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The dynamic behavior of potentials and currents in porous electrodes is crucial for optimizing the computational speed of lithium-ion battery models. Pseudo-two-dimensional (P2D) models, based on partial differential equations, offer insight but pose computational challenges. P2D equations are tackled with iterative algorithms, like the Newton or the shooting method. Yet, initiating the algorithm with random guesses for solid and electrolyte potentials can cause diverging ionic current values inside the electrolyte phase, increasing the computation time required to converge to the final solution. This study proposes a novel model order reduction using a galvanic pseudo-potential to prevent the occurrence of diverging currents. By sidestepping infinite values for ionic current inside the electrolyte phase, the method streamlines math and speeds up the shooting method used for solving battery model equations.

In this paper, we study the number of steady solutions of a non-linear model arising in Climatology. By applying a shooting method we show the existence of infinitely many steady solutions for some values of a parameter (the solar constant). This method allows us to determine how many times a solution attains the critical temperature (-10degreesC) at which the coalbedo is assumed to be discontinuous., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

The (nonlinear) behaviour of laser beams can be described with Nonlinear Schrödinger equations (NLS). The purpose of this thesis is to shed light on two mathematical papers that give existence and uniqueness results for an NLS equation called the soliton equation. The contribution is to expand the level of detail with which the analysis and some proofs are presented, and to unify the notation where possible., Applied Sciences

The wave propagation in open waveguides filled with chiral media is considered using a variational approach and numerical simulations.

We build and analyze a mathematical model of a system for biogas upgrading: an absorption column connected to a microalgae culture. The construction of the model is based on a chemical engineering approach and classical models of microalgae cultures. In our analysis, we prove the existence and uniqueness of solutions of a boundary value problem describing the absorption column. Some properties of this solution are also proved. Then, we use these results to study the long-term behavior of the mathematical model describing the coupled system. We state sufficient conditions for the existence of a globally attracting steady state characterized by the presence of microalgae. Finally, we find numerically the steady states of the system to test the efficiency of the system for purifying biogas.

In this paper we prove the existence and the multiplicity of radial positive oscillatory solutions for a nonlinear problem governed by the mean curvature operator in the Lorentz-Minkowski space. The problem is set in an N-dimensional ball and is subject to Neumann boundary conditions. The main tool used is the shooting method for ODEs.

We build and analyze a mathematical model of a system for biogas upgrading: an absorption column connected to a microalgae culture. The construction of the model is based on a chemical engineering approach and classical models of microalgae cultures. In our analysis, we prove the existence and uniqueness of solutions of a boundary value problem describing the absorption column. Some properties of this solution are also proved. Then, we use these results to study the long-term behavior of the mathematical model describing the coupled system. We state sufficient conditions for the existence of a globally attracting steady state characterized by the presence of microalgae. Finally, we find numerically the steady states of the system to test the efficiency of the system for purifying biogas.

We build and analyze a mathematical model of a system for biogas upgrading: an absorption column connected to a microalgae culture. The construction of the model is based on a chemical engineering approach and classical models of microalgae cultures. In our analysis, we prove the existence and uniqueness of solutions of a boundary value problem describing the absorption column. Some properties of this solution are also proved. Then, we use these results to study the long-term behavior of the mathematical model describing the coupled system. We state sufficient conditions for the existence of a globally attracting steady state characterized by the presence of microalgae. Finally, we find numerically the steady states of the system to test the efficiency of the system for purifying biogas.

In this paper, numerical study of Casson non-Newtonian fluid over a nonlinear stretching sheet using shooting method will be presented. The governing nonlinear partial differential equation is converted to an ordinary differential equation  by using similarity transformations. This ordinary differential equation is handled numerically with the use of well-known shooting technique aided by Runge-Kutta method. For comparison of the results, MATLAB in-built solver BVP4C is used. The discussion of the results is provided in the forms of tables as well as graphically.

In this paper, numerical study of Casson non-Newtonian fluid over a nonlinear stretching sheet using shooting method will be presented. The governing nonlinear partial differential equation is converted to an ordinary differential equation  by using similarity transformations. This ordinary differential equation is handled numerically with the use of well-known shooting technique aided by Runge-Kutta method. For comparison of the results, MATLAB in-built solver BVP4C is used. The discussion of the results is provided in the forms of tables as well as graphically.

