Comparison for accurate solutions of nonlinear Hammerstein fuzzy integral equations

In this paper, efficient numerical techniques have been proposed to solve nonlinear Hammerstein fuzzy integral equations. The proposed methods are based on Bernsteinpolynomials and Legendre wavelets approximation. Usually, nonlinear fuzzy integral equations are very difficult to solve both analytically and numerically. The present methods applied to the integral equations is reduced to solve the system of nonlinear algebraic equations. Again, this system has been solved by Newton’s method. The numerical results obtained by present methods have been compared with those of the homotopy analysis method. Illustrative examples have been discussed to demonstrate the validity and applicability of the presented methods

Numerical Approximate Methods for Solving Linear and Nonlinear Integral Equations

Integral equation has been one of the essential tools for various area of applied mathematics. In this work, we employed different numerical methods for solving both linear and nonlinear Fredholm integral equations. A goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations. Integral equations can be viewed as equations which are results of transformation of points in a given vector spaces of integrable functions by the use of certain specific integral operators to points in the same space. If, in particular, one is concerned with function spaces spanned by polynomials for which the kernel of the corresponding transforming integral operator is separable being comprised of polynomial functions only, then several approximate methods of solution of integral equations can be developed. This work, specially, deals with the development of different wavelet methods for solving integral and intgro-differential equations. Wavelets theory is a relatively new and emerging area in mathematical research. It has been applied in a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for waveform representations and segmentations, time frequency analysis, and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms. Wavelets can be separated into two distinct types, orthogonal and semi-orthogonal. The preliminary concept of integral equations and wavelets are first presented in Chapter 1. Classification of integral equations, construction of wavelets and multi-resolution analysis (MRA) have been briefly discussed and provided in this chapter. In Chapter 2, different wavelet methods are constructed and function approximation by these methods with convergence analysis have been presented. In Chapter 3, linear semi-orthogonal compactly supported B-spline wavelets together with their dual wavelets have been applied to approximate the solutions of Fredholm integral equations (both linear and nonlinear) of the second kind and their systems. Properties of these wavelets are first presented; these properties are then utilized to reduce the computation of integral equations to some algebraic equations. Convergence analysis of B-spline method has been discussed in this chapter. Again, in Chapter 4, system of nonlinear Fredholm integral equations have been solved by using hybrid Legendre Block-Pulse functions and xiii Bernstein collocation method. In Chapter 5, two practical problems arising from chemical phenomenon, have been modeled as Fredholm- Hammerstein integral equations and solved numerically by different numerical techniques. First, COSMO-RS model has been solved by Bernstein collocation method, Haar wavelet method and Sinc collocation method. Second, Hammerstein integral equation arising from chemical reactor theory has been solved by B-spline wavelet method. Comparison of results have been demonstrated through illustrative examples. In Chapter 6, Legendre wavelet method and Bernoulli wavelet method have been developed to solve system of integro-differential equations. Legendre wavelets along with their operational matrices are developed to approximate the solutions of system of nonlinear Volterra integro-differential equations. Also, nonlinear Volterra weakly singular integro-differential equations system has been solved by Bernoulli wavelet method. The properties of these wavelets are used to reduce the system of integral equations to a system of algebraic equations which can be solved numerically by Newton's method. Rigorous convergence analysis has been done for these wavelet methods. Illustrative examples have been included to demonstrate the validity and applicability of the proposed techniques. In Chapter 7, we have solved the second order Lane-Emden type singular differential equation. First, the second order differential equation is transformed into integro-differential equation and then solved by Legendre multi-wavelet method and Chebyshev wavelet method. Convergence of these wavelet methods have been discussed in this chapter. In Chapter 8, we have developed a efficient collocation technique called Legendre spectral collocation method to solve the Fredholm integro-differential-difference equations with variable coefficients and system of two nonlinear integro-differential equations which arise in biological model. The proposed method is based on the Gauss-Legendre points with the basis functions of Lagrange polynomials. The present method reduces this model to a system of nonlinear algebraic equations and again this algebraic system has been solved numerically by Newton's method. The study of fuzzy integral equations and fuzzy differential equations is an emerging area of research for many authors. In Chapter 9, we have proposed some numerical techniques for solving fuzzy integral equations and fuzzy integro-differential equations. Fundamentals of fuzzy calculus have been discussed in this chapter. Nonlinear fuzzy Hammerstein integral equation has been solved by Bernstein polynomials and Legendre wavelets, and then compared with homotopy analysis method. We have solved nonlinear fuzzy Hammerstein Volterra integral equations with constant delay by Bernoulli wavelet method and then compared with B-spline wavelet method. Finally, fuzzy integro-differential equation has been solved by Legendre wavelet method and compared with homotopy analysis method. In fuzzy case, we have applied two-dimensional numerical methods which are discussed in chapter 2. Convergence analysis and error estimate have been also provided for Bernoulli wavelet method. xiv The study of fractional calculus, fractional differential equations and fractional integral equations has a great importance in the field of science and engineering. Most of the physical phenomenon can be best modeled by using fractional calculus. Applications of fractional differential equations and fractional integral equations create a wide area of research for many researchers. This motivates to work on fractional integral equations, which results in the form of Chapter 10. First, the preliminary definitions and theorems of fractional calculus have been presented in this chapter. The nonlinear fractional mixed Volterra-Fredholm integro-differential equations along with mixed boundary conditions have been solved by Legendre wavelet method. A numerical scheme has been developed by using Petrov-Galerkin method where the trial and test functions are Legendre wavelets basis functions. Also, this method has been applied to solve fractional Volterra integro-differential equations. Uniqueness and existence of the problem have been discussed and the error estimate of the proposed method has been presented in this work. Sinc Galerkin method is developed to approximate the solution of fractional Volterra-Fredholm integro-differential equations with weakly singular kernels. The proposed method is based on the Sinc function approximation. Uniqueness and existence of the problem have been discussed and the error analysis of the proposed method have been presented in this chapte

