methods like finite element and finite difference methods yields an unstable solution for a fairly large open channel. Multiple Knot B-Spline Wavelets (MKBSW) are in the class of semi-orthogonal wavelets that have compact support. Hence, thesè basis functions are suitable for solving the Saint-Venant equations. However, solving the Saint-Venant equations by MKBSW method requires a long CPU time. In this paper, we present an adaptive wavelet method to solve the Saint-Venant equation in a fairly short time. In fact, we first solve the problem in a few first moments and then by statistical methods of time series and regression, where the active wavelets are predicted in the next moments. Moreover, by this adaptive method, the cumulative errors (that are produced by solving the discretized system, numerically) are decreased for large open channels. Two numerical examples are given to support our results. [ABSTRACT FROM AUTHOR]

We generalize Ringström’s global future causal stability results (Ringström 2009) [11] for certain expanding cosmological solutions of the Einstein-scalar field equations to solutions of the Einstein–Maxwell-scalar field system. In particular, after noting that the power law inflationary spacetimes considered by Ringström (2009) in [11] are solutions of the Einstein–Maxwell-scalar field system (with exponential potential) as well as of the Einstein-scalar field system (with the same exponential potential), we consider (nonlinear) perturbations of initial data sets of these spacetimes which include electromagnetic perturbations as well as gravitational and scalar perturbations. We show that if (as in Ringström (2009) [11]) we focus on pairs of relatively scaled open sets on an initial slice of , and if we choose a set of perturbed data which on is sufficiently close to that of , then in the maximal globally hyperbolic spacetime development of this data via the Einstein–Maxwell-scalar field equations, all causal geodesics emanating from are future complete (just as in ). We also verify that, in a certain sense, the future asymptotic behavior of the fields in the spacetime developments of the perturbed data sets does not differ significantly from the future asymptotic behavior of . [ABSTRACT FROM AUTHOR]

We investigate theoretically inertial waves inside a liquid confined between two co-rotating coaxial cylinders of finite length. We consider the case of small viscosity and high angular velocity (i.e., small Ekman numbers), a parameter range of interest for many geophysical applications. In this case, inertial waves propagating in the container show multiple reflections at the walls before the waves can be damped by weak diffusion. We allow for the inner cylinder wall to be parallel or inclined with respect to the annulus' vector of rotation (truncated cone). For the limit of zero viscosity, the wave propagation is governed by a boundary value problem that is composed of a linear second-order hyperbolic partial differential equation and the impermeability boundary conditions. For the special case of vertical cylinder walls (no inclination of the inner cylinder), this boundary value problem is separable, the corresponding eigenmodes can analytically be found and they are regular. However, when the inner cylinder wall is inclined, the hyperbolicity of the governing equation leads to internal shear layers (corresponding to singularities for the inviscid case). The geometrical structure of the shear layers can be explained by inertial waves, trapped on limit cycles denoted as wave attractors. The shape of the limit cycles depends on the wave frequency. In fact, the spectrum of regular modes, existing for the case of vertical cylinder walls, vanishes almost completely when the inner wall is inclined. Instead of a spectrum of discrete frequencies and regular eigenmodes, a spectrum of wave attractor frequency bands and singular eigenmodes exist. The question addressed here is whether the spectrum of wave attractor intervals collapses to the discrete frequency spectrum when the inclination angle of the inner cylinder goes to zero. To answer this question, the attractor frequency intervals are evaluated numerically for a series of decreasing cylinder inclination angles and are compared with the analytically found eigenspectrum for the case of zero inclination. Goal is to better understand the asymptotic behavior of the problem for decreasing inclination angles. This understanding helps to interpret results from laboratory experiments with geometries that differ from the perfect annulus with parallel cylinder walls. [ABSTRACT FROM AUTHOR]

