scholarly journals Relationship between Sumudu and Some Efficient Integral Transforms

2020 ◽  
Vol 9 (3) ◽  
pp. 153-159 ◽  
Keyword(s):  
Infinite Series ◽  
Initial Value Problems ◽  
Integral Transforms ◽  
Renewal Equation ◽  
Physical Explanation ◽  
Initial Value ◽  
Gamma Functions ◽  
Improper Integrals ◽  
The Relationship

Nowadays integral transforms are most appropriate techniques for finding the solution of typical problems because these techniques convert them into simpler problems. Finding the solution of initial value problems is the main use of integral transforms. However, there are so many other applications of integral transforms in different areas of mathematics and statistics such as in solving improper integrals of first kind, evaluating the sum of the infinite series, developing the relationship between Beta and Gamma functions, solving renewal equation etc. In this paper, scholars established the relationship between Sumudu and some efficient integral transforms. The application section of this paper has tabular representation of integral transforms of some regularly used functions to demonstrate the physical explanation of relationship between Sumudu and mention integral transforms.

An Application of Modified Predictor-Corrector Method

2004 ◽  
Author(s):  
Mladen Mesˇtrovic´
Keyword(s):  
Initial Value Problems ◽  
Numerical Solutions ◽  
Integration Method ◽  
Convex Combination ◽  
Numerical Algorithms ◽  
Initial Value ◽  
Linear Convex Combination ◽  
Weighting Coefficients ◽  
Predictor Corrector ◽  
The Relationship

The explicit numerical integration method, introduced and proposed in the paper given by Chiou and Wu [1], is further developed. The method is based on the relationship that m-step Adams-Moulton method is linear convex combination of the (m − 1)-step Adams-Moulton and m-step Adams-Bashforth method with a fixed weighting coefficients. The general form taken from Chiou and Wu [1] is used to evaluate the recurrence expressions using the different number of previous mesh points. The explicit expressions are given for modified 3-step predictor-corrector method. The numerical algorithms are given for first and second-order nonlinear initial value problems and for system of ordinary differential equations. Some numerical examples, for different kind of problems, are used to demonstrate the efficiency and the accuracy of the proposed numerical method. The calculated numerical solutions show superiority of presented modified predictor-corrector method to standard Adams-Bashforth-Moulton predictor-corrector method.

scholarly journals On Effects of a New Method for Fractional Initial Value Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Hülya Kodal Sevindir ◽  
Süleyman Çetinkaya ◽  
Ali Demir
Keyword(s):  
Iterative Method ◽  
Fractional Derivative ◽  
Infinite Series ◽  
Initial Value Problems ◽  
Analytic Solution ◽  
Initial Value ◽  
Transform Method ◽  
Series Form ◽  
Time Fractional Derivative ◽  
A New Technique

The aim of this study is to analyze nonlinear Liouville-Caputo time-fractional problems by a new technique which is a combination of the iterative and ARA transform methods and is denoted by IAM. First, the ARA transform method and its inverse are utilized to get rid of time fractional derivative. Later, the iterative method is applied to establish the solution of the problem in infinite series form. The main advantages of this method are that it converges to analytic solution of the problem rapidly and implementation of method is easy. Finally, outcomes of the illustrative examples prove the efficiency and accuracy of the method.

Mathematical Models of Papermaking

2001 ◽  
Vol 6 (1) ◽  
pp. 9-19 ◽  
Author(s):  
A. Buikis ◽  
J. Cepitis ◽  
H. Kalis ◽  
A. Reinfelds ◽  
A. Ancitis ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Initial Value Problems ◽  
Moisture Transfer ◽  
Bound Water ◽  
Drying Process ◽  
Initial Value ◽  
First Order ◽  
Heat And Moisture ◽  
The Mathematical Model ◽  
The Web

The mathematical model of wood drying based on detailed transport phenomena considering both heat and moisture transfer have been offered in article. The adjustment of this model to the drying process of papermaking is carried out for the range of moisture content corresponding to the period of drying in which vapour movement and bound water diffusion in the web are possible. By averaging as the desired models are obtained sequence of the initial value problems for systems of two nonlinear first order ordinary differential equations. 

scholarly journals Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2023
Author(s):  
Christopher Nicholas Angstmann ◽  
Byron Alexander Jacobs ◽  
Bruce Ian Henry ◽  
Zhuang Xu
Keyword(s):  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Evolution Equations ◽  
Integral Transform ◽  
Initial Value ◽  
Intrinsic Nature ◽  
Recent Interest ◽  
Special Cases ◽  
Singular Kernels

There has been considerable recent interest in certain integral transform operators with non-singular kernels and their ability to be considered as fractional derivatives. Two such operators are the Caputo–Fabrizio operator and the Atangana–Baleanu operator. Here we present solutions to simple initial value problems involving these two operators and show that, apart from some special cases, the solutions have an intrinsic discontinuity at the origin. The intrinsic nature of the discontinuity in the solution raises concerns about using such operators in modelling. Solutions to initial value problems involving the traditional Caputo operator, which has a singularity inits kernel, do not have these intrinsic discontinuities.

scholarly journals On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1303
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Faraidun Kadir Hamasalh
Keyword(s):  
Time Scale ◽  
Initial Value Problems ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Discrete Fractional Calculus ◽  
Initial Value ◽  
Forward Difference ◽  
The Mean ◽  
Monotonicity Analysis ◽  
Monotonicity Results

Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) for each z∈Ma+h,h. Conversely, if y(a+υh) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) on Ma,h. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale hZ utilizing the monotonicity results.

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