Commonly, in Ordinary Differential Equations courses, equations with impulses or discontinuous forcing functions are studied. In this context, the Laplace Transform of the Dirac delta function and unit step function is taught, which are used as forcing functions in theoretical equations. However, application in real situations is also an important part of the learning process. In this sense, most books are flawed regarding the practical applications of this type of equation. Therefore, the purpose of this work is to study the solution of differential equations under the action of discontinuous forcing functions or impulses, contextualized in Physics or Engineering problems, using Laplace transforms. For this, this article analyzes some physical systems that are not so explored in the literature, such as the galvanometer, a circuit used inside an ammeter or voltmeter to measure current or voltage. In addition, we also studied an R-L-C circuit (Resistor, Inductor and Capacitor), using the Laplace Transform to find the capacitor voltage, demonstrating an extremely useful way for solving electrical circuits in series or in parallel.

ABSTRACT In this work, the solution of Poisson’s equation in a semi-disc under a Dirichlet boundary condition at the base and a Neumann boundary condition on the circumference is calculated. The solution is determined in terms of Green’s function, which is calculated in two ways, by the method of images and by solving its equation. In the particular case of Laplace’s equation, it is presented a second way to solve it, which uses separation of variables and a Fourier transform. work Poissons Poisson s semidisc semi disc Greens Green function ways Laplaces Laplace transform

A equação de Laplace é uma equação diferencial que pode descrever diferentes fenômenos físicos. Neste trabalho discutimos como ela pode ser resolvida numericamente através do método da relaxação. O operador laplaciano é escrito em termos de diferenças finitas, estabelecendo uma relação entre o valor da função em um ponto e a média aritmética da função em pontos vizinhos. A relação obtida é aplicada iterativamente e soluções numéricas são obtidas. Aplicamos o algoritmo para a solução da equação de calor considerando estados estacionários e para a determinação de potencial eletrostático entre dois cilindros condutores coaxiais. O acordo entre soluções analíticas e numéricas é muito bom, o algoritmo é computacionalmente barato e pode ser implementado com conhecimentos básicos de algoritmos em poucas linhas de código. Discutimos como essa ferramenta didática pode ser aplicada em diferentes contextos de ensino. físicos relaxação finitas vizinhos obtidas coaxiais bom código ensino

Laplace’s equation is a differential equation that can describe different physical phenomena. In this work we discuss how it can be numerically solved using the relaxation method. The Laplacian operator is written in terms of finite differences, establishing a relationship between the value of the function at one point and the arithmetic mean of the function at neighboring points. The obtained relationship is applied iteratively and numerical solutions are obtained. We apply the algorithm for the solution of the heat equation considering steady states and for the determination of electrostatic potential between two coaxial conducting cylinders. The agreement between analytical and numerical solutions is very good, the algorithm is computationally cheap and can be implemented with basic knowledge of algorithms in a few lines of code. We discuss how this didactic tool can be applied in different teaching contexts. Laplaces Laplace s phenomena method differences points cylinders good code contexts

Abstract In this paper, we consider a Caputo space-time fractional Schrödinger equation for the delta potential. To solve the equation, we use the joint Laplace and Fourier transforms on the spatial and time coordinates, respectively. After applying the integral transformations, we use the special initial and boundary physical conditions obtained by trial and error; these special initial conditions involve considering the initial spatial wave function in terms of the Mittag-Leffler function. Consequently, using the fractional calculus, we obtain the wave functions and corresponding eigenvalues. Finally, to verify the solution, we recover the standard case corresponding to α →1 and β → 1.

ABSTRACT Disturbance of the blood supply to the brain is the main cause of cerebrovascular disease. This disturbance may be due to ischemia (thrombosis, arterial embolism) or to hemorrhage. In this study, the feasibility and potentialities of the electrical bioimpedance technique for monitoring unilateral brain damage was analyzed using a computerized model. The simulations were performed on a hemispheric numerical model of the human head. The electric potentials were obtained by solving Laplace's equation in the frequency domain. A hemispheric electrical asymmetry index was introduced that allows determining the approximate location of the damaged region within the damaged hemisphere, and its evaluation was performed for several conductivities and lesion volumes. A negative value was obtained for ischemia (potential amplitude in the ischemic hemisphere greater than in the intact hemisphere), whereas for hemorrhage a positive value was found. The simulations revealed that variations in electrical conductivity and the volume of damaged tissue have an effect on the hemispheric asymmetry of electrical potential at the surface.