A two-layer, shallow-water model for three-dimensional (3D) gravity currents is proposed. The formulation results from the shallow-water-equations for two layers of immiscible liquids, subjected by the rigid-lid condition, so that the upper surface of the lighter layer remains perfectly flat during the motion. The arising pressure must be determined by solving the equations of motion, which is no problem for two-dimensional and axisymmetric gravity currents because the pressure is easily eliminated. In 3D gravity currents, the pressure is determined by solving a Poisson equation, together with momentum and mass balance equations. By means of a suitable scaling and a perturbation expansion, the equations are uncoupled from each other so that the problem is considerably simplified. Numerical results are compared with 3D lock-exchange release experiments. A comparison between numerical and experimental results of the gravity current indicates a fairly good agreement, whereas the results concerning the upper layer field variables shows that the numerical results are consistent with the experiments. [ABSTRACT FROM AUTHOR]

Abstract: Solving the Saint-Venant equations by using numerical schemes like finite difference and finite element methods leads to some unwanted oscillations in the water surface elevation. The reason for these oscillations lies in the method used for the approximation of the nonlinear terms. One of the ways of smoothing these oscillations is by adding artificial viscosity into the scheme. In this paper, by using a suitable discretization, we first solve the one-dimensional Saint-Venant equations by a finite element method and eliminate the unwanted oscillations without using an artificial viscosity. Second, our main discussion is concentrated on numerical stabilization of the solution in detail. In fact, we first convert the systems resulting from the discretization to systems relating to just water surface elevation. Then, by using M-matrix properties, the stability of the solution is shown. Finally, two numerical examples of critical and subcritical flows are given to support our results. [Copyright &y& Elsevier]

In the present paper, the solutions of the hyperbolic partial differential equation with fractional time derivative of order α (1<α≤2) are obtained with the help of approximate analytical method of nonlinear problems called the homotopy analysis method. By using initial values, the explicit solutions of the equations for different particular cases have been derived which demonstrate the effectiveness, validity, potentiality and reliability of the method in reality. Numerical results for different particular cases are presented graphically. The numerical solutions show that only a few iterations are needed to obtain accurate approximate solutions. [ABSTRACT FROM AUTHOR]

In the application of feedback controls, a delay may appear as a perturbation caused by the computation of the controls. For vibrating systems, this delay can destroy the stabilizing effect of the control. To avoid this problem, we consider feedback laws where a certain delay is included as a part of the control law and not as a perturbation. We consider systems that are governed by the wave equation. As a first system, we consider a string that is fixed at one end and stabilized with a boundary feedback with constant delay at the other end. As a second example, we consider a circular string where both ends of the string are coupled by a feedback law. For both systems, we show the exponential stability of the proposed feedback with retarded input. Moreover, for the first system, we show robustness with respect to variations in time of the feedback parameter. [ABSTRACT FROM PUBLISHER]

Abstract: The purpose of this paper is to present a new family of numerical methods for the approximation of second order hyperbolic partial differential equations submitted to a convex constraint on the solution. The main application is dynamic contact problems. The principle consists in the use of a singular mass matrix obtained by the mean of different discretizations of the solution and of its time derivative. We prove that the semi-discretized problem is well-posed and energy conserving. Numerical experiments show that this is a crucial property to build stable numerical schemes. [Copyright &y& Elsevier]

Abstract: In this paper we study the spatial behaviour of solutions for the three-phase-lag heat equation on a semi-infinite cylinder. The theory of three-phase-lag heat conduction leads to a hyperbolic partial differential equation with a fourth-order derivative with respect to time. First, we investigate the spatial evolution of solutions of an initial boundary-value problem with zero boundary conditions on the lateral surface of the cylinder. Under a boundedness restriction on the initial data, an energy estimate is obtained. An upper bound for the amplitude term in this estimate in terms of the initial and boundary data is also established. For the case of zero initial conditions, a more explicit estimate is obtained which shows that solutions decay exponentially along certain spatial–time lines. A class of non-standard problems is also considered for which the temperature and its first two time derivatives at a fixed time are assumed proportional to their initial values. These results are relevant in the context of the Saint–Venant Principle for heat conduction problems. [Copyright &y& Elsevier]

The main objective of this paper is to study the existence, uniqueness and other properties of solutions of a certain hyperbolic partial differential equation in two independent variables. The Banach fixed point theorem and a certain integral inequality with explicit estimate are used to establish the results. [ABSTRACT FROM AUTHOR]