In this paper, numerical study of Casson non-Newtonian fluid over a nonlinear stretching sheet using shooting method will be presented. The governing nonlinear partial differential equation is converted to an ordinary differential equation  by using similarity transformations. This ordinary differential equation is handled numerically with the use of well-known shooting technique aided by Runge-Kutta method. For comparison of the results, MATLAB in-built solver BVP4C is used. The discussion of the results is provided in the forms of tables as well as graphically.

We consider another non-classical of optimal control problem that is spurred by some current research on the nonlinear income issue in the field of financial matters. This class of issue can be set up as a maximizing issue in the area of Optimal Control. In any case, the state value at the final fixed time, y(T), is priori unknown and the integrand is an element of the unknown y(T). This is a non-classical optimal control problem. In this paper we apply the new costate value conditions p(T) in the definition of the optimal control problem. We solve some examples in this issue using the numerical shooting method to illuminate the subsequent Two Point Boundary Value Problem (TPBVP) and join the free y(T) as an additionally unknown. Basically similar outcomes are obtained through the nonlinear programming (NP) discrete-time results.

For 1 < p < ∞, we consider the following problem - Δ p u = f(u), u > 0 in Ω, ∂ ν u = 0 on ∂Ω, where Ω ⊂ ℝ N is either a ball or an annulus. The nonlinearity f is possibly supercritical in the sense of Sobolev embeddings; in particular our assumptions allow to include the prototype nonlinearity f(s) = -s p-1 + s q-1 for every q > p. We use the shooting method to get existence and multiplicity of non-constant radial solutions. With the same technique, we also detect the oscillatory behavior of the solutions around the constant solution u ≡ 1. In particular, we prove a conjecture proposed in [D. Bonheure, B. Noris and T. Weth, Ann. Inst. Henri Poincaré Anal. Non Linéaire 29 (2012) 573-588], that is to say, if p = 2 and f′ (1) > λ rad k + 1 ,with λ rad k + 1 the (k + 1)-th radial eigenvalue of the Neumann Laplacian, there exists a radial solution of the problem having exactly k intersections with u ≡ 1, for a large class of nonlinearities., SCOPUS: ar.j, DecretOANoAutActif, info:eu-repo/semantics/published

This paper deals with existence and multiplicity of positive solutions for a quasilinear problem with Neumann boundary conditions. The problem is set in a ball and admits at least one constant non-zero solution; moreover, it involves a nonlinearity that can be supercritical in the sense of Sobolev embeddings. The main tools used are variational techniques and the shooting method for ODE's. These results are contained in A. Boscaggin, F. Colasuonno, B. Noris. Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions. ESAIM Control Optim. Calc. Var., DOI: 10.1051/cocv/2016064 (2017; F. Colasuonno, B. Noris. A p-Laplacian supercritical Neumann problem. Discrete Contin. Dyn. Syst., 37 (2017) 3025-3057.

We consider another non-classical of optimal control problem that is spurred by some current research on the nonlinear income issue in the field of financial matters. This class of issue can be set up as a maximizing issue in the area of Optimal Control. In any case, the state value at the final fixed time, y(T), is priori unknown and the integrand is an element of the unknown y(T). This is a non-classical optimal control problem. In this paper we apply the new costate value conditions p(T) in the definition of the optimal control problem. We solve some examples in this issue using the numerical shooting method to illuminate the subsequent Two Point Boundary Value Problem (TPBVP) and join the free y(T) as an additionally unknown. Basically similar outcomes are obtained through the nonlinear programming (NP) discrete-time results.

We consider another non-classical of optimal control problem that is spurred by some current research on the nonlinear income issue in the field of financial matters. This class of issue can be set up as a maximizing issue in the area of Optimal Control. In any case, the state value at the final fixed time, y(T), is priori unknown and the integrand is an element of the unknown y(T). This is a non-classical optimal control problem. In this paper we apply the new costate value conditions p(T) in the definition of the optimal control problem. We solve some examples in this issue using the numerical shooting method to illuminate the subsequent Two Point Boundary Value Problem (TPBVP) and join the free y(T) as an additionally unknown. Basically similar outcomes are obtained through the nonlinear programming (NP) discrete-time results.

The problem of the brachistochronic motion of a holonomic scleronomic mechanical system is analyzed. The system moves in an arbitrary field of known potential forces. The problem is formulated as an optimal control task, where generalized speeds are taken as control variables. The problem considered is reduced to solving the corresponding two-point boundary-value problem (TPBVP). In order to determine the global minimal solution of the TPBVP, an appropriate numerical procedure based on the shooting method is presented. The global minimal solution represents the solution with the minimum time of motion. The procedure is illustrated by an example of determining the brachistochronic motion of a disk that performs plane motion in a vertical plane in a homogeneous field of gravity.

