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Related Concept Videos

Approximate Integration01:24

Approximate Integration

194
In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
194
Rationalizing Substitutions01:29

Rationalizing Substitutions

172
Integrals involving non-rational functions are often difficult to evaluate using standard techniques, especially when radicals appear in the integrand. Rationalizing substitution provides a systematic method for simplifying such integrals by converting them into rational forms that are easier to handle.Consider a rod whose linear mass density depends on a constant linear density, a characteristic length, and the distance from the left end of the rod. Determining the total mass requires...
172
Integration by Parts: Indefinite Integrals01:26

Integration by Parts: Indefinite Integrals

598
Integration by parts is a fundamental technique in calculus for evaluating integrals involving the product of two functions. It is particularly useful when direct integration is not feasible. The method is based on the product rule for differentiation, which states that the derivative of a product equals the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating this identity and rearranging terms, the integration by parts...
598
Integration by Parts: Definite Integrals01:23

Integration by Parts: Definite Integrals

296
Definite integrals involving the product of two functions over a fixed interval can be evaluated using integration by parts. This method rewrites the integral as the difference of a product evaluated at the endpoints and a remaining definite integral that is often simpler to compute.A representative example is the definite integral of the inverse tangent function. Since there is no direct integration formula for arctan ⁡x, the integrand is rewritten as a product of arctan⁡ x and the...
296
Substitution Rule Applied to Definite Integrals01:24

Substitution Rule Applied to Definite Integrals

235
When evaluating a definite integral whose integrand matches the structure of a composite function, the substitution method provides an efficient way to simplify the calculation. This method is based on reversing the chain rule from differentiation, allowing a complicated expression to be rewritten in a simpler form. When the integrand contains an inner function and its derivative, substitution naturally reduces the complexity of the problem.The core idea of substitution for definite integrals...
235
Substitution Rule Applied to Indefinite Integrals01:27

Substitution Rule Applied to Indefinite Integrals

223
When a force is applied to a linear spring, the restoring force increases proportionally with the amount of displacement. This behavior is described by Hooke’s law, which allows the work done on the spring to be determined directly from the force–displacement relationship. In this case, the force varies in a simple and predictable manner, making the calculation relatively simple.On the other hand, a nonlinear spring does not obey Hooke’s law. Its restoring force depends on...
223

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Approximate series solution of nonlinear singular boundary value problems arising in physiology.

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Approximate solution of Urysohn integral equations using the Adomian decomposition method.

Randhir Singh1, Gnaneshwar Nelakanti1, Jitendra Kumar1

  • 1Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India.

Thescientificworldjournal
|February 28, 2014
PubMed
Summary

The Adomian decomposition method (ADM) offers a direct recursive approach for approximating solutions to Urysohn integral equations. This study demonstrates its accuracy and applicability through numerical examples and convergence analysis.

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Area of Science:

  • Numerical Analysis
  • Integral Equations
  • Applied Mathematics

Background:

  • Urysohn integral equations are a class of nonlinear integral equations with applications in various scientific fields.
  • Approximate analytical solutions are often required due to the complexity of exact solutions.
  • The Adomian decomposition method (ADM) is a powerful technique for solving nonlinear equations.

Purpose of the Study:

  • To apply the Adomian decomposition method (ADM) for finding approximate series solutions to Urysohn integral equations.
  • To establish a direct recursive scheme for efficient approximate solutions.
  • To analyze the convergence properties and error bounds of the ADM for these equations.

Main Methods:

  • The Adomian decomposition method (ADM) is employed to derive a recursive formula for the series solution.
  • Components of the series solution are calculated iteratively.
  • Convergence and error analysis are performed theoretically.

Main Results:

  • The ADM provides a straightforward recursive scheme for generating approximate series solutions.
  • The components of the series solution are easily computable.
  • Numerical examples confirm the accuracy and practical utility of the ADM.

Conclusions:

  • The Adomian decomposition method is an effective technique for solving Urysohn integral equations.
  • The method offers a balance between accuracy and computational simplicity.
  • The convergence and error analysis provide theoretical support for the method's reliability.