Abstract: Hyperbolic partial differential equations with an integral condition serve as models in many branches of physics and technology. Recently, much attention has been expended in studying these equations and there has been a considerable mathematical interest in them. In this work, the solution of the one-dimensional nonlocal hyperbolic equation is presented by the method of lines. The method of lines (MOL) is a general way of viewing a partial differential equation as a system of ordinary differential equations. The partial derivatives with respect to the space variables are discretized to obtain a system of ODEs in the time variable and then a proper initial value software can be used to solve this ODE system. We propose two forms of MOL for solving the described problem. Several numerical examples and also some comparisons with finite difference methods will be investigated to confirm the efficiency of this procedure. [Copyright &y& Elsevier]

Abstract: First we assume a simple neuronal model based on Stein’s Model [Richard B. Stein, A theoretical analysis of neuronal variability, Biophys. J. 5 (1965) 173–194] in which, after a refractory period, excitatory and inhibitory exponentially decaying inputs of constant size occur at random intervals and sum until a threshold is reached. We briefly discuss the distribution of time intervals between successive neuronal firings, the firing rate as a function of input frequency, the strength–duration curve and the role of inhibition. Then a first-order partial differential-difference equation for the distribution of neuronal firing intervals is derived and a numerical scheme based on finite difference is constructed for solving such type of initial and boundary value problems. The proposed method is analyzed for stability and convergence. Finally, some test examples are given to validate convergence and the computational efficiency of the present scheme. [Copyright &y& Elsevier]

We justify a method for reducing a wide class of nonlinear equations (including several partial differential equations) to ordinary differential equations in locally convex spaces. The possibilities of this method are demonstrated by an example of a class of nonlinear hyperbolic partial differential equations. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the nonlinear shallow shell model introduced in 1966 by W.T. Koiter in [On the nonlinear theory of thin elastic shells. III, Nederl. Akad. Wetensch. Proc. Ser. B 69 (1966) 33–54, Section 11] and later studied in [M. Bernadou, J.T. Oden, An existence theorem for a class of nonlinear shallow shell problems, J. Math. Pures Appl. (9) 60(3) (1981) 285–308]. We consider a version of this model which is based upon the intrinsic shell modeling techniques introduced by Michel Delfour and Jean-Paul Zolésio. We show existence and uniqueness of both regular and weak solutions to the dynamical model and that the solutions are continuous with respect to the initial data. While existence and uniqueness of regular solutions to nonlinear dynamic shell equations has been known, full Hadamard well-posedness of weak solutions, as shown in this paper, is a new result which solves an old open problem in the field. [Copyright &y& Elsevier]

This paper presents for the first time numerical predictions of mechanical blood hemolysis obtained by solving a hyperbolic partial differential equation (PDE) modelling the hemolysis in a Eulerian frame of reference. This provides hemolysis predictions over the entire computational domain as an alternative to the Lagrangian approach consisting in evaluating cell hemolysis along their trajectories. The solution of a PDE over a computational domain, such as in the approach presented herein, yields a unique solution. This is a clear advantage over the Lagrangian approach, which requires the human-made choice of a limited number of trajectories for integration and inevitably results in the incomplete coverage of the computational domain. The hyperbolic hemolysis model is solved with a Discontinuous Galerkin finite element method. The solution algorithm also includes adaptive remeshing to provide high accuracy simulations. Predictions of the modified index of hemolysis (MIH) are presented for flows in dialysis cannulae and sudden contractions. MIH predictions for cannulae differ significantly from those obtained by other authors using the Lagrangian approach. The predictions for flows in sudden contractions are used, along with our own experimental measurements, to assess the value of the threshold shear stress required for hemolysis that is included in the hemolysis model. [ABSTRACT FROM AUTHOR]