RESUMEN La alteración del suministro de sangre al cerebro es la principal causa de la enfermedad cerebrovascular. Esta alteración puede deberse a una isquemia (trombosis, embolismo arterial) o a una hemorragia. En este estudio se analizó la viabilidad y las potencialidades de la técnica de bioimpedancia eléctrica para monitorear el daño cerebral unilateral utilizando un modelo computacional. Las simulaciones se realizaron en un modelo numérico hemisférico de la cabeza humana. Los potenciales eléctricos se obtuvieron mediante la solución de la ecuación de Laplace en el dominio de la frecuencia. Se introdujo un índice de asimetría eléctrica hemisférica que permite determinar la localización aproximada de la región dañada dentro del hemisferio dañado, el cual fue evaluado para varias conductividades y volúmenes de lesión. Se obtuvo un valor negativo para la isquemia (amplitud del potencial en el hemisferio isquémico mayor que en el hemisferio intacto), mientras que para la hemorragia se obtuvo un valor positivo. Las simulaciones evidenciaron que las variaciones en la conductividad eléctrica y el volumen del tejido dañado tienen un efecto sobre la asimetría hemisférica del potencial eléctrico en la superficie.

Abstract: The equipotential lines of the field generated by the cell used in dielectric measurements of whole fruit tomato are plotted. This cell consists of two inclined electrodes, at inverse potentials, and is an adaptation made in the Dielectric Laboratory of the cell used by Varlan-Sansen. To obtain the graphs, the Laplace equation was solved by finite differences and a program in Fortran language was written, which performs the calculations by numerical iteration. The field generated by the cell at different electrode separation distances was analyzed.

Resumen: Se grafican las líneas equipotenciales del campo generado por la celda utilizada en mediciones dieléctricas de tomate fruta entera. Esta celda consta de dos electrodos inclinados, a potenciales inversos, y es una adaptación realizada en el Laboratorio de Dieléctricos de la celda utilizada por Varlan-Sansen. Para obtener las gráficas se resolvió la ecuación de Laplace por diferencias finitas y se escribió un programa en lenguaje Fortran, que realiza los cálculos por iteración numérica. Se analizó el campo generado por la celda a diferentes distancias de separación de los electrodos.

Desde que James Clerck Maxwell publicou seu brilhante trabalho unificando a eletricidade e magnetismo surgiram, a partir de então, inúmeras aplicações práticas do mesmo. Este artigo discute o potencial elétrico bidimensional e tridimensional, obtidos a partir da solução da equação de Laplace, assim como a solução para os campos elétrico e magnético para a equação de Helmholtz com condições iniciais específicas. Explicitamos a utilização do teorema de superposição em um exemplo claro de eletrostática. Em particular, foi estudado o problema do potencial elétrico em uma caixa retangular com condições de contorno de Dirichlet, fazendo uma aplicação em guias de onda e cavidades ressonantes. O software Mathematica foi usado para obter os gráficos do potencial e campo elétrico.

Ever since James Clerck Maxwell published his brilliant work unifying electricity and magnetism, numerous practical applications of it have arisen. This paper discusses the two- and three-dimensional electric potentials obtained from the solution of Laplace’s equation, as well as the solution to the electric and magnetic fields for the Helmholtz equation with specific initial conditions. We explain the use of the superposition theorem in a clear electrostatics example. In particular, the problem of the electric potential in a rectangular box with Dirichlet boundary conditions was studied, making an application to waveguides and resonant cavities. Mathematica software was used to obtain the plots of the potential and electric field.

Quase todos os problemas relevantes na física envolvem equações diferenciais. Dentre esses, uma classe especial e muito importante de problemas envolvem equações diferenciais parciais de segunda ordem. Um bom exemplo é a equação de Laplace que é fundamental em eletrostática, dinâmica de fluidos e termodinâmica, por exemplo. Em geral, a solução da equação de Laplace não é tarefa fácil e muitos são os métodos desenvolvidos para resolvê-la. Separação de variáveis e método das imagens são bons exemplos desses métodos. Nesse trabalho, apresentaremos as transformações conformes como uma ferramenta muito útil na solução da equação de Laplace. Esse método é pouco utilizado por não ser apresentado aos estudantes em um curso regular de exatas, tal omissão é ainda mais acentuada nos cursos de licenciatura. Nosso objetivo é ajudar a sanar essa lacuna apresentando um texto didático, introdutório e de fácil compreensão ao público interessado.

