Approximate solutions for mixed nonlinear Volterra–Fredholm type integral equations via modified block-pulse functions

AbstractIn this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.

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On Complex-Valued Triple Controlled Metric Spaces and Applications

Journal of Function Spaces ◽

10.1155/2021/5563456 ◽

2021 ◽

Vol 2021 ◽

pp. 1-7

Author(s):

Nabil Mlaiki ◽

Thabet Abdeljawad ◽

Wasfi Shatanawi ◽

Hassen Aydi ◽

Yaé Ulrich Gaba

Keyword(s):

Fixed Point ◽

Integral Equations ◽

Metric Spaces ◽

Polynomial Equations ◽

Degree Polynomial ◽

Fredholm Type ◽

Type Integral ◽

Type Spaces ◽

Complex Valued

In this manuscript, we introduce the concept of complex-valued triple controlled metric spaces as an extension of rectangular metric type spaces. To validate our hypotheses and to show the usability of the Banach and Kannan fixed point results discussed herein, we present an application on Fredholm-type integral equations and an application on higher degree polynomial equations.

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New Explicit Bounds on Gamidov Type Integral Inequalities for Functions in Two Variables and Their Applications

Abstract and Applied Analysis ◽

10.1155/2014/539701 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Kelong Cheng ◽

Chunxiang Guo

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Integral Inequalities ◽

Fredholm Integral Equations ◽

Uniqueness Of Solutions ◽

Fredholm Type ◽

Type Integral ◽

Explicit Bounds ◽

Linear And Nonlinear

Some linear and nonlinear Gamidov type integral inequalities in two variables are established, which can give the explicit bounds on the solutions to a class of Volterra-Fredholm integral equations. Some examples of application are presented to show boundedness and uniqueness of solutions of a Volterra-Fredholm type integral equation.

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Monte-Carlo for solving large linear systems of ordinary diﬀerential equations

Vestnik of Saint Petersburg University Mathematics Mechanics Astronomy ◽

10.21638/spbu01.2021.104 ◽

2021 ◽

Vol 8 (1) ◽

pp. 37-48

Author(s):

Sergey M. Ermakov ◽

◽

Maxim G. Smilovitskiy ◽

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Differential Equations ◽

Integral Equations ◽

Virtual Machines ◽

Variable Coefficients ◽

Fredholm Type ◽

Type Integral ◽

Constant Coefficients ◽

Linear Differential

Monte-Carlo approach towards solving Cauchy problem for large systems of linear differential equations is being proposed in this paper. Firstly, a quick overlook of previously obtained results from applying the approach towards Fredholm-type integral equations is being made. In the main part of the paper, a similar method is being applied towards a linear system of ODE. It is transformed into an equivalent system of Volterra-type integral equations, which relaxes certain limitations being present due to necessary conditions for convergence of majorant series. The following theorems are being stated. Theorem 1 provides necessary compliance conditions that need to be imposed upon initial and transition distributions of a required Markov chain, for which an equality between estimate’s expectation and a desirable vector product would hold. Theorem 2 formulates an equation that governs estimate’s variance, while theorem 3 states a form for Markov chain parameters that minimise the variance. Proofs are given, following the statements. A system of linear ODEs that describe a closed queue made up of ten virtual machines and seven virtual service hubs is then solved using the proposed approach. Solutions are being obtained both for a system with constant coefficients and time-variable coefficients, where breakdown intensity is dependent on t. Comparison is being made between Monte-Carlo and Rungge Kutta obtained solutions. The results can be found in corresponding tables.

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On the Solutions of a Class of Integral Equations Pertaining to Incomplete H-Function and Incomplete H-Function

Mathematics ◽

10.3390/math8050819 ◽

2020 ◽

Vol 8 (5) ◽

pp. 819

Author(s):

Manish Kumar Bansal ◽

Devendra Kumar ◽

Jagdev Singh ◽

Kottakkaran Sooppy Nisar

Keyword(s):

Integral Equation ◽

Fractional Calculus ◽

Integral Equations ◽

Real World ◽

Mellin Transform ◽

Fredholm Type ◽

Type Integral ◽

Real World Problems

The main aim of this article is to study the Fredholm-type integral equation involving the incomplete H-function (IHF) and incomplete H-function in the kernel. Firstly, we solve an integral equation associated with the IHF with the aid of the theory of fractional calculus and Mellin transform. Next, we examine an integral equation pertaining to the incomplete H-function with the help of theory of fractional calculus and Mellin transform. Further, we indicate some known results by specializing the parameters of IHF and incomplete H-function. The results computed in this article are very general in nature and capable of giving many new and known results connected with integral equations and their solutions hitherto scattered in the literature. The derived results are very useful in solving various real world problems.

