Numerical Investigation of Volterra Integral Equations of Second Kind using Optimal Homotopy Asymptotic Method

Chu, Yu‐Ming; Ullah, Saif; Ali, Muzaher; Tuzzahrah, Ghulam Fatima; Munir, Taj

doi:10.1016/j.amc.2022.127304

Cited by 9 publications

(6 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The first 11 Bernstein polynomials of degree ten over the interval [a,b], are given below: 𝐵 0,10 (𝑥) = (𝑏 − 𝑥) 10 /(𝑏 − 𝑎) 10 𝐵 6,10 (𝑥) = 210(𝑏 − 𝑥) 4 (𝑥 − 𝑎) 6 /(𝑏 − 𝑎) 10 𝐵 1,10 (𝑥) = 10(𝑏 − 𝑥) 9 (𝑥 − 𝑎)/(𝑏 − 𝑎) 10…”

Section: Bernstein Polynomialsunclassified

“…Zarnan [9] used Bernstein Polynomials to solve a Love's integral Equations. Yu-Ming Chu and et al in [10] are offers Numerical investigation of Volterra integral Equations of second type using most effective Homotopy Asymptotic technique. Khidir in [11] is present a brand new Numerical approach for solving Volterra integral Equations using Chebyshev Spectral technique.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

New Solution Method for Solving Volterra Integral Equation Using Bernstein Polynomials

Zarnan,

Hameed

2024

Preprint

View full text Add to dashboard Cite

Linear and nonlinear differential equations extensively used mathematical models for many interesting and important phenomena observed in numerous areas of science and technology. They are inspired by problems in diverse fields such as economics, biology, fluid dynamics, physics, differential geometry, engineering, control theory, materials science, and quantum mechanics. This special issue aims to highlight recent developments in methods and applications of linear and nonlinear differential equations. In addition, there are papers analyzing equations that arise in engineering, classical and fluid mechanics, and finance. In this paper a novel technique implementing Bernstein polynomials is introduced for the numerical solution of a Volterra integral equation. The obtained solutions are novel, and previous literature lacks such derivations. For this aim, we derive a simple and efficient matrix formulation using Bernstein polynomials as trial functions. Numerical examples are considered to verify the effectiveness of the proposed derivations and numerical solutions are compared with the existing methods available in the literature. The Volterra integral equation can be used to describe the capacitance of the parallel plate capacitor (PPC) in the electrostatic field. It is shown that the numerical results are excellent. The efficiency of the proposed numerical technique is exhibited through graphical illustrations and results drafted in tabular form for specific values of the parameters to validate the numerical investigation.

show abstract

Section: Bernstein Polynomialsunclassified

Section: Introductionmentioning

confidence: 99%

New Solution Method for Solving Volterra Integral Equation Using Bernstein Polynomials

Zarnan,

Hameed

2024

Preprint

View full text Add to dashboard Cite

show abstract

“…Moreover, researchers have focused on solving fractional NLEEs, employing diverse techniques such as the Homotopy Analysis Method [ 32 ], VIM (Ref. [ 33 ]), He’s HPM [ 34 ], ADM [ 35 , 36 ], homotopy asymptotic scheme [ 37 ], and reduced differential transform method [ 16 ]. Ren recently contributed results related to Caputo-type partial differential equations [ 38 ].…”

Section: Introductionmentioning

confidence: 99%

Exploring fractional-order new coupled Korteweg-de Vries system via improved Adomian decomposition method

Arshad,

Aldosary,

Batool

et al. 2024

PLoS ONE

View full text Add to dashboard Cite

This paper aims to extend the applications of the projected fractional improved Adomian Decomposition method (fIADM) to the fractional order new coupled Korteweg-de Vries (cKdV) system. This technique is significantly recognized for its effectiveness in addressing nonlinearities and iteratively handling fractional derivatives. The approximate solutions of the fractional-order new cKdV system are obtained by employing the improved ADM in fractional form. These solutions play a crucial role in designing and optimizing systems in engineering applications where accurate modeling of wave phenomena is essential, including fluid dynamics, plasma physics, nonlinear optics, and other mathematical physics domains. The fractional order new cKdV system, integrating fractional calculus, enhances accuracy in modeling wave interactions compared to the classical cKdV system. Comparison with exact solutions demonstrates the high accuracy and ease of application of the projected method. This proposed technique proves influential in resolving fractional coupled systems encountered in various fields, including engineering and physics. Numerical results obtained using Mathematica software further verify and demonstrate its efficacy.

show abstract

“…These mathematicians contributed significantly to fractional calculus and its many applications. For further information on fractional calculus, see [15][16][17][18][19][20][21][22][23][24][25][26][27]. In the modern era, fractional calculus is frequently used to describe a variety of phenomena, such as the fractional conservation of mass, and the fractional Schrodinger equation in quantum theory.…”

Section: Introductionmentioning

confidence: 99%

New Versions of Fuzzy-Valued Integral Inclusion over p-Convex Fuzzy Number-Valued Mappings and Related Fuzzy Aumman’s Integral Inequalities

Alreshidi,

Khan,

Breaz

et al. 2023

Symmetry

View full text Add to dashboard Cite

It is well known that both concepts of symmetry and convexity are directly connected. Similarly, in fuzzy theory, both ideas behave alike. It is important to note that real and interval-valued mappings are exceptional cases of fuzzy number-valued mappings (FNVMs) because fuzzy theory depends upon the unit interval that make a significant contribution to overcoming the issues that arise in the theory of interval analysis and fuzzy number theory. In this paper, the new class of p-convexity over up and down (UD) fuzzy relation has been introduced which is known as UD-p-convex fuzzy number-valued mappings (UD-p-convex FNVMs). We offer a thorough analysis of Hermite–Hadamard-type inequalities for FNVMs that are UD-p-convex using the fuzzy Aumann integral. Some previous results from the literature are expanded upon and broadly applied in our study. Additionally, we offer precise justifications for the key theorems that Kunt and İşcan first deduced in their article titled “Hermite–Hadamard–Fejer type inequalities for p-convex functions”. Some new and classical exceptional cases are also discussed. Finally, we illustrate our findings with well-defined examples.

show abstract

Numerical Investigation of Volterra Integral Equations of Second Kind using Optimal Homotopy Asymptotic Method

Cited by 9 publications

References 11 publications

New Solution Method for Solving Volterra Integral Equation Using Bernstein Polynomials

New Solution Method for Solving Volterra Integral Equation Using Bernstein Polynomials

Exploring fractional-order new coupled Korteweg-de Vries system via improved Adomian decomposition method

New Versions of Fuzzy-Valued Integral Inclusion over p-Convex Fuzzy Number-Valued Mappings and Related Fuzzy Aumman’s Integral Inequalities

Contact Info

Product

Resources

About