Science Explorer Back to mapMap
Discussion
Topic · Modeling and Simulation

Fractional Differential Equations Solutions

Fractional Differential Equations Solutions is a research topic within Modeling and Simulation. Science Explorer counts 37k research works in it since 1951. 9.9% of them reached the world's top 10% most cited for their field and year.

This cluster of papers focuses on the modeling, analysis, and applications of anomalous diffusion using fractional calculus, nonlinear equations, and the homotopy analysis method. It explores phenomena such as random walk, fractional derivatives, time-fractional diffusion equations, single particle tracking in biological cells, and diffusion in crowded environments.

  • Anomalous Diffusion
  • Fractional Calculus
  • Nonlinear Equations
  • Homotopy Analysis Method
  • Random Walk
  • Fractional Derivatives
  • Time-Fractional Diffusion Equation
  • Single Particle Tracking
  • Biological Cells
  • Crowded Environments
Research works
37k
fractional, since 1951
In the world top 10%
3.6k
per year above
Top-10% rate
9.9%
share of its works in the world top 10%
Growth, 2018-21 → 2022-25
+17%
the tick is no change
Largest producer
China
31% of the 10 largest, 2022-now
Fractional Differential Equations Solutions

Which countries lead Fractional Differential Equations Solutions research?

By volume, China and India publish the most (2.3k and 1.4k works in 2022-now).

By volume, 2022-now

  1. 1 China 2.3k works
  2. 2 India 1.4k works
  3. 3 Saudi Arabia 910 works
  4. 4 Türkiye 555 works
  5. 5 Pakistan 530 works
  6. 6 Iran 437 works
  7. 7 Egypt 361 works
  8. 8 United States 361 works
  9. 9 Iraq 284 works
  10. 10 Algeria 236 works

How concentrated that is

The same countries as shares of everything the list above accounts for. A node where two countries do two thirds of the work and one spread evenly across twelve read alike as a ranking and not at all alike here.

China: 30.9%India: 19.3%Saudi Arabia: 12.3%Türkiye: 7.5%6 others listed: 30.0%31%largest
China2,277 · 30.9%India1,425 · 19.3%Saudi Arabia910 · 12.3%Türkiye555 · 7.5%6 others listed2,210 · 30.0%

Shares of the rows listed above, not of the whole node.

Which institutions lead Fractional Differential Equations Solutions research?

By volume in 2022-now, Vellore Institute of Technology University publishes the most Fractional Differential Equations Solutions research, followed by King Saud University and King Abdulaziz University.

By volume, 2022-now

  1. 1 Vellore Institute of Technology UniversityIndia 89 works
  2. 2 King Saud UniversitySaudi Arabia 71 works
  3. 3 King Abdulaziz UniversitySaudi Arabia 69 works
  4. 4 Université Sultan Moulay SlimaneMorocco 66 works
  5. 5 Islamic Azad University, TehranIran 66 works
  6. 6 Prince Sattam Bin Abdulaziz UniversitySaudi Arabia 65 works
  7. 7 Qassim UniversitySaudi Arabia 64 works
  8. 8 Russian Academy of SciencesRussia 63 works
  9. 9 Imam Mohammad ibn Saud Islamic UniversitySaudi Arabia 57 works
  10. 10 China Medical UniversityTaiwan 52 works

Who are the leading researchers in Fractional Differential Equations Solutions?

The most-cited researchers publishing on Fractional Differential Equations Solutions include Sudhir Kumar, Harry Eugene Stanley and Seyedali Mirjalili.

  1. 1 Sudhir Kumar United States 295k citations
  2. 2 Harry Eugene Stanley United States 198k citations
  3. 3 Seyedali Mirjalili Australia 154k citations
  4. 4 Arun Kumar Germany 150k citations
  5. 5 Paula Moraga Saudi Arabia 143k citations
  6. 6 Tasawar Hayat Pakistan 135k citations

Ranked by citations received across their whole record, among researchers with at least three works on this topic.

What are the most cited papers in Fractional Differential Equations Solutions?

The most cited paper in Fractional Differential Equations Solutions is "The random walk's guide to anomalous diffusion: a fractional dynamics approach" (Ralf Metzler, J. Klafter, 2000), with 9,129 citations. Next come "Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon" (5,387) and "User’s guide to viscosity solutions of second order partial differential equations" (5,020).

Most cited papers
#PaperReleasedCitations
1 The random walk's guide to anomalous diffusion: a fractional dynamics approach Ralf Metzler, J. Klafter · Physics Reports Dec 2000 9,129
2 Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon Thomas A. Witten, Leonard M. Sander · Physical Review Letters 9 Nov 1981 5,387
3 User’s guide to viscosity solutions of second order partial differential equations Michael G. Crandall, Hitoshi Ishii, Pierre-Louis Lions · Bulletin of the American Mathematical Society Open access 1992 5,020
4 Dual variational methods in critical point theory and applications Antonio Ambrosetti, Paul H. Rabinowitz · Journal of Functional Analysis Open access Dec 1973 4,419
5 On certain integrals of Lipschitz-Hankel type involving products of bessel functions G. Eason, Ben Noble, Ian Naismith Sneddon · Philosophical Transactions of the Royal Society of London Series A Mathematical and Physical Sciences 19 Apr 1955 4,401
All 100 most cited papers, with filters and sorting

Where is Fractional Differential Equations Solutions research done?