The paper aims to research the orientation possibilities of an object in the horizontal plane, from its start course into a required orientating recourse, by using simplified navigation methods. The object's directed motion uses controlling powering impulses, variable distributed in time, during constrained motion time. Three logical decision-making methods are designed for calculating the best maneuvering trajectory with minimal error on the target. Computing the powering impulses, their execution instances, as well as their types, are ensured by the methods and presented in the paper. The developed controlling methods are: a modified multiple shooting method, a new control law, called in this paper, current error orientation, as well as a fuzzy logic method. These methods are designed as decision-making software implemented in an electronic hardware as a predefined programmable controller. This provides pre-programmable orientation of the object at the very beginning of course motion, towards a targeted point settled out of initial direction. The controlling methods use optimal diversification of full elapsed determining time to execute, in sequences, appropriate types and number of powering impulses. Simulation tests of the methods, as well as the designed hardware, are also presented in this paper as a contribution to the development researches of horizontal motion control.

The problem of the brachistochronic motion of a holonomic scleronomic mechanical system is analyzed. The system moves in an arbitrary field of known potential forces. The problem is formulated as an optimal control task, where generalized speeds are taken as control variables. The problem considered is reduced to solving the corresponding two-point boundary-value problem (TPBVP). In order to determine the global minimal solution of the TPBVP, an appropriate numerical procedure based on the shooting method is presented. The global minimal solution represents the solution with the minimum time of motion. The procedure is illustrated by an example of determining the brachistochronic motion of a disk that performs plane motion in a vertical plane in a homogeneous field of gravity.

This paper presents an investigation for boundary layer flow of viscoelastic fluids past a flat plate. Fractional-order Blasius equation with spatial fractional Riemann–Liouville derivative is derived firstly by using Lie group transformation. The solution is obtained numerically by the generalized shooting method, employing the shifted Grünwald formula and classical fourth order Runge–Kutta method as the iterative scheme. The effects of the order of fractional derivative and the generalized Reynolds number on the velocity profiles are analyzed and discussed. Numerical results show that the smaller the value of the fractional order derivative leads to the faster velocity of viscoelastic fluids near the plate but not to hold near the outer flow. As the Reynolds number increases, the fluid is moving faster in the whole boundary layer consistently.

It is well known that differential equations (DEs) play an important role in many sciences. They are mathematical representations of many physical systems. By studying such DEs, one gains many important insights about the physical system. Solutions of DEs provide information on the physical system behavior. As many physical systems are nonlinear in nature, this naturally gives rise to nonlinear differential equations (NLDEs). Such NLDEs are, in most cases, hard or sometimes impossible to solve analytically. In such situations, we resort to numerical techniques to approximate the solutions. The purpose of this thesis is to consider nonlinear multi-layer boundary value problems and seek approximate solutions. Many methods exist in the literature to numerically solve nonlinear boundary value problems. However, only few papers dealt with nonlinear multi- layer boundary value problems. In this work, we employ the homotopic analysis method (HAM) as the method of choice. We consider a real physical system dealing with the fluid flow in multi-channel porous media whose governing equations is exactly a nonlinear multi-layer boundary value problem.

This paper analyzes the problem of brachistochronic planar motion of a mechanical system with nonlinear nonholonomic constraint. The nonholonomic system is represented by two Chaplygin blades [3,4,5,6] of negligible dimensions, which impose nonlinear constraint in the form of perpendicularity of velocities. The brachistrochronic planar motion is considered, with specified initial and terminal positions, at unchanged value of mechanical energy during motion. Differential equations of motion, where the reactions of nonholonomic constraints and control forces figure, are obtained on the basis of general theorems of mechanics [8]. Here, this is more convenient to use than some other methods of analytical mechanics applied to nonholonomic mechanical systems [9], where a subsequent physical interpretation of the multipliers of constraints is required. The formulated brachistochrone problem, with adequately chosen quantities of state, is solved as simple a task of optimal control as possible in this case [6,7] by applying the Pontryagin maximum principle [1]. The corresponding two-point boundary value problem of the system of ordinary nonlinear differential equations is obtained, which has to be numerically solved [2]. Numerical procedure for solving the two-point boundary value problem is performed by the method of shooting. On the basis of thus obtained brachistochronic motion, the active control forces, along with the reactions of nonholonomic constraints, are defined. Using the Coulomb friction laws [8,9], a minimum required value of the coefficient of sliding friction is defined [10], so that the considered system is moving in accordance with nonholonomic bilateral constraints.