Most of the relevant physics problems involve differential equations. A special class of problems is those that involve second-order partial differential equations. The Laplace equation, for example, is fundamental in electrostatics, fluid dynamics, and thermodynamics. In many cases, obtaining the solution to the Laplace equation is difficult with the frequently used methods, namely: separation of variables and the method of images. In this work, we will present the conformal transformations as an additional tool that, in certain cases, is much simpler to use. As this tool is often overlooked in regular exact courses, our goal is to help fill this gap by presenting a didactic, introductory, and an easily understood text to the interested public.

Abstract The aim of this paper is to investigate the description of the q-deformed multiple-trapping equation for charge carrier transport in amorphous semiconductors. We first modify the multiple-trapping model of charge carriers in amorphous semiconductors from time-of-flight transient photo-current in the framework of the q-derivative formalism, and then we construct our simulated current by using an approach based on the Laplace method. It is implemented in a program proposed recently by [14] which allows us to construct a current using the Padé approximation expansion. Furthermore, we study the influence of the parameter q of the q-calculus formalism on the drift mobility.

Abstract We explicitly show that the groups of 2 × 2 unitary matrices with determinant equal to 1 whose entries are double or dual numbers are homomorphic to ${\rm SO}(2,1)$ or to the group of rigid motions of the Euclidean plane, respectively, and we introduce the corresponding two-component spinors. We show that with the aid of the double numbers we can find generating functions for separable solutions of the Laplace equation in the ( 2 + 1 ) Minkowski space, which contain special functions that also appear in the solution of the Laplace equation in the three-dimensional Euclidean space, in spheroidal and toroidal coordinates.

Resumen Mostramos explícitamente que los grupos de matrices unitarias 2 × 2 con determinante igual a 1 cuyas entradas son números dobles o números duales son homomorfos a ${\rm SO}(2,1)$ o al grupo de movimientos rígidos del plano euclideano, respectivamente, e introducimos los espinores de dos componentes correspondientes. Mostramos que con la ayuda de los números dobles podemos hallar funciones generatrices para soluciones separables de la ecuación de Laplace en el espacio de Minkowski ( 2 + 1 ), las cuales contienen funciones especiales que también aparecen en la solución de la ecuación de Laplace en el espacio euclideano tridimensional, en coordenadas esferoidales y toroidales.

Abstract This paper deals with an interdisciplinary research work between Mathematical sciences and Electrical engineering to develop fractional model of Resistance-Inductance circuit (RL circuit). Authors obtained the analytical solution of this fractional model in terms of Mittag Leffler function. Graphical interpretation of solution also discussed in this paper.

Resumo Neste trabalho, a equação de Laplace é resolvida analiticamente no plano complexo para o campo de pressões hidrodinâmicas gerado pelo movimento de corpo rígido da barragem na presença de um meio fluido infinito e incompressível (um reservatório). A força exercida pelo fluido do reservatório na face da estrutura da barragem é então determinada através da integração da pressão hidrodinâmica no plano complexo, e os efeitos conservativos (parte real desta força) que traduzem os aspectos inerciais da interação barragem-reservatório, e os efeitos dissipativos (parte imaginária desta força) que traduzem os aspectos de amortecimento desta interação são analisados em função de um parâmetro escalar característico de fluxo de superfície livre (número de Froude). É feita, também, a apresentação das soluções assintóticas para os efeitos citados.

Abstract In this paper, the Laplace’s equation is solved analytically in the complex plane for the field of hydrodynamic pressures generated by the rigid body movement of a dam against a reservoir with infinite domain and incompressible fluid. The force the reservoir fluid exerts on the face of the dam is determined through the integration of the hydrodynamic pressure in the complex plane. The conservative effects (real part) and dissipative effects (imaginary part) of the force are analyzed as a function of the Froude number. The asymptotic solution of the aforementioned effects are also presented in this paper.

ABSTRACT The field of differential equations has recently gained attention due to recent developments in science and technology. For this reason, the analysis for the use of new methodologies to solve them has become important. Based on the combination of Laplace Transform method (LT) and Perturbation Method (PM) this article pro- poses the Laplace transform-Perturbation Method (LT-PM) which finds its motivation on the application of LT to linear ordinary differential equations. The goal of this work is to propose a modification of PM - the LT-PM), in order to solve nonlinear perturbative problems with boundary conditions defined on finite intervals. The proposed methodology consisted on the application of LT to the differential equation to solve and then, assuming that its solutions can be expressed as a series of perturbative parameter powers. Thus, the solution of the problem is obtained by systematically applying the transformed inverse LT. The main results of this paper were shown through two case studies, where LT-PM is identified as potentially useful for finding multiple solutions to nonlinear problems. Additionally, the LT-PM enhances the applicability of PM, in some cases of mixed and Neumann boundary conditions, where PM is unsuitable to provide the results. With the purpose of verifying the accuracy of the obtained results, the Square Residual Error (SRE) was calculated. The resulting value was extremely low, which showed the precision and potential of LT-PM. We conclude that, although the proposed method resulted efficient for the case studies presented in this article, it is expected that LT-PM can be a potentially useful tool for other case studies. Particularly those related to the practical applications of science and engineering.