Abstract: We study the enzymatic degradation of xenobiotic polymers mathematically. As a mathematical model, we derive a linear second-order hyperbolic partial differential equation which governs the evolution of the weight distribution with respect to the molecular weight. Given an initial weight distribution and a final weight distribution, we formulate a problem to determine a degradation rate. We establish a necessary and sufficient condition for which the problem has a local solution. We also introduce a numerical technique based on our analysis, and present a numerical result that we obtained applying weight distributions before and after enzymatic degradation of polyvinyl alcohol. [Copyright &y& Elsevier]

In this article, we propose a new approach for solving an initial-boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study. [ABSTRACT FROM AUTHOR]

• Addresses external convective heat transfer due to flow past a flat plate. • Solves the problem when plate heat flux is a function of time and space. • Represents a generalization of past work on external convective heat transfer. • Results are shown to reduce to past work for several special cases. • Results may be applicable for several practical engineering systems. Convective heat transfer due to laminar fluid flow past a flat plate is a standard problem in heat transfer. While constant heat flux or temperature along the plate is often assumed for solving such problems, there may be several practical scenarios where the heat flux along the plate varies as a function of both space and time. Developing an analytical solution for the resulting plate temperature distribution is important for understanding and optimizing the thermal performance of such systems. While some work exists on analyzing problems with time-dependent or space-dependent heat flux, there is a lack of work on the general problem where the heat flux is a function of both space and time. This paper presents a solution for this problem by solving the integral form of the energy equation, along with the use of fourth-order Karman-Pohlhausen polynomials for velocity and temperature distributions in the momentum and thermal boundary layers. A non-linear, first order, hyperbolic partial differential equation for the plate temperature is derived in response to the time- and space-varying plate heat flux. This equation is shown to agree well with results from past work for several special cases. Numerical solutions for the generalized equation are presented. Based on this approach, the plate temperature distribution is predicted for several heat flux profiles that may be of interest in practical applications. Results from this work improve our understanding of unsteady convective heat transfer, and contribute towards modeling and optimization of practical engineering systems where such phenomena occur. [ABSTRACT FROM AUTHOR]

• Addresses external convective heat transfer due to flow past a flat plate. • Solves the problem when plate heat flux is a function of time and space. • Represents a generalization of past work on external convective heat transfer. • Results are shown to reduce to past work for several special cases. • Results may be applicable for several practical engineering systems. Convective heat transfer due to laminar fluid flow past a flat plate is a standard problem in heat transfer. While constant heat flux or temperature along the plate is often assumed for solving such problems, there may be several practical scenarios where the heat flux along the plate varies as a function of both space and time. Developing an analytical solution for the resulting plate temperature distribution is important for understanding and optimizing the thermal performance of such systems. While some work exists on analyzing problems with time-dependent or space-dependent heat flux, there is a lack of work on the general problem where the heat flux is a function of both space and time. This paper presents a solution for this problem by solving the integral form of the energy equation, along with the use of fourth-order Karman-Pohlhausen polynomials for velocity and temperature distributions in the momentum and thermal boundary layers. A non-linear, first order, hyperbolic partial differential equation for the plate temperature is derived in response to the time- and space-varying plate heat flux. This equation is shown to agree well with results from past work for several special cases. Numerical solutions for the generalized equation are presented. Based on this approach, the plate temperature distribution is predicted for several heat flux profiles that may be of interest in practical applications. Results from this work improve our understanding of unsteady convective heat transfer, and contribute towards modeling and optimization of practical engineering systems where such phenomena occur. [ABSTRACT FROM AUTHOR]

In this paper, we will apply the operator method to prove the generalized Hyers-Ulam stability of the wave equation, u t t (x , t) − c 2 ▵ u (x , t) = f (x , t) , for a class of real-valued functions with continuous second partial derivatives. Finally, we will discuss the stability more explicitly by giving examples. [ABSTRACT FROM AUTHOR]

We apply the method of a kind of dilation invariance to prove the generalized Hyers-Ulam stability of the (inhomogeneous) wave equation with a source, u t t (x , t) − c 2 ▵ u (x , t) = f (x , t) , for a class of real-valued functions with continuous second partial derivatives in each of spatial and the time variables. [ABSTRACT FROM AUTHOR]