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Topic · Numerical Analysis

Differential Equations and Numerical Methods

Differential Equations and Numerical Methods is a research topic within Numerical Analysis. Science Explorer counts 36k research works in it since 1950. 10.0% of them reached the world's top 10% most cited for their field and year.

This cluster of papers focuses on numerical methods and analysis for solving singularly perturbed problems, particularly in the context of convection-diffusion and reaction-diffusion equations. The research covers topics such as boundary layers, finite difference schemes, asymptotic analysis, parameter-robust methods, and adaptive meshes to accurately solve these challenging problems.

  • Singular Perturbation
  • Numerical Methods
  • Convection-Diffusion Problems
  • Boundary Layers
  • Finite Difference Schemes
  • Asymptotic Analysis
  • Parameter-Robust Methods
  • Reaction-Diffusion Equations
  • Adaptive Meshes
  • Error Analysis
Research works
36k
fractional, since 1950
In the world top 10%
3.6k
per year above
Top-10% rate
10.0%
share of its works in the world top 10%
Growth, 2018-21 → 2022-25
+8%
the tick is no change
Largest producer
China
34% of the 10 largest, 2022-now
Differential Equations and Numerical Methods

Which countries lead Differential Equations and Numerical Methods research?

By volume, China and Russia publish the most (1.3k and 733 works in 2022-now).

By volume, 2022-now

  1. 1 China 1.3k works
  2. 2 Russia 733 works
  3. 3 India 555 works
  4. 4 United States 270 works
  5. 5 Saudi Arabia 225 works
  6. 6 Türkiye 186 works
  7. 7 Iran 160 works
  8. 8 Uzbekistan 149 works
  9. 9 Ukraine 124 works
  10. 10 Egypt 115 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: 34.2%Russia: 19.2%India: 14.5%United States: 7.0%6 others listed: 25.1%34%largest
China1,309 · 34.2%Russia733 · 19.2%India555 · 14.5%United States270 · 7.0%6 others listed958 · 25.1%

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

Which institutions lead Differential Equations and Numerical Methods research?

By volume in 2022-now, Russian Academy of Sciences publishes the most Differential Equations and Numerical Methods research, followed by Lomonosov Moscow State University and Centre National de la Recherche Scientifique.

By volume, 2022-now

  1. 1 Russian Academy of SciencesRussia 212 works
  2. 2 Lomonosov Moscow State UniversityRussia 75 works
  3. 3 Centre National de la Recherche ScientifiqueFrance 42 works
  4. 4 National Academy of Sciences of UkraineUkraine 31 works
  5. 5 National University of UzbekistanUzbekistan 28 works
  6. 6 Institute of Mathematics and Mathematical ModelingKazakhstan 27 works
  7. 7 Academy of Sciences Republic of UzbekistanUzbekistan 26 works
  8. 8 National Institute of Technology RourkelaIndia 26 works
  9. 9 Qassim UniversitySaudi Arabia 25 works
  10. 10 Peoples' Friendship University of RussiaRussia 25 works

Who are the leading researchers in Differential Equations and Numerical Methods?

The most-cited researchers publishing on Differential Equations and Numerical Methods include John H. Seinfeld, James E. Gunn and Tasawar Hayat.

  1. 1 John H. Seinfeld United States 140k citations
  2. 2 James E. Gunn United States 136k citations
  3. 3 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 Differential Equations and Numerical Methods?

The most cited paper in Differential Equations and Numerical Methods is "Deterministic Nonperiodic Flow" (Edward N. Lorenz, 1963), with 19,616 citations. Next come "Perturbation Methods" (4,699) and "An iteration method for the solution of the eigenvalue problem of linear differential and integral operators" (4,448).

Most cited papers
#PaperReleasedCitations
1 Deterministic Nonperiodic Flow Edward N. Lorenz · Journal of the Atmospheric Sciences Open access Mar 1963 19,616
2 Perturbation Methods A. H. Nayfeh, Vimal Singh · IEEE Transactions on Systems Man and Cybernetics May 1978 4,699
3 An iteration method for the solution of the eigenvalue problem of linear differential and integral operators Cornelius Lanczos · Journal of research of the National Bureau of Standards Oct 1950 4,448
4 An integrable shallow water equation with peaked solitons Roberto Camassa, Darryl D. Holm · Physical Review Letters Open access 13 Sep 1993 3,754
5 Difference Methods for Initial-Value Problems Robert D. Richtmyer, E. H. Dill · Physics Today Open access Apr 1959 3,233
All 99 most cited papers, with filters and sorting

Where is Differential Equations and Numerical Methods research done?

The largest centres of Differential Equations and Numerical Methods research in 2022-now are Moscow (Russia), Beijing (China), Tashkent (Uzbekistan) and Shanghai (China). Among places with at least 20 works in it, it is an unusually large share of all research in Jimma.

Largest cities, 2022-now

  1. 1 MoscowRussia 287 works
  2. 2 BeijingChina 120 works
  3. 3 TashkentUzbekistan 94 works
  4. 4 ShanghaiChina 84 works
  5. 5 JinanChina 70 works
  6. 6 RiyadhSaudi Arabia 63 works
  7. 7 NanjingChina 62 works
  8. 8 AlmatyKazakhstan 60 works
  9. 9 GuangzhouChina 56 works
  10. 10 KyivUkraine 53 works

Where it is the local speciality

  1. JimmaET · 23.2 works33×
← less than its size predicts1×more →

Location quotient: how much more of its research is in Differential Equations and Numerical Methods than the world average.

See Differential Equations and Numerical Methods on the map

Where is the best place to study Differential Equations and Numerical Methods?

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

0%50%100%mean 29.44%fractional works in this field (log) →share of work in the world top 10% →Hunan University of …Imam Mohammad ibn Sa…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.

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
1Hunan University of TechnologyChina 69.271.7%28.4×15 +207.9%
2Prince Sattam Bin Abdulaziz UniversitySaudi Arabia 56.431.5%10.9×14 +116.7%
3Prince Sultan UniversitySaudi Arabia 54.333.5%21.9×12 +30.9%
4Imam Mohammad ibn Saud Islamic UniversitySaudi Arabia 53.834.6%14.8×14 +201.4%
5University of MacauChina 51.926.4%7.4×12 +172.7%
6Lomonosov Moscow State UniversityRussia 49.24.7%17.3×75 -8.8%
7National Institute of Technology RourkelaIndia 48.422.4%21.8×26 +11.3%
8King Faisal UniversitySaudi Arabia 48.226.8%10.0×13 +592.6%
9China Medical UniversityTaiwan 47.131.7%15.1×11 -1.7%
10Qassim UniversitySaudi Arabia 46.811.1%18.9×25 +176.2%

Universities only. The score blends 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 Differential Equations and Numerical Methods, country by country

Is Differential Equations and Numerical Methods research growing?

Output in 2022-2025 was 8% higher than in 2018-2021, peaking in 2024. The fastest-growing topics are Differential Equations and Numerical Methods.

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 Differential Equations and Numerical Methods as a percentage of every work published that year, 0.029% in 2024.

still counting0.025%0.050%0.075%Differential Equations and Numerical Methods0.022% in 202519801990200020102020

Fractional works of Differential Equations and Numerical Methods 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%Differential Equations and Numerical Methods10.1% 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.

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