Numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types)

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Jiao Wang
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Numerical Algorithm ◽  
Numerical Solutions ◽  
Integral Transformation ◽  
Operational Matrix ◽  
Fredholm Integral Equations ◽  
Two Dimensional ◽  
Content Type ◽  
Mixed Types

Purpose This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types). Design/methodology/approach The main idea of the presented algorithm is to combine Bernoulli polynomials approximation with Caputo fractional derivative and numerical integral transformation to reduce the studied two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations to easily solved algebraic equations. Findings Without considering the integral operational matrix, this algorithm will adopt straightforward discrete data integral transformation, which can do good work to less computation and high precision. Besides, combining the convenient fractional differential operator of Bernoulli basis polynomials with the least-squares method, numerical solutions of the studied equations can be obtained quickly. Illustrative examples are given to show that the proposed technique has better precision than other numerical methods. Originality/value The proposed algorithm is efficient for the considered two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations. As its convenience, the computation of numerical solutions is time-saving and more accurate.

An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

2018 ◽  
Vol 28 (11) ◽  
pp. 2620-2649 ◽  
Author(s):  
Rajni Rohila ◽  
R.C. Mittal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Diffusion Equations ◽  
Spline Functions ◽  
Two Dimensional ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
Advection Diffusion

Purpose This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method. Design/methodology/approach A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids. Findings Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also. Originality/value ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

scholarly journals A Novel and Efficient Numerical Algorithm for Solving 2D Fredholm Integral Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
H. Bin Jebreen
Keyword(s):  
Approximate Solution ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Numerical Algorithm ◽  
Numerical Experiments ◽  
Operational Matrix ◽  
Fredholm Integral Equations ◽  
Scaling Functions ◽  
Algebraic Equations

A novel and efficient numerical method is developed based on interpolating scaling functions to solve 2D Fredholm integral equations (FIE). Using the operational matrix of integral for interpolating scaling functions, FIE reduces to a set of algebraic equations that one can obtain an approximate solution by solving this system. The convergence analysis is investigated, and some numerical experiments confirm the accuracy and validity of the method. To show the ability of the proposed method, we compare it with others.

Operational matrix of two dimensional Chebyshev wavelets and its applications in solving nonlinear partial integro-differential equations

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Yaser Rostami
Keyword(s):  
Differential Equations ◽  
Design Methodology ◽  
Operational Matrix ◽  
Cross Product ◽  
High Accuracy ◽  
Operational Matrices ◽  
Two Dimensional ◽  
Chebyshev Wavelets ◽  
Content Type ◽  
Collocation Points

Purpose This paper aims to present a new method for the approximate solution of two-dimensional nonlinear Volterra–Fredholm partial integro-differential equations with boundary conditions using two-dimensional Chebyshev wavelets. Design/methodology/approach For this purpose, an operational matrix of product and integration of the cross-product and differentiation are introduced that essentially of Chebyshev wavelets. The use of these operational matrices simplifies considerably the structure of the computation used for a set of the algebraic system has been obtained by using the collocation points and solved. Findings Theorem for convergence analysis and some illustrative examples of using the presented method to show the validity, efficiency, high accuracy and applicability of the proposed technique. Some figures are plotted to demonstrate the error analysis of the proposed scheme. Originality/value This paper uses operational matrices of two-dimensional Chebyshev wavelets and helps to obtain high accuracy of the method.

ψ-Haar wavelets method for numerically solving fractional differential equations

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Amjid Ali ◽  
Teruya Minamoto ◽  
Umer Saeed ◽  
Mujeeb Ur Rehman
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Approximation ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Nonlinear Problems ◽  
Mathematical Tool ◽  
Content Type ◽  
Fractional Models ◽  
Fractional Differential

Purpose The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear fractional differential equations involving ψ-Caputo derivative. Design/methodology/approach An operational matrix to find numerical approximation of ψ-fractional differential equations (FDEs) is derived. This study extends the method to nonlinear FDEs by using quasi linearization technique to linearize the nonlinear problems. Findings The error analysis of the proposed method is discussed in-depth. Accuracy and efficiency of the method are verified through numerical examples. Research limitations/implications The method is simple and a good mathematical tool for finding solutions of nonlinear ψ-FDEs. The operational matrix approach offers less computational complexity. Originality/value Engineers and applied scientists may use the present method for solving fractional models appearing in applications.

A stochastic operational matrix method for numerical solutions of mixed stochastic Volterra–Fredholm integral equations

2019 ◽  
Vol 18 (02) ◽  
pp. 2050005
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Stochastic Integral ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Chebyshev Wavelets ◽  
Operational Matrix Method ◽  
Stochastic Operational Matrix ◽  
Stochastic Integral Equations

In this paper, we have studied space-time Brownian motion and its applications to mixed type stochastic integral equations. Approximate solutions of mixed stochastic integral equations have been obtained by using two-dimensional (2D) second kind Chebyshev wavelets (CWs). Furthermore, some examples have been presented to justify the efficiency of 2D second kind CWs.

Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-spline finite elements

2015 ◽  
Vol 32 (5) ◽  
pp. 1275-1306 ◽  
Author(s):  
R C Mittal ◽  
Amit Tripathi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Spline Functions ◽  
Parabolic Partial Differential Equations ◽  
Two Dimensional ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
Burgers Equations

Purpose – The purpose of this paper is to develop an efficient numerical scheme for non-linear two-dimensional (2D) parabolic partial differential equations using modified bi-cubic B-spline functions. As a test case, method has been applied successfully to 2D Burgers equations. Design/methodology/approach – The scheme is based on collocation of modified bi-cubic B-Spline functions. The authors used these functions for space variable and for its derivatives. Collocation form of the partial differential equation results into system of first-order ordinary differential equations (ODEs). The obtained system of ODEs has been solved by strong stability preserving Runge-Kutta method. The computational complexity of the method is O(p log(p)), where p denotes total number of mesh points. Findings – Obtained numerical solutions are better than those available in literature. Ease of implementation and very small size of computational work are two major advantages of the present method. Moreover, this method provides approximate solutions not only at the grid points but also at any point in the solution domain. Originality/value – First time, modified bi-cubic B-spline functions have been applied to non-linear 2D parabolic partial differential equations. Efficiency of the proposed method has been confirmed with numerical experiments. The authors conclude that the method provides convergent approximations and handles the equations very well in different cases.

A novel technique based on Bernoulli wavelets for numerical solutions of two-dimensional Fredholm integral equation of second kind

2019 ◽  
Vol 36 (6) ◽  
pp. 1798-1819
Author(s):  
S. Saha Ray ◽  
S. Behera
Keyword(s):  
Integral Equation ◽  
Fredholm Integral Equation ◽  
Numerical Solutions ◽  
Bernoulli Polynomials ◽  
Fredholm Integral Equations ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Content Type ◽  
Good Convergence ◽  
Novel Technique

Purpose A novel technique based on Bernoulli wavelets has been proposed to solve two-dimensional Fredholm integral equation of second kind. Bernoulli wavelets have been created by dilation and translation of Bernoulli polynomials. This paper aims to introduce properties of Bernoulli wavelets and Bernoulli polynomials. Design/methodology/approach To solve the two-dimensional Fredholm integral equation of second kind, the proposed method has been used to transform the integral equation into a system of algebraic equations. Findings Numerical experiments shows that the proposed two-dimensional wavelets technique can give high-accurate solutions and good convergence rate. Originality/value The efficiency of newly developed two-dimensional wavelets technique has been validated by different illustrative numerical examples to solve two-dimensional Fredholm integral equations.

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