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Numerical Solution of Nonlinear Fredholm Integrodifferential Equations of Fractional Order by Using Hybrid of Block-Pulse Functions and Chebyshev Polynomials

Mathematical Problems in Engineering ◽

10.1155/2011/341989 ◽

2011 ◽

Vol 2011 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Changqing Yang

Keyword(s):

Integral Equations ◽

Chebyshev Polynomials ◽

Mathematical Physics ◽

Fredholm Integral Equations ◽

Chebyshev Series ◽

Fredholm Type ◽

Numerical Examples ◽

Block Pulse Functions ◽

Hybrid Function ◽

Nonlinear Fredholm Integral Equations

A numerical method for solving nonlinear Fredholm integral equations of second kind is proposed. The Fredholm-type equations, which have many applications in mathematical physics, are then considered. The method is based upon hybrid function approximate. The properties of hybrid of block-pulse functions and Chebyshev series are presented and are utilized to reduce the computation of nonlinear Fredholm integral equations to a system of nonlinear. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method.

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Solving of Linear Volterra-Fredholm Integral Equations via Modification of Block Pulse Functions

Jurnal Matematika Integratif ◽

10.24198/jmi.v17.n1.32003.33-42 ◽

2021 ◽

Vol 17 (1) ◽

pp. 33

Author(s):

Ayyubi Ahmad

Keyword(s):

Approximate Solution ◽

Integral Equations ◽

Operational Matrix ◽

Approximate Solutions ◽

Computational Method ◽

Fredholm Integral Equations ◽

Triangular System ◽

Algebraic Equations ◽

Block Pulse Functions ◽

Linear Algebraic Equations

A computational method based on modification of block pulse functions is proposed for solving numerically the linear Volterra-Fredholm integral equations. We obtain integration operational matrix of modification of block pulse functions on interval [0,T). A modification of block pulse functions and their integration operational matrix can be reduced to a linear upper triangular system. Then, the problem under study is transformed to a system of linear algebraic equations which can be used to obtain an approximate solution of linear Volterra-Fredholm integral equations. Furthermore, the rate of convergence is O(h) and error analysis of the proposed method are investigated. The results show that the approximate solutions have a good of efficiency and accuracy.

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Fixed point theorems for Geraghty-type mappings applied to solving nonlinear Volterra-Fredholm integral equations in modular G-metric spaces

Arab Journal of Mathematical Sciences ◽

10.1108/ajms-10-2020-0098 ◽

2021 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Godwin Amechi Okeke ◽

Daniel Francis

Keyword(s):

Fixed Point ◽

Integral Equations ◽

Existence And Uniqueness ◽

Metric Spaces ◽

Fredholm Integral Equations ◽

Fredholm Type ◽

Content Type ◽

Special Cases ◽

Type Integral ◽

Social Applications

PurposeThe authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. The authors apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend generalize compliment and include several known results as special cases.Design/methodology/approachThe results of this paper are theoretical and analytical in nature.FindingsThe authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend, generalize, compliment and include several known results as special cases.Research limitations/implicationsThe results are theoretical and analytical.Practical implicationsThe results were applied to solving nonlinear integral equations.Social implicationsThe results has several social applications.Originality/valueThe results of this paper are new.

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The Solutions of Mixed Monotone Fredholm-Type Integral Equations in Banach Spaces

Discrete Dynamics in Nature and Society ◽

10.1155/2013/604105 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Hua Su

Keyword(s):

Boundary Value Problem ◽

Banach Spaces ◽

Unique Solution ◽

Integral Equations ◽

Nonlinear Ordinary Differential Equations ◽

Point Boundary ◽

Fredholm Type ◽

Monotone Iterative ◽

Type Integral ◽

Ordered Banach Spaces

By introducing new definitions ofϕconvex and-φconcave quasioperator andv0quasilower andu0quasiupper, by means of the monotone iterative techniques without any compactness conditions, we obtain the iterative unique solution of nonlinear mixed monotone Fredholm-type integral equations in Banach spaces. Our results are even new toϕconvex and-φconcave quasi operator, and then we apply these results to the two-point boundary value problem of second-order nonlinear ordinary differential equations in the ordered Banach spaces.

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