The largest centres of Fractional Differential Equations Solutions research in 2022-now are Riyadh (Saudi Arabia), Beijing (China), Cairo (Egypt) and Shanghai (China). Among places with at least 20 works in it, it is an unusually large share of all research in Malakand, Beni Mellal and Zeytinburnu.

Largest cities, 2022-now

  1. 1 RiyadhSaudi Arabia 263 works
  2. 2 BeijingChina 186 works
  3. 3 CairoEgypt 134 works
  4. 4 ShanghaiChina 129 works
  5. 5 TehranIran 114 works
  6. 6 NanjingChina 112 works
  7. 7 JeddahSaudi Arabia 105 works
  8. 8 LahorePakistan 104 works
  9. 9 IslamabadPakistan 100 works
  10. 10 VelloreIndia 95 works

Where it is the local speciality

  1. MalakandPK · 36.3 works73×
  2. Beni MellalMA · 66.4 works44×
  3. ZeytinburnuTR · 21.5 works37×
← less than its size predicts1×more →

Location quotient: how much more of its research is in Fractional Differential Equations Solutions than the world average.

See Fractional Differential Equations Solutions on the map

Where is the best place to study Fractional Differential Equations Solutions?

Among universities, judged by research, Henan Polytechnic University, Prince Sultan University and Prince Sattam Bin Abdulaziz University score highest, combining citation excellence, specialisation, size, growth and international reach. Research strength is one signal when choosing where to study; it does not measure teaching.

0%20%40%mean 28.02%fractional works in this field (log) →share of work in the world top 10% →Henan Polytechnic Un…Lebanese American Un…Prince Sultan Univer…Prince Sattam Bin Ab…
above the meannear itbelow it

One dot per university in the first ten of the table below, at the default weights. The upper left is the interesting corner: small places doing unusually strong work.

Citation excellenceshare of its work in the world top 10%, 2022-now and all time averaged
40%
Specialisationhow much of its work is in this field, counted up to 10× the world share
10%
Sizeits output in this field, on a log scale
30%
Growthoutput growth, counted from -20% to +20%
10%
Internationalshare of its work with a co-author abroad
10%
Each slider sets a weight; the share it gets of the score is beside it.
#UniversityScoreTop 10%SpecialisationWorksGrowth
1Henan Polytechnic UniversityChina 81.144.4%19.0×51 +34.7%
2Prince Sultan UniversitySaudi Arabia 72.920.1%48.0×49 +48.6%
3Prince Sattam Bin Abdulaziz UniversitySaudi Arabia 72.617.7%26.9×65 +93.5%
4Lebanese American UniversityLebanon 71.425.0%44.9×47 -
5Al-Balqa Applied UniversityJordan 70.226.5%30.2×30 +22.6%
6Hunan University of TechnologyChina 68.447.6%14.7×15 +149.5%
7Taibah UniversitySaudi Arabia 68.327.6%19.3×28 +22.7%
8King Khalid UniversitySaudi Arabia 67.917.7%13.1×46 +182.9%
9Near East UniversityCyprus 67.221.5%34.0×42 +527.3%
10Al Baha UniversitySaudi Arabia 67.032.1%25.0×18 +768.8%

Universities only. The score blends citation excellence (40%), size (30%), specialisation (10%), growth (10%), international reach (10%), 2022-now; growth compares 2018-21 with 2022-25 and counts up to +20%, since a jump from a small base is mostly noise.

Every institution in Fractional Differential Equations Solutions, country by country

Is Fractional Differential Equations Solutions research growing?

Output in 2022-2025 was 17% higher than in 2018-2021, peaking in 2022. The fastest-growing topics are Fractional Differential Equations Solutions.

still counting19801990200020102020
grewheldshrank

The same series as a ribbon: one cell per year, darker for more. The line above answers how much; this answers when.

Its share of all research

Growing output can still be a shrinking share: all of science grows. This is Fractional Differential Equations Solutions as a percentage of every work published that year, 0.043% in 2024.

still counting0.020%0.040%0.060%Fractional Differential Equations Solutions0.040% in 202519801990200020102020

Fractional works of Fractional Differential Equations Solutions over all fractional works that year. How the balance of science has shifted.

How much of it reaches the world top 10%

Year by year.

still counting5%10%15%Fractional Differential Equations Solutions9.8% in 2025200020102020

Of each year's works in it, the share that reached the world's 10% most cited for their field and year. A field is the world for its own topics, so there is no other line to hold it against.

Which topics inside it are moving

Growth and decline on one axis around a shared zero. Two lists side by side hide the thing that matters: whether the growth dwarfs the decline, or the other way round.

Discussion

Ask a question or share a note about Fractional Differential Equations Solutions. Keep it to the research: no personal attacks, advertising or links. Any reader can report a post, which collapses it for everyone until it is opened.