RESUMEN El campo de las ecuaciones diferenciales ha cobrado auge en la actualidad por el desarrollo científico y tecnológico. Por esta situación, el estudio de nuevas metodologías para solucionarlas se ha vuelto importante. A partir de la combinación del método de Laplace Transform (LT) y el método de perturbación (PM) este trabajo presenta el método LT-PM, y su motivación se encuentra en la aplicación conocida de la LT a ecuaciones diferenciales ordinarias lineales. El objetivo de este trabajo fue presentar una modificación del método de perturbación (PM), el método de perturbación con transformada de Laplace (LT-PM), con el fin de resolver problemas perturbativos no lineales, con condiciones a la frontera definidas en intervalos finitos. La metodología consistió en aplicar LT a la ecuación diferencial por resolver y después de asumir que la solución de la misma se puede expresar como una serie de potencias de un parámetro perturbativo, se obtiene la solución del problema aplicando sistemáticamente la transformada inversa de Laplace. Los principales resultados de este trabajo se muestran a partir de dos casos de estudio presentados, donde se observa que LT-PM es potencialmente útil para encontrar soluciones múltiples de problemas no lineales. Además, LT-PM mejora la aplicabilidad del método de perturbación en algunos casos de condiciones a la frontera mixtas y de Neumann, donde PM simplemente no funciona. Con el fin de verificar la exactitud de los resultados obtenidos, se calculó su error residual cuadrático (SRE), el cual resultó muy bajo, de donde se dedujo su precisión y la potencialidad de LT-PM. Se concluye que si bien el método propuesto resulta eficiente en los casos particulares presentados, se espera que sea una herramienta potencialmente eficiente y útil para otros casos de estudio, particularmente, en aquellos relacionados con aplicaciones prácticas en ciencias e ingeniería.

RESUMO Nesse artigo a equação de Laplace foi utilizada para representar uma distribuição de temperaturas estacionárias no primeiro quadrante no plano cartesiano com diferentes condições de fronteira, tendo sido examinada com detalhes, a luz da transformada de Fourier em seno e cosseno. Após obter a solução formal para cada exemplo, foi possível, usando as equações de Cauchy-Riemann obter cada campo de escoamento de calor. Em um dos exemplos analisados, o campo de velocidade do escoamento tem a forma de um vórtice livre com centro na origem, e desse modo, foi estabelecida uma relação adimensional entre a magnitude do vórtice e a condição de Dirichlet imposta na fronteira. Um exemplo, em particular, foi incluído para mostrar a limitação do uso do método utilizado nesse estudo para a obtenção de soluções explícitas para a equação de Laplace.

ABSTRACT In this paper the Laplace equation was used to represent a distribution of stationary temperatures in the first quadrant in the Cartesian plane with different boundary conditions, having been examined in detail, the light of the sine and cosine Fourier transform. After obtaining the formal solution for each example, it was possible, using the Cauchy-Riemann equations to obtain each field of heat flow. In one of the examples analyzed, the velocity field of the flow is in the form of a free vortex with center at the origin, and an dimensionless relationship between the vortex magnitude and the Dirichlet condition imposed at the boundary has been established. An example, in particular, was included to show the limitation of the this method to obtain explicit solutions for the Laplace equation

Abstract In this paper, we obtain analytical solutions for the time-fractional diffusion and time-fractional convection-diffusion equations. These equations are obtained from the standard equations by replacing the time derivative with a fractional derivative of order α. Fractional operators of type Liouville-Caputo, Atangana-Baleanu-Caputo, fractional conformable derivative in Liouville-Caputo sense, and Atangana-Koca-Caputo were used to model diffusion and convection-diffusion equation. The Laplace and Fourier transforms were applied to obtain analytical solutions for the fractional order diffusion and convection-diffusion equations. The solutions obtained can be useful to understand the modeling of anomalous diffusion, subdiffusive systems and super-diffusive systems, transport processes, random walk and wave propagation phenomenon.