We study asymptotically positively homogeneous first order systems in the plane, with boundary conditions which are positively homogeneous, as well. Defining a generalized concept of Fučík spectrum which extends the usual one for the scalar second order equation, we prove existence and multiplicity of solutions. In this way, on one hand we extend to the plane some known results for scalar second order equations (with Dirichlet, Neumann or Sturm-Liouville boundary conditions), while, on the other hand, we investigate some other kinds of boundary value problems, where the boundary points are chosen on a polygonal line, or in a cone. Our proofs rely on the shooting method. © 2013 Juliusz Schauder Centre for Nonlinear Studies Nicolaus Copernicus University.

The paper analyzes the problem of brachistochronic motion for the general case of holonomic scleronomic mechanical system. The problem is formulated as an optimal control task, where the generalized velocities are taken as a control, provided that the value of the total mechanical energy of the system is given. The initial and final values of generalized coordinates are specified. The corresponding two-point boundary value problem of the system of ordinary nonlinear differential equations is obtained, which it is, in a general case, necessary to solve numerically. In solving it by the shooting method it is necessary to evaluate the intervals in which initial values of the conjugate vector are, so as to achieve the specified values of generalized coordinates. This presents a problem, because the conjugate vector coordinates do not represent the physical quantities and their values cannot be easily estimated, and due to the homogeneity of a conjugate system they are directly proportional to the chosen value of the negative coordinates. The work provides the procedures for evaluation of all possible values of the coordinates of the conjugate vector, so outside these intervals there are not certain solutions of the two-point boundary value problem. Also, it is given the procedure for determining the global minimum for the systems with three degrees of freedom, where it is possible to provide three-dimensional graphic representation of the space, where the third dimension is the ultimate moment. In the case of several extremal solutions of the principle of maximum the global optimum is the lowest value of the solutions obtained for the ultimate moment. The procedure is demonstrated on the problem of the brachistochronic spherical rigid body motion, in which the center of mass is not at the fixed point.

It is well known that excitable media can sustain "fast" stable traveling pulses of excitation. In models for which analysis is tractable, the fast pulse solution is known to appear through a saddlenode bifurcation accompanied by a "slow" unstable traveling pulse. Furthermore, the uniform rest state is also a stable solution. It is generally assumed that the boundary between the basins of attractions of the rest state and fast pulse (i.e., threshold) consists of the stable manifold of the slow pulse. We use numerical experiments to explore this issue. Our results indicate that, near the saddle-node bifurcation, the stable manifold of the slow pulse does indeed act as the threshold between the rest state and fast pulse. However, further away from the saddle-node bifurcation, a global bifurcation involving the heteroclinic connections between slow and fast pulses occurs. This bifurcation gives rise to an unstable periodic solution that has been referred to as a one-dimensional spiral wave. Beyond the bifurcation point, the stable manifold of the one-dimensional spiral wave appears to act as the threshold between the rest state and fast pulse. © 2009 Society for Industrial and Applied Mathematics.

It is well known that excitable media can sustain "fast" stable traveling pulses of excitation. In models for which analysis is tractable, the fast pulse solution is known to appear through a saddlenode bifurcation accompanied by a "slow" unstable traveling pulse. Furthermore, the uniform rest state is also a stable solution. It is generally assumed that the boundary between the basins of attractions of the rest state and fast pulse (i.e., threshold) consists of the stable manifold of the slow pulse. We use numerical experiments to explore this issue. Our results indicate that, near the saddle-node bifurcation, the stable manifold of the slow pulse does indeed act as the threshold between the rest state and fast pulse. However, further away from the saddle-node bifurcation, a global bifurcation involving the heteroclinic connections between slow and fast pulses occurs. This bifurcation gives rise to an unstable periodic solution that has been referred to as a one-dimensional spiral wave. Beyond the bifurcation point, the stable manifold of the one-dimensional spiral wave appears to act as the threshold between the rest state and fast pulse. © 2009 Society for Industrial and Applied Mathematics.

A closure of the Sturm sequence algorithm is found, leading to shooting method for the nth eigenvalue and eigenfunction of a Sturm-Liouville operator. The method works for general (separated) boundary conditions, and provides an a-posteriori error estimate for the approximate value of the nth eigenvalue. A related method, which involves critical lengths in the invariant imbedding method, is shown to be incorrect for general boundary conditions.