Solutions to Uncertain Volterra Integral Equations by Fitted Reproducing Kernel Hilbert Space Method

We present an efficient modern strategy for solving some well-known classes of uncertain integral equations arising in engineering and physics fields. The solution methodology is based on generating an orthogonal basis upon the obtained kernel function in the Hilbert space 1 2 [ , ] in order to formulate the analytical solutions in a rapidly convergent series form in terms of their -cut representation. The approximation solution is expressed by -term summation of reproducing kernel functions and it is convergent to the analytical solution. Our investigations indicate that there is excellent agreement between the numerical results and the RKHS method, which is applied to some computational experiments to demonstrate the validity, performance, and superiority of the method. The present work shows the potential of the RKHS technique in solving such uncertain integral equations

System Engineering Applied to Fuenmayor Karst Aquifer (San Julián de Banzo, Huesca) and Collins Glacier (King George Island, Antarctica)

La ingeniería de sistemas, definida generalmente como arte y ciencia de crear soluciones integrales a problemas complejos, se aplica en el presente documento a dos sistemas naturales, a saber, un sistema acuífero kárstico y un sistema glaciar, desde una perspectiva hidrológica. Las técnicas de identificación, desarrolladas típicamente en ingeniería para representar sistemas artificiales por medio de modelos lineales y no lineales, pueden aplicarse en el estudio de los sistemas naturales donde se producen fenómenos de acoplamiento entre el clima y la hidrosfera. Los métodos evolucionan para afrontar nuevos campos de identificación donde se requieren estrategias para encontrar el modelo idóneo adaptado a las peculiaridades del sistema. En este sentido, se han considerado especialmente las herramientas basadas en la transformada wavelet utilizadas en la preparación de series temporales, suavizado de señales, análisis espectral, correlación cruzada y predicción, entre otros. Bajo este enfoque, una aplicación a mencionar entre las tratadas en esta tesis, es la determinación analítica del núcleo efectivo estacional (SEC) a través del estudio de la coherencia wavelet entre temperatura del aire y la descarga del glaciar, que establece un conjunto de períodos de muestreo aceptablemente coherentes, a partir del cual se crearán los modelos del sistema glacial. El estudio está dirigido específicamente a estimar la influencia de la precipitación sobre la descarga del acuífero kárstico de Fuenmayor, en San Julián de Banzo, Huesca, España. De la misma manera, se ocupa de las consecuencias de la temperatura del aire en la fusión del hielo glaciar, que se manifiesta en la corriente de drenaje del glaciar Collins, isla King George, Antártida. En el proceso de identificación paramétrica y no paramétrica se buscan los modelos que mejor representen la dinámica interna del sistema. Eso conduce a pruebas iterativas, donde se van creando modelos que se verifican sistemáticamente con los datos reales del muestreo, de acuerdo a un criterio de eficiencia dado. La solución mejor valorada según los resultados obtenidos en los casos tratados apuntan a estructuras de modelos en bloques. Esta tesis significa una exposición formal de la metodología de identificación de sistemas propios de la ingeniería en el contexto de los sistemas naturales, que mejoran los resultados obtenidos en muchos casos de la hidrología kárstica que comúnmente usaban métodos ad hoc ocasionales de carácter estadístico; así mismo, los enfoques propuestos en los casos de glaciología con el análisis wavelet y los modelos orientados a datos raramente considerados en la literatura, revelan información esencial ante la imposibilidad de precisar la totalidad de la física que rige el sistema. Notables resultados se derivan en la caracterización de la respuesta del manantial de Fuenmayor y su correlación con la precipitación, desde la perspectiva de un sistema lineal, que se complementa con los métodos de identificación basados en técnicas no lineales. Así mismo, la implementación del modelo para el glaciar Collins, obtenido también mediante métodos de identificación de caja negra, puede revelar una inestabilidad de los límites de los periodos activos de la descarga, y consecuentemente la variabilidad en la tendencia actual en el cambio climático global

Tools for Modelling and Identification with Bond Graphs and Genetic Programming

The contributions of this work include genetic programming grammars for bond graph modelling and for direct symbolic regression of sets of differential equations; a bond graph modelling library suitable for programmatic use; a symbolic algebra library specialized to this use and capable of, among other things, breaking algebraic loops in equation sets extracted from linear bond graph models. Several non-linear multi-body mechanics examples are pre- sented, showing that the bond graph modelling library exhibits well-behaved simulation results. Symbolic equations in a reduced form are produced au- tomatically from bond graph models. The genetic programming system is tested against a static non-linear function identification problem using type- less symbolic regression. The direct symbolic regression grammar is shown to have a non-deceptive fitness landscape: perturbations of an exact pro- gram have decreasing fitness with increasing distance from the ideal. The planned integration of bond graphs with genetic programming for use as a system identification technique was not successfully completed. A catego- rized overview of other modelling and identification techniques is included as context for the choice of bond graphs and genetic programming

Hägusad teist liiki integraalvõrrandid

Käesolevas doktoritöös on uuritud hägusaid teist liiki integraalvõrrandeid. Need võrrandid sisaldavad hägusaid funktsioone, s.t. funktsioone, mille väärtused on hägusad arvud. Me tõestasime tulemuse sileda tuumaga hägusate Volterra integraalvõrrandite lahendite sileduse kohta. Kui integraalvõrrandi tuum muudab märki, siis integraalvõrrandi lahend pole üldiselt sile. Nende võrrandite lahendamiseks me vaatlesime kollokatsioonimeetodit tükiti lineaarsete ja tükiti konstantsete funktsioonide ruumis. Kasutades lahendi sileduse tulemusi tõestasime meetodite koonduvuskiiruse. Me vaatlesime ka nõrgalt singulaarse tuumaga hägusaid Volterra integraalvõrrandeid. Uurisime lahendi olemasolu, ühesust, siledust ja hägusust. Ülesande ligikaudseks lahendamiseks kasutasime kollokatsioonimeetodit tükiti polünoomide ruumis. Tõestasime meetodite koonduvuskiiruse ning uurisime lähislahendi hägusust. Nii analüüs kui ka numbrilised eksperimendid näitavad, et gradueeritud võrke kasutades saame parema koonduvuskiiruse kui ühtlase võrgu korral. Teist liiki hägusate Fredholmi integraalvõrrandite lahendamiseks pakkusime uue lahendusmeetodi, mis põhineb kõigi võrrandis esinevate funktsioonide lähendamisel Tšebõšovi polünoomidega. Uurisime nii täpse kui ka ligikaudse lahendi olemasolu ja ühesust. Tõestasime meetodi koonduvuse ja lähislahendi hägususe.In this thesis we investigated fuzzy integral equations of the second kind. These equations contain fuzzy functions, i.e. functions whose values are fuzzy numbers. We proved a regularity result for solution of fuzzy Volterra integral equations with smooth kernels. If the kernel changes sign, then the solution is not smooth in general. We proposed collocation method with triangular and rectangular basis functions for solving these equations. Using the regularity result we estimated the order of convergence of these methods. We also investigated fuzzy Volterra integral equations with weakly singular kernels. The existence, regularity and the fuzziness of the exact solution is studied. Collocation methods on discontinuous piecewise polynomial spaces are proposed. A convergence analysis is given. The fuzziness of the approximate solution is investigated. Both the analysis and numerical methods show that graded mesh is better than uniform mesh for these problems. We proposed a new numerical method for solving fuzzy Fredholm integral equations of the second kind. This method is based on approximation of all functions involved by Chebyshev polynomials. We analyzed the existence and uniqueness of both exact and approximate fuzzy solutions. We proved the convergence and fuzziness of the approximate solution.https://www.ester.ee